Electromagnetic Theory
Syllabus Electromagnetic Theory UNIT I: Electrostatics Laplace and Poisson equation – Boundary value problems - boundary conditions and uniqueness theorem – Laplace equation in three dimensions – Solution in Cartesian and spherical polar co ordinates – Examples of solutions for boundary value problems- Polarization and displacement vectors - Boundary conditions -Dielectric sphere in a uniform field –Molecular polarisability and electrical susceptibility -Electrostatic energy in the presence of dielectric – Multipole expansion. UNIT II : Magnetostatics Biot - Savart Law - Ampere's law - Magnetic vector potential and magnetic field of a localised current distribution - Magnetic moment, force and torque on a current distribution in an external field - Magnetostatic energy - Magnetic induction and magnetic field in macroscopic media - Boundary conditions Uniformly magnetised sphere. UNIT III : Maxwell Equations Faraday's laws of Induction - Maxwell's displacement current - Maxwell's equations – free space and linear isotropic media - Vector and scalar potentials Gauge invariance - Wave equation and plane wave solution - Coulomb and Lorentz gauges - Energy and momentum of the field - Poynting's theorem -Lorentz force - Conservation laws for a system of charges and electromagnetic fields. UNIT IV : Electromagnetic Waves Plane waves in non-conducting media - Linear and circular polarization, reflection and refraction at a plane interface - Fresnel’s law, interference,
coherence and diffraction -Waves in a conducting medium - Propagation of waves in a rectangular wave guide - Inhomogeneous wave equation and retarded potentials - Radiation from a localized source - Oscillating electric dipole.
Subject Introduction
UNIT I: Electrostatics Laplace and Poisson equation
– Boundary value problems -
boundary conditions and uniqueness theorem – Laplace equation in three dimensions – Solution in Cartesian and spherical polar co ordinates – Examples of solutions for boundary value problems- Polarization and displacement vectors - Boundary conditions
-Dielectric sphere in a uniform field –Molecular
polarisability and electrical susceptibility -Electrostatic energy in the presence of dielectric – Multipole expansion.
Classical Electrodynamics Classical electrodynamics deals with electric and magnetic fields and interactions caused by macroscopic distributions of electric charges and currents. This means that the concepts of localised charges and currents assume the validity of certain mathematical limiting processes in which it is considered possible for the charge and current distributions to be localised in infinitesimally small volumes of space. This is in obvious contradiction to electromagnetism on a microscopic scale, where charges and currents are known to be spatially extended objects. However, the limiting processes yield results which are correct on a macroscopic scale. In this Chapter we start with the force interactions in classical electrostatics and classical magnetostatics and introduce the static electric and magnetic fields and find two uncoupled systems of equations for them. Then we see how the conservation of electric charge and its relation to electric current leads to the dynamic connection between electricity and magnetism and how the two can be unified in one theory, classical electrodynamics, described by one system of coupled dynamic field equations.
Electrostatics The theory that describes physical phenomena related to the interaction between stationary electric charges or charge distributions in space is called electrostatics. Coulomb’s law It has been found experimentally that in classical electrostatics the interaction between two stationary electrically charged bodies can be described in terms of a
mechanical force. Let us consider the simple case depicted in figure where F denotes the force acting on a charged particle with charge q located at x, due to the presence of a charge q_located at x_. According to Coulomb’s law this force is, in vacuum, given by the expression
where we have used results from Example. In SI units, which we shall use throughout, the force F is measured in Newton (N), the charges q and q_in Coulomb (C) [= Ampere-seconds (As)], and the length in meters (m). The constant vacuum permittivity and
Farad per meter (F/m) is the m/s is the speed of light in vacuum.
In CGS units and the force is measured in dyne, the charge in stat coulomb, and length in centimeters (cm).
The electrostatic field Instead of describing the electrostatic interaction in terms of a “force action at a distance,” it turns out that it is often more convenient to introduce the concept of a field and to describe the electrostatic interaction in terms of a static vectorial electric field Estat defined by the limiting process
where F is the electrostatic force, as defined in equation on the facing page, from a net charge q_on the test particle with a small electric net charge q. Since the purpose of the limiting process is to assure that the test charge q does not influence the field, the expression for Estat does not depend explicitly on q but only on the charge q_and the relative radius vector x _x_. This means that we can say that any net electric charge produces an electric field in the space that surrounds it, regardless of the existence of a second charge anywhere in this space.1 we find that the electrostatic field Estat at the field point x (also known as the observation point), due to a field-producing charge q_at the source point x_, is given by
In the presence of several field producing discrete charges q_i, at x_i, i _1_2_3____, respectively, the assumption of linearity of vacuum2 allows us to superimpose their individual E fields into a total E field
If the discrete charges are small and numerous enough, we introduce the charge density r located at x_and write the total field as
where, in the last step, we used formula equation. We emphasize that equation (1.5) above is valid for an arbitrary distribution of charges, including discrete charges, in which case r can be expressed in terms of one or more Dirac delta functions.
I.e., Estat is an irrotational field.
Taking the divergence of the general Estat expression for an arbitrary charge distribution, equation, and using the representation of the Dirac delta function, equation we find that
which is Gauss’s law in differential form.
Magnetostatics While electrostatics deals with static charges, magnetostatics deals with stationary currents, i.e., charges moving with constant speeds, and the interaction between these currents.
Ampère’s law Experiments on the interaction between two small current loops have shown that they interact via a mechanical force, much the same way that charges interact. Let F denote such a force acting on a small loop C carrying a current J located at x, due to the presence of a small loop C_carrying a current J_located at x_. According to Ampère’s law this force is, in vacuum, given by the expression.
Here dl and dl_are tangential line elements of the loops C and C_, respectively, and, in SI units
H/m is the vacuum
permeability. From the definition of e0 and m0 (in SI units) we observe that
which is a useful relation. At first glance, equation above appears to be unsymmetrical in terms of the loops and therefore to be a force law which is in contradiction with Newton’s third law. However, by applying the vector triple product “bac-cab” formula on page, we can rewrite in the following way
Recognising the fact the integrand in the first integral is an exact differential so that this integral vanishes, we can rewrite the force expression, equation above, in
the following symmetric way
This clearly exhibits the expected symmetry in terms of loops C and C_. The magnetostatic field In analogy with the electrostatic case, we may attribute the magnetostatic interaction to a vectorial magnetic field Bstat. I turns out that Bstat can be defined through
Gauss’s Law and Applications Though Coulomb’s law is fundamental, one finds it cumbersome to use it to calculate electric field due to a continuous charge distribution because the integrals involved can be quite difficult. An alternative but completely equivalent formulation is Gauss’s Law which is very useful in situations which exhibit certain symmetry. Electric Lines of Force : Electric lines of force (also known as field lines) is a pictorial representation of the electric field. These consist of directed lines indicating the direction of electric field at various points in space. • There is no rule as to how many lines are to be shown. However, it is customary to draw number of lines proportional to the charge. Thus if N number of lines are drawn from or into a charge Q, 2N number of lines would be drawn for charge 2Q. • The electric field at a point is directed along the tangent to the field lines. A positive charge at this point will move along the tangent in a direction indicated by the arrow. • Lines are dense close to a source of the electric field and become sparse as one moves away. • Lines originate from a positive charge and end either on a negative charge or move to infinity.
• Lines of force due to a solitary negative charge is assumed to start at infinity and end at the negative charge. • Field lines do not cross each other. ( if they did, the field at the point of crossing will not be uniquely defined.) • A neutral point is a point at which field strength is zero. This occurs because of cancellation of electric field at such a point due to multiple charges. Exercise : Draw field lines and show the neutral point for a charge +4Q located at (1, 0) and −Q located at (−1, 0).
Electric Flux The concept of flux is borrowed from flow of water through a surface. The amount of water flowing through a surface depends on the velocity of water, the area of the surface and the orientation of the surface with respect to the direction of velocity of water. Though an area is generally considered as a scalar, an element of area may be considered to be a vector because : • It has magnitude (measured in m2).
• If the area is infinitisimally small, it can be considered to be in a plane. We can then associate a direction with it. For instance, if the area element lies in the x-y plane, it can be considered to be directed along the z–direction. (Conventionally, the direction of the area is taken to be along the outward normal.)
We define the flux of the electric field through an area d~S to be given by the scalar Product
For an arbitrary surface S, the flux is obtainted by integrating over all the surface Elements
If the electric field is uniform, the angle θ is constant and we have
Thus the flux is equal to the product of magnitude of the electric field and the projection of area perpendicular to the field.
Unit of flux is N-m2/C. Flux is positive if the field lines come out of the surface and is negative if they go into it. Solid Angle : The concept of solid angle is a natural extension of a plane angle to three dimensions. Consider an area element dS at a distance r from a point P. Let ˆn be the unit vector along the outward normal to dS. The element of the solid angle subtended by the area element at P is defined as
where dS⊥ is the projection of d~S along a direction perpendicular to ~r. If α is the angle between ˆr and ˆn, then,
Solid angle is dimensionless. However, for practical reasons it is measured in terms of a unit called steradian (much like the way a planar angle is measured in terms of degrees). The maximum possible value of solid angle is 4π, which is the angle subtended by an area which encloses the point P completely. Example 1: A right circular cone has a semi-vertical angle α. Calculate the solid angle at the apex P of the cone. Solution : The cap on the cone is a part of a sphere of radius R, the slant length of the cone.
Using spherical polar coordinates, an area element on the cap is R2 sin θdθdφ, where θ is the polar angle and φ is the azimuthal angle. Here, φ goes from 0 to 2π while θ goes from 0 to α. Thus the area of the cap is
Example 2 An wedge in the shape of a rectangular box is kept on a horizontal floor. The two triangular faces and the rectangular face ABFE are in the vertical plane. The electric field is horizontal, has a magnitude 8 × 104 N/C and enters the wedge through the face ABFE, as shown. Calculate the flux through each of the faces and through the entire surface of the wedge.
Solution : The outward normals to the triangular faces AED, BFC, as well as the normal to the base are perpendicular to ~E . Hence the flux through each of these faces is zero. The vertical rectangular face ABFE has an area 0.06 m2. The outward normal to this face is perpendicular to the electric field. The flux is entering through this face and is negative. Thus flux through ABFE is
To find the flux through the slanted face, we need the angle that the normal to this face makes with the horizontal electric field. Since the electric field is perpendicular to the side ABFE, this angle is equal to the angle between AE and AD, which is cos−1(.3/.5). The area of the slanted face ABCD is 0.1 m2. Thus the flux through ABCD is
GAUSS’S LAW - Integral form The flux calculation done in Example 4 above is a general result for flux out of any closed surface, known as Gauss’s law. Total outward electric flux φ through a closed surface S is equal to 1/ǫ0 times the charge enclosed by the volume defined by the surface S
Mathematicaly, the surface integral of the electric field over any closed surface is equal to the net charge enclosed divided by ǫ0
• The law is valid for arbitry shaped surface, real or imaginary. • Its physical content is the same as that of Coulomb’s law. • In practice, it allows evaluation of electric field in many practical situations by forming imagined surfaces which exploit symmetry of the problem. Such surfaces are called Gaussian surfaces.
GAUSS’S LAW - Differential form The integral form of Gauss’s law can be converted to a differential form by using the divergence theorem. If V is the volume enclosed by the surface S,
Direct Calculation of divergence from Coulomb’s Law : We will use the field expression
to directly evaluate the divergence of the electric field. Since the differentiation is with respect to ~r while the integration is with respect to ~r′, we can take the divergence inside the integral,
Divergence of ~r is equal to 3
i.e. ∇ ・ ~E = 0, which violates Gauss’s Law, that ∇ ・ ~E = ρ/ǫ0. The problem arises because the function 1/ | ~r −~r′ | has a singularity at ~r = ~r′. This point has to be taken care of with care. Except at this point the divergence of the integral is indeed zero. Applications of Gauss’s Law Field due to a uniformly charged sphere of radius R with a charge Q By symmetry, the field is radial. Gaussian surface is a concentric sphere of radius r. The outward normals to the Gaussian surface is parallel to the field ~E at every point. Hence
The field outside the sphere is what it would be if all the charge is concentrated at the origin of the sphere.
Field due to an infinite line charge of linear charge density λ Gaussian surface is a cylinder of radius r and length L. By symmetry, the field has the same magnitude at every point on the curved surface and is directed outwards. At the end caps, ~E is perpendicular to d~S everywhere and the flux is zero. For the curved surface, ~E and d~S are parallel,
where ˆρ is a unit vector perpendicular to the line,directed outward for positive line charge and inward for negative line charge.
UNIT II : Magnetostatics Biot - Savart Law - Ampere's law - Magnetic vector potential and magnetic field of a localised current distribution - Magnetic moment, force and torque on a current distribution in an external field Magnetostatic energy - Magnetic induction and magnetic field in macroscopic media - Boundary conditions - Uniformly magnetised sphere.
Magnetostatics Whereas electrostatics deals with static electric charges (electric charges that do not move), and the interaction between these charges, magnetostatics deals with static electric currents (electric charges moving with constant speeds), and the interaction between these currents. Here we shall discuss the theory of magnetostatics in some detail.
The Biot-Savart Law In this Section we will discuss the magnetic field produced by a steady current. A steady current is a flow of charge that has been going on forever, and will be going on forever. These currents produce magnetic fields that are constant in time. The magnetic field produced by a steady line current is given by the Biot-Savart Law:
where
is an element of the wire,
wire and P, and
is the vector connecting the element of the
is the permeability constant which is equal to
The unit of the magnetic field is the Tesla (T). For surface and volume currents the Biot-Savart law can be rewritten as
and
Example:Problem Find the magnetic field at point P for each of the steady current configurations shown in Figure a) The total magnetic field at P is the vector sum of the magnetic fields produced by the four segments of the current loop. Along the two straight sections of the loop,
and
are parallel or opposite, and thus
. Therefore, the magnetic
field produced by these two straight segments is equal to zero. Along the two circular segments
and
are perpendicular. Using the right-hand rule it is easy
to show that
and
where equal to
is pointing out of the paper. The total magnetic field at P is therefore
b) The magnetic field at P produced by the circular segment of the current loop is equal to
where
is pointing out of the paper. The magnetic field produced at P by each of
the two linear segments will also be directed along the negative z axis. The magnitude of the magnetic field produced by each linear segment is just half of the field produced by an infinitely long straight wire :
The total field at P is therefore equal to
Example:Problem Suppose you have two infinite straight-line charges λ, a distance d apart, moving
along at a constant v (see Figure ). How fast would v have to be in order for the magnetic attraction to balance the electrical repulsion?
Figure Problem When a line charge moves it looks like a current of magnitude I = λv. The two parallel currents attract each other, and the attractive force per unit length is
and is attractive. The electric generated by one of the wires can be found using Gauss' law and is equal to
The electric force per unit length acting on the other wire is equal to
and is repulsive (like charges). The electric and magnetic forces are balanced when
or
This requires that
This requires that the speed v is equal to the speed of light, and this can therefore never be achieved. Therefore, at all velocities the electric force will dominate.
Ampère’s law Experiments on the force interaction between two small loops that carry static electric currents I and I’ (i.e. the currents I and I’ do not vary in time) have shown that the loops interact via a mechanical force, much the same way that static electric charges interact. Let Fms.(x) denote the magnetostatic force on a loop C, with tangential line vector element dl, located at x and carrying a current I in the direction of dl, due to the presence of a loop C ‘ , with tangential line element dl’, located at x’ and carrying a current I’ in the direction of dl’ in otherwise empty space. This spatial configuration is illustrated in graphical form in figure : According to Ampère’s law the magnetostatic force in question is given by
Here μ’ is in hennery per metre (Hm-1) is the permeability of free space. From the definition of ε’ and μ’ (in SI units) we observe that :
which is most useful relation. At first glance, equation above may appear asymmetric in terms of the loops and therefore be a force law that does not obey Newton’s third law. However, by applying the vector triple product ‘bac-cab’ formula )
Ampère’s law postulates how a small loop C, carrying a static electric current I directed along the line element dl at x, experiences a magnetostatic force Fms (x) from a small loop C’, carrying a static electric current I ‘ directed along the line element dl ’ located at x’. we can rewrite above equation as :
Since the integrand in the first integral over C is an exact differential, this integral vanishes and we can rewrite the force expression, formula on the preceding page, in the following symmetric way
Which clearly exhibits the expected interchange symmetry between loops C and C’
The Vector Potential The magnetic field generated by a static current distribution is uniquely defined by the so-called Maxwell equations for magnetostatics:
Similarly, the electric field generated by a static charge distribution is uniquely defined by the so-called Maxwell equations for electrostatics:
The fact that the divergence of
is equal to zero suggests that there are no point
charges for . Magnetic field lines therefore do not begin or end anywhere (in
contrast to electric field lines that start on positive point charges and end on negative point charges). Since a magnetic field is created by moving charges, a magnetic field can never be present without an electric field being present. In contrast, only an electric field will exist if the charges do not move. Maxwell's equations for magnetostatics show that if the current density is known, both the divergence and the curl of the magnetic field are known. The Helmholtz theorem indicates that in that case there is a vector potential
such that
However, the vector potential is not uniquely defined. We can add to it the gradient of any scalar function f without changing its curl:
The divergence of
is equal to
It turns out that we can always find a scalar function f such that the vector potential that
is divergence-less. The main reason for imposing the requirement is that it simplifies many equations involving the vector potential. For
example, Ampere's law rewritten in terms of
is
or
This equation is similar to Poisson's equation for a charge distribution ρ:
Therefore, the vector potential
can be calculated from the current
similar to how we obtained V from ρ. Thus
in a manner
Note: these solutions require that the currents go to zero at infinity (similar to the requirement that ρ goes to zero at infinity).
Example:Problem Find the magnetic vector potential of a finite segment of straight wire carrying a current I. Check that your answer is consistent with eq. The current at infinity is zero in this problem, and therefore we can use the expression for
in terms of the line integral of the current I. Consider the wire
located along the z axis between z1 and z2 (see Figure 5.6) and use cylindrical coordinates. The vector potential at a point P is independent of φ (cylindrical symmetry) and equal to
Here we have assumed that the origin of the coordinate system is chosen such that P has z = 0. The magnetic field at P can be obtained from the vector potential and is equal to
where θ1 and θ2 are defined in Figure. This result is identical to the result of Example 5 in Griffiths.
Figure . Problem Example:Problem If
is uniform, show that
the point in question. That is check that The curl of
is equal to
, where
is the vector from the origin to and
.
Since
is uniform it is independent of r, θ, and φ and therefore the second and
third term on the right-hand side of this equation are zero. The first term, expressed in Cartesian coordinates, is equal to
The fourth term, expressed in Cartesian coordinates, is equal to
Therefore, the curl of
The divergence of
is equal to
is equal to
Example:Problem Find the vector potential above and below the plane surface current of Example in
Griffiths.
In Example of Griffiths a uniform surface current is flowing in the xy plane, directed parallel to the x axis:
However, since the surface current extends to infinity, we can not use the surface integral of
to calculate
and an alternative method must be used to
obtain . Since Example 8 showed that surface current and
is uniform above the plane of the
is uniform below the plane of the surface current, we can
use the result of Problem to calculate :
In the region above the xy plane (z > 0) the magnetic field is equal to
Therefore,
In the region below the xy plane (z < 0) the magnetic field is equal to
Therefore,
We can verify that our solution for
is correct by calculating the curl of
must be equal to the magnetic field). For z > 0:
(which
The
vector
potential
example,
and
is
however
not
uniquely
defined.
For
are also possible solutions that generate
the same magnetic field. These solutions also satisfy the requirement that
.
The Three Fundamental Quantities of Magnetostatics Our discussion of the magnetic fields produced by steady currents has shown that there are three fundamental quantities of magnetostatics: 1. The current density 2. The magnetic field 3. The vector potential These three quantities are related and if one of them is known, the other two can be calculated. The following table summarizes the relations between , , and :
Magnetic Moment : The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will
exert on it. A loop of electric current, a bar magnet, an electron, a molecule, and a planet all have magnetic moments. Both the magnetic moment and magnetic field may be considered to be vectors having a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet. The magnetic field produced by a magnet is proportional to its magnetic moment as well. More precisely, the term magnetic moment normally refers to a system's magnetic dipole moment, which produces the first term in the multipole expansion of a general magnetic field. The dipole component of an object's magnetic Two Defination of moment : The preferred definition of a magnetic moment has changed over time. Before the 1930s, textbooks defined the moment using magnetic poles. Since then, most have defined it in terms of Ampèrian currents. field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse cube of the distance from the object. Magnetic pole definition The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. Consider a bar magnet which has magnetic poles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since magnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: the strength p of its poles (the magnetic pole strength), and the vector ℓ separating them. The moment is defined as It points in the direction from South to North pole. The analogy with electric dipoles should not be taken too far because magnetic dipoles are associated
withangular momentum (see Magnetic moment and angular momentum). Nevertheless, magnetic poles are very useful for magnetostatic calculations, particularly in applications to ferromagnets. Practitioners using the magnetic pole approach generally represent the magnetic field by the irrotational field H, in analogy to theelectric field E.
An electrostatic analogue for a magnetic moment: two opposing charges separated by a finite distance.
Current loop definition Suppose a planar closed loop carries an electric current I and has vector area S (x, y, and z coordinates of this vector are the areas of projections of the loop onto the yz, zx, and xy planes). Its magnetic moment m, vector, is defined as: By convention, the direction of the vector area is given by the right hand grip rule (curling the fingers of one's right hand in the direction of the current around the loop, when the palm of the hand is "touching" the loop's outer edge, and the straight thumb indicates the direction of the vector area and thus of the magnetic moment). If the loop is not planar, the moment is given as
where × is the vector cross product. In the most general case of an arbitrary current distribution in space, the magnetic moment of such a distribution can be found from the following equation:
where r is the position vector pointing from the origin to the location of the volume element, and J is the current density vector at that location. The above equation can be used for calculating a magnetic moment of any assembly of moving charges, such as a spinning charged solid, by substituting where ρ is the electric charge density at a given point and v is the instantaneous linear velocity of that point. For example, the magnetic moment produced by an electric charge moving along a circular path is , where r is the position of the charge q relative to the center of the circle and v is the instantaneous velocity of the charge.
Moment μ of a planar current having magnitude I and enclosing an area S Practitioners using the current loop model generally represent the magnetic field by the solenoidal field B, analogous to the electrostatic field D.
Magnetic moment of a solenoid
3-D image of a solenoid A generalization of the above current loop is a multi-turn coil, or solenoid. Its moment is the vector sum of the moments of individual turns. If the solenoid has N identical turns (single-layer winding) and vector area S
Units of Magnetic Moment : The unit for magnetic moment is not a base unit in the International System of Units (SI) and it can be represented in more than one way. For example, in the current loop definition, the area is measured in square meters and I is measured in amperes, so the magnetic moment is measured in ampere–square meters (A m2). In the equation for torque on a moment, the torque is measured in joules and the magnetic field in tesla, so the moment is measured in Joules per Tesla (J⋅T−1). These two representations are equivalent: 1 A·m2 = 1 J·T−1. In the CGS system, there are several different sets of electromagnetism units, of which the main ones are ESU, Gaussian, and EMU. Among these, there are two alternative (non-equivalent) units of magnetic dipole moment in CGS: (ESU CGS) 1 stat A·cm2 = 3.33564095×10−14 (A·m2 or J·T−1) and (more frequently used) (EMU CGS and Gaussian-CGS) 1 erg/G = 1 abA·cm2 = 10−3 (m2·A or J/T). The ratio of these two non-equivalent CGS units (EMU/ESU) is equal exactly to the speed of light in free space, expressed in cm·s−1. All formulas in this article are correct in SI units, but in other unit systems, the formulas may need to be changed. For example, in SI units, a loop of current with current I and area A has magnetic moment I×A (see below), but in Gaussian units the magnetic moment is I×A/c.
Magnetic moment angular momentum : The magnetic moment has a close connection with angular momentum called the gyromagnetic effect. This effect is expressed on a macroscopic scale in the Einstein-de Haas effect, or "rotation by magnetization," and its inverse, the Barnett effect, or "magnetization by rotation."In particular, when a magnetic moment is subject to a torque in a magnetic field that tends to align it with the applied magnetic field, the moment precesses (rotates about the axis of the
applied field). This is a consequence of the angular momentum associated with the moment. Viewing a magnetic dipole as a rotating charged sphere brings out the close connection between magnetic moment and angular momentum. Both the magnetic moment and the angular momentum increase with the rate of rotation of the sphere. The ratio of the two is called the gyromagnetic ratio, usually denoted by the symbol γ. For a spinning charged solid with a uniform charge density to mass density ratio, the gyromagnetic ratio is equal to half the charge-to-mass ratio. This implies that a more massive assembly of charges spinning with the same angular momentum will have a proportionately weaker magnetic moment, compared to its lighter counterpart. Even though atomic particles cannot be accurately described as spinning charge distributions of uniform charge-to-mass ratio, this general trend can be observed in the atomic world, where the intrinsic angular momentum (spin) of each type of particle is a constant: a small half-integer times the reduced Planck constant ħ. This is the basis for defining the magnetic moment units
of Bohr
magneton (assuming charge-to-mass
ratio of
theelectron)
and nuclear magneton (assuming charge-to-mass ratio of the proton).
Force and Torque on a current distribution in an external field : Force on a moment See also: force between magnets A magnetic moment in an externally-produced magnetic field has a potential energy U: In a case when the external magnetic field is non-uniform, there will be a force, proportional to the magnetic field gradient, acting on the magnetic moment itself. There has been some discussion on how to calculate the force acting on a magnetic dipole. There are two expressions for the force acting on a magnetic dipole, depending on whether the model used for the dipole is a current loop or
two monopoles (analogous to the electric dipole). The force obtained in the case of a current loop model is
In the case of a pair of monopoles being used (i.e. electric dipole model)
and one can be put in terms of the other via the relation
In all these expressions m is the dipole and B is the magnetic field at its position. Note that if there are no currents or time-varying electrical fields ∇ × B = 0 and the two expressions agree. An electron, nucleus, or atom placed in a uniform magnetic field will precess with a frequency known as the Larmor frequency. Torque on a moment The
magnetic
moment
can also
be defined
as
a vector relating
the
aligning torque on the object from an externally applied magnetic field to the field vector itself. The relationship is given by where τ is the torque acting on the dipole and B is the external magnetic field. External magnetic field produced by a magnetic dipole moment : Any system possessing a net magnetic dipole moment m will produce a dipolar magnetic field (described below) in the space surrounding the system. While the net magnetic field produced by the system can also have higherorder multipole components, those will drop off with distance more rapidly, so that only the dipolar component will dominate the magnetic field of the system at distances far away from it.
Magnetic field lines around a "magnetostatic dipole" the magnetic dipole itself is in the center and is seen from the side. The vector potential of magnetic field produced by magnetic moment m is
and magnetic flux density is
Alternatively one can obtain the scalar potential first from the magnetic pole perspective,
and hence magnetic field strength is
The magnetic field of an ideal magnetic dipole is depicted on the left. Internal magnetic field of a dipole The two models for a dipole (current loop and magnetic poles) give the same predictions for the magnetic field far from the source. However, inside the source region they give different predictions. The magnetic field between poles (see figure for Magnetic pole definition) is in the opposite direction to the magnetic
moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right). Clearly, the limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material
The magnetic field of a current loop . If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is
Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole. If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer together but keeping the product of magnetic pole-charge and distance constant, the limiting field is
These fields are related by
, where
is the magnetization. Forces between two magnetic dipoles As discussed earlier, the force exerted by a dipole loop with moment m1 on another with moment m2 is where B2 is the magnetic field due to moment 2. The result of calculating the gradient is[8][9]
where
is the unit vector pointing from magnet 1 to magnet 2 and r is the
distance. An equivalent expression is
The force acting on m1 is in opposite direction. The torque of magnet 2 on magnet 1 is
Examples of Magnetic Moments : Two kinds of magnetic sources Fundamentally, contributions to any system's magnetic moment may come from sources of two kinds: (1) motion of electric charges, such as electric currents, and (2) the intrinsic magnetism ofelementary particles, such as the electron. Contributions due to the sources of the first kind can be calculated from knowing the distribution of all the electric currents (or, alternatively, of all the electric charges and their velocities) inside the system, by using the formulas below. On the other hand, the magnitude of each elementary particle's intrinsic magnetic moment is a fixed number, often measured experimentally to a great precision. For
example,
any
electron's
magnetic
moment
is
measured
to
be
−9.284764×10−24 J/T.The direction of the magnetic moment of any elementary
particle is entirely determined by the direction of its spin (the minus in front of the value above indicates that any electron's magnetic moment is antiparallel to its spin). The net magnetic moment of any system is a vector sum of contributions from one or both types of sources. For example, the magnetic moment of an atom of hydrogen-1 (the lightest hydrogen isotope, consisting of a proton and an electron) is a vector sum of the following contributions: 1. the intrinsic moment of the electron, 2. the orbital motion of the electron around the proton, 3. the intrinsic moment of the proton. Similarly, the magnetic moment of a bar magnet is the sum of the intrinsic and orbital magnetic moments of the unpaired electrons of the magnet's material.
Magnetic moment of an atom For an atom, individual electron spins are added to get a total spin, and individual orbital angular momenta are added to get a total orbital angular momentum. These two then are added usingangular momentum coupling to get a total angular momentum. The magnitude of the atomic dipole moment is then
where J is the total angular momentum quantum number, gJ is the Landé g-factor, and μB is the Bohr magneton. The component of this magnetic moment along the direction of the magnetic field is then
where m is called the magnetic quantum number or the equatorial quantum number, which can take on any of 2J+1 values: . The negative sign occurs because electrons have negative charge. Due to the angular momentum, the dynamics of a magnetic dipole in a magnetic field differs from that of an electric dipole in an electric field. The field does exert
a torque on the magnetic dipole tending to align it with the field. However, torque is proportional to rate of change of angular momentum, so precession occurs: the direction of spin changes. This behavior is described by theLandau-LifshitzGilbert equation:
where
is gyromagnetic ratio, m is magnetic moment, λ is damping coefficient
and Heff is effective magnetic field (the external field plus any self-field). The first term describes precession of the moment about the effective field, while the second is a damping term related to dissipation of energy caused by interaction with the surroundings.
Magnetic moment of an electron Electrons and many elementary particles also have intrinsic magnetic moments, an explanation of which requires a quantum mechanical treatment and relates to the intrinsic angular momentumof the particles as discussed in the article electron magnetic dipole moment. It is these intrinsic magnetic moments that give rise to the macroscopic effects of magnetism, and other phenomena, such as electron paramagnetic resonance. The magnetic moment of the electron is
where μB is the Bohr magneton, S is electron spin, and the g-factor gS is 2 according to Dirac's theory, but due to quantum electrodynamic effects it is slightly larger in reality: 2.002 319 304 36. The deviation from 2 is known as the anomalous magnetic dipole moment. Again it is important to notice that m is a negative constant multiplied by the spin, so the magnetic moment of the electron is antiparallel to the spin. This can be understood with the following classical picture: if we imagine that the spin angular momentum is created by the electron mass spinning around some axis, the electric current that this rotation creates circulates in the oppositedirection,
because of the negative charge of the electron; such current loops produce a magnetic moment which is antiparallel to the spin. Hence, for a positron (the antiparticle of the electron) the magnetic moment is parallel to its spin.
Magnetic moment of a nucleus The nuclear system is a complex physical system consisting of nucleons, i.e., protons and neutrons. The quantum mechanical properties of the nucleons include the spin among others. Since the electromagnetic moments of the nucleus depend on the spin of the individual nucleons, one can look at these properties with measurements of nuclear moments, and more specifically the nuclear magnetic dipole moment. Most common nuclei exist in their ground state, although nuclei of some isotopes have long-lived excited states. Each energy state of a nucleus of a given isotope is characterized by a well-defined magnetic dipole moment, the magnitude of which is a fixed number, often measured experimentally to a great precision. This number is very sensitive to the individual contributions from nucleons, and a measurement or prediction of its value can reveal important information about the content of the nuclear wave function. There are several theoretical models that predict the value of the magnetic dipole moment and a number of experimental techniques aiming to carry out measurements in nuclei along the nuclear chart. Magnetic moment of a molecule Any molecule has a well-defined magnitude of magnetic moment, which may depend on the molecule's energy state. Typically, the overall magnetic moment of a molecule is a combination of the following contributions, in the order of their typical strength: •
magnetic
moments
due
to
its
unpaired electron
spins (paramagnetic contribution), if any •
orbital motion of its electrons, which in the ground state is often proportional to the external magnetic field (diamagnetic contribution)
•
the combined magnetic moment of its nuclear spins, which depends on the nuclear spin configuration.
Examples of molecular magnetism •
Oxygen molecule, O2, exhibits strong paramagnetism, due to unpaired spins of its outermost two electrons.
•
Carbon
dioxide molecule,
CO2,
mostly
exhibits diamagnetism, a much weaker magnetic moment of the electron orbitals that is proportional to the external magnetic
field.
In
the
rare
instance
when
a
magnetic isotope, such as 13C or 17O, is present, it will contribute its nuclear magnetism to the molecule's magnetic moment. •
Hydrogen molecule, H2, in a weak (or zero) magnetic field exhibits nuclear magnetism, and can be in a para- or an ortho- nuclear spin configuration.
Magnetostatic Energy : This is the magnetic potential energy generated when a magnetic body is present in a magnetic field. For a strongly magnetic body, even in the absence of an external magnetic field, magnetostatic energy is generated within the magnetic body by the internal magnetic field generated in the opposite direction from the magnetization (and called the diamagnetic field). The amount is the amount of work required for the magnetic poles to exist counter to the internal magnetic field at both ends of the magnetic body. A strongly magnetic body is given a magnetic domain structure in order to minimize this magnetostatic energy. Now we will provide the expression for the contribution of magnetostatic interactions to the free energy of the system. The derivation of such expression is quite straightforward if one assumes that the energy density of magnetostatic field is given by:
m
where
is the whole space. In fact, by expressing the magnetostatic field as
Eq. becomes:
m The first term in Eq. vanishes owing to the integral orthogonality of the solenoidal field
m and the conservative field
remaining part, remembering that
over the whole space . The
is nonzero only within the region
, is the
magnetostatic free energy:
m We observe that magnetostatic energy expresses a nonlocal interaction, since the magnetostatic field functionally depends, through the boundary value problem , on the whole magnetization vector field, as we anticipated in the beginning of the section. The latter equation has the physical meaning of an interaction energy of an assigned continuous magnetic moments distribution, namely it can be obtained by computing the work, made against the magnetic field generated by the continuous distribution, to bring an elementary magnetic moment infinity to its actual position within the distribution
from
Magnetostatic self-energy : The origin of domains still cannot be explained by the two energy terms above. Another contribution comes from the magnetostatic self-energy, which originates from the classical interactions between magnetic dipoles. For a continuous material it is described by Maxwell's equations
In our magnetostatic problem, we do not have any electric fields
or free
currents . Thus, there are two remaining equations
The magnetic induction
is given by
. A general solution is
given by
where
is the magnetic scalar potential. Inserting the expressions for
gives :
inside magnetic bodies and outside in air or vacuum. These equations have to be solved with the boundary conditions
and
on the surface of the magnet to obtain
and derive from it
.
is the unit
normal to the magnetic body, taken to be positive in outward direction. In micromagnetics, the magnetization distribution
is given. With relation
the magnetic scalar potential can be calculated from the magnetization distribution. The demagnetizing field
is then obtained .
Finally the magnetostatic energy is given by
Magnetic induction and magnetic field in macroscopic media : Usually quantum mechanics deals with matter on the scale of atoms and atomic particles. However, at low temperatures, there are phenomena that are manifestations of quantum mechanics on a macroscopic scale. The most wellknown effects are superfluidity of helium and superconductivity which both show spectacular behavior. E.g. in both cases matter can flow with zero flow resistance. In rotating helium so-called quantum vortices are formed which are all equally strong and which can organize in beautiful patterns. A similar effect shows up in superconductors where an applied magnetic field is squeezed in bundles each containing the same amount of magnetic flux. Macroscopic quantum effects are among the most elegant phenomena in physics. Chapter 21 of the Feynman Lectures on Physics on this topic starts with "This lecture is only for entertainment." In the period from 1996 to 2003 four Nobel prizes were given for work which was related to macroscopic quantum phenomena. Macroscopic quantum phenomena can be observed in superfluid helium and in superconductors, but also in dilute quantum gases and in laser light. Although these media are very different, their behavior is very similar as they all show macroscopic quantum behavior.
The
phenomena
which
are
called macroscopic
quantum
phenomena are macroscopic in two ways: 1. The quantum states are occupied by a large number of particles (typically Avogadro's number). 2. The quantum states involved are macroscopic in size (up to km size in superconducting wires).
Maxwell’s macroscopic theory Under certain conditions, for instance for small magnitudes of the primary field strengths E and B, we may assume that the response of a substance to the fields can be approximated by a linear one so that the electric displacement vector D.(t, x) is only linearly dependent on the electric field E(t, x)., and the magnetising field H(t, x). is only linearly dependent on the magnetic field B. (t, x). In this chapter we derive these linearised forms, and then consider a simple, explicit linear model for a medium from which we derive the expression for the dielectric permittivity ".ε(t, x), the magnetic susceptibility μ(t, x) , and the refractive index or index of refraction n. (t, x) of this medium. Using this simple model, we study certain interesting aspects of the propagation of electromagnetic particles and waves in the medium
Magnetisation and the magnetising field An analysis of the properties of magnetic media and the associated currents shows that three such types of currents exist: 1. In analogy with true charges for the electric case, we may have true currents jtrue, i.e. a physical transport of true (free) charges. 2. In analogy with the electric polarisation P there may be a form of charge transport associated with the changes of the polarisation with time. Such currents, induced by an external field, are called polarisation currents and are identified with :
3. There may also be intrinsic currents of a microscopic, often atomistic, nature that are inaccessible to direct observation, but which may produce net effects at discontinuities and boundaries. These magnetisation currents are denoted jM. Magnetic monopoles have not yet been unambiguously identified in experiments. So there is no correspondence in the magnetic case to the electric monopole moment, formula
The lowest order magnetic moment, corresponding to the
electric dipole moment, is the magnetic dipole moment
Analogously to the electric case, one may, for a distribution of magnetic dipole moments in a volume, describe this volume in terms of its magnetisation, or magnetic dipole moment per unit volume, M. Via the definition of the vector potential A one can show that the magnetisation current and the magnetization is simply related:
In a stationary medium we therefore have a total current which is (approximately) the sum of the three currents enumerated above:
One might then be led to think that the right-hand side (RHS) of the ∆ × B
Maxwell equation
However, moving the term ∆ × M from the right hand side (RHS) to the left hand side (LHS) and introducing the magnetising field (magnetic field intensity, Ampère-turn density) as
and using the definition for D, equation, we find that
Hence, in this simplistic view, we would pick up a term
which makes
the equation inconsistent: the divergence of the left hand side vanishes while the divergence of the right hand side does not! Maxwell realised this and to overcome this inconsistency he was forced to add his famous displacement current term which precisely compensates for the last term the RHS expression. We may, in analogy with the electric case, introduce a magnetic susceptibility for the medium. Denoting it Xm, we can write
where, approximately
and
is the relative permeability. In the case of anisotropy, Km will be a tensor, but it is still only a linear approximation
Macroscopic Maxwell equations : Field equations, expressed in terms of the derived, and therefore in principle superfluous, field quantities D and H are obtained from the Maxwell-Lorentz microscopic equations, by replacing the E and B in the two source equations by using the approximate relations formula,respectively:
This set of differential equations, originally derived by Maxwell himself, are called Maxwell’s macroscopic equations. Together with the boundary conditions and the constitutive relations, they describe uniquely (but only approximately) the properties of the electric and magnetic fields in matter and are convenient to use in certain simple cases, particularly in engineering applications. However, the structure of these equations rely on certain linear approximations and there are many situations where they are not useful or even applicable. Therefore, these equations, which are the original Maxwell equations (albeit expressed in their modern vector form as introduced by OLIVER HEAVISIDE), should be used with some care.
Boundary conditions in general the fields , B, D and H (vectors) are discontinuous at points where ε, μ and σ also are. Hence the field vectors will be discontinuous at a boundary between two media with different constitutive parameters. The integral form of Maxwell’s equations can be used to determine the relations, called boundary conditions, of the normal and tangential components of the fields at the interface between two regions with different constitutive parameters ε, μ and σ where surface density of sources may exist along the boundary. The boundary condition for D(vector) can be calculated using a very thin, small pillbox that crosses the interface of the two media, as shown in Fig. . Applying the divergence theorem7 to we have
outward flux of D(vector) over them is (Dn1 −Dn2)ds = ( D1 − D2)·ˆnds, where these Dn are the normal components of D(vector), ds is the area of each base, and ˆn is the unit normal drawn from medium 2 to medium 1. At the limit, by taking a shallow enough pillbox, we can disregard the flux over the curved surface, whereupon the sources of D(vector) reduce to the density of surface free charge ρs on the interface
Figure : Derivation of boundary conditions at the interface of two media. Hence the normal component of D(vector) changes discontinously across the interface by an amount equal to the free charge surface density ρs on the surface boundary. Similarly the boundary condition for B(vector) can be established using the Gauss’ law for magnetic fields . Since the magnetic field is solenoidal, it follows that the normal components of
B(vector) are continuous across the
interface between two media
The behavior of the tangential components of E(vector) can be determined using a infinitesimal rectangular loop at the interface which has sides of lengh dh, normal to the interface, and sides of lengh dl parallel to it . From the integral form of the Faraday’s law, and defining ˆt as the unit tangent vector parallel to the direction of integration on the upper side of the loop, we Have
In the limit, as dh → 0, the area ds = dldh bounded by the loop approaches zero and, since B(vector) is finite, the flux of B(vector) vanishes. Hence( E1 − E2) (vector) · ˆt = 0 and we conclude that the tangential components of E(vector) are
continuous across the interface between two media. In terms of the normal ˆn to the boundary, this can be written as :
Analogously, using the same infinitesimal rectangular loop, it can be deduced from the generalized Ampère’s law, (1.2d), that
where, since D(vector) is finite, its flux vanishes. Nevertheless, the flux of the surface current can have a non-zero value when the integration loop is reduced to zero, if the conductivity σ of the medium 2, and consequently Js(vector), is infinite. This requires the surface to be a perfect conductor. Thus
the tangential component of H(vector) is discontinuous by the amount of surface current density Js(vector). For finite conductivity, the tangential magnetic field is continuous across the boundary. A summary of the boundary conditions, given in , are particularized in for the case when the medium 2 is a perfect conductor (σ2 → ∞). General boundary conditions :
Boundary Conditions when the mediun 2 is a perfect conductor (σ2 → ∞).
A uniformly magnetized sphere Consider
a
sphere
magnetization
of
radius
,
with
a
uniform
permanent
, surrounded by a vacuum region. The simplest way of
solving this problem is in terms of the scalar magnetic potential introduced in Eqs. , it is clear that
satisfies Laplace's equation,
since there is zero volume magnetic charge density in a vacuum or a uniformly magnetized magnetic medium. However, according to Eq. , there is a magnetic surface charge density,
on the surface of the sphere. One of the matching conditions at the surface of the sphere is that the tangential component of
must be continuous. It follows from
Eq. that the scalar magnetic potential must be continuous at
, so that
Integrating Eq. over a Gaussian pill-box straddling the surface of the sphere yields
In other words, the magnetic charge sheet on the surface of the sphere gives rise to a iscontinuity in the radial gradient of the magnetic scalar potential at
.
The most general axisymmetric solution to Eq. which satisfies physical boundary conditions at
and
is
for
, and
for
. The boundary condition yields
for all . The boundary condition gives
for all , since
for
, and
Thus,
for
, and
. It follows that
for
. Since there is a uniqueness theorem associated with Poisson's
equation, we can be sure that this axisymmetric potential is the only solution to the problem which satisfies physical boundary conditions at
and infinity.
In the vacuum region outside the sphere
It is easily demonstrated from Eq. that
where
This, of course, is the magnetic field of a magnetic dipole
. Not surprisingly,
the net dipole moment of the sphere is equal to the integral of the magnetization of the sphere.
(which is the dipole moment per unit volume) over the volume
Figure 4: Schematic demagnetization curve for a permanent magnet Inside the sphere we have
and
, giving
and
Thus, both the
and
fields are uniform inside the sphere. Note that the
magnetic intensity is oppositely directed to the magnetization. In other words, the
field acts to demagnetize the sphere. How successful it is at achieving this
depends on the shape of the hysteresis curve in the negative
and positive
quadrant. This curve is sometimes called the demagnetization curve of the magnetic material which makes up the sphere. Figure 4 shows a schematic demagnetization curve. The curve is characterized by two quantities: the retentivity
(i.e., the residual magnetic field strength at zero magnetic
intensity) and the coercivity
(i.e., the negative magnetic intensity required
to demagnetize the material: this quantity is conventionally multiplied by
to
give it the units of magnetic field strength). The operating point (i.e., the values of
and
inside the sphere) is obtained from the intersection of the
demagnetization curve and the curve
. It is clear from Eqs. and that
for a uniformly magnetized sphere in the absence of external fields. The magnetization inside the sphere is easily calculated once the operating point has been determined. In fact,
. It is clear from Fig. 4 that for a
magnetic material to be a good permanent magnet it must possess both a large retentivity and a large coercivity. A material with a large retentivity but a small coercivity is unable to retain a significant magnetization in the absence of a strong external magnetizing field.
UNIT III : Maxwell Equations Faraday's laws of Induction - Maxwell's displacement current Maxwell's equations – free space and linear isotropic media Vector and scalar potentials - Gauge invariance - Wave equation and plane wave solution - Coulomb and Lorentz gauges - Energy and momentum of the field - Poynting's theorem -Lorentz force Conservation laws for a system of charges and electromagnetic fields.
Faraday’s law of induction : Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.
Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current which flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. This version of Faraday's law strictly holds only when the closed circuit is a loop of infinitely thin wire, and is invalid in other circumstances as discussed below. A different version, the Maxwell–Faraday equation , is valid in all circumstances.
in physics, a quantitative relationship between a changing magnetic field and the electric field created by the change, developed on the basis of experimental observations made in 1831 by the English scientist Michael Faraday. The phenomenon called electromagnetic induction was first noticed and investigated by Faraday; the law of induction is its quantitative expression. Faraday discovered that, whenever the magnetic field about an electromagnet was made to grow and collapse by closing and opening the electric circuit of which it was a part, an electric current could be detected in a separate conductor nearby. Moving a permanent magnet into and out of a coil of wire also induced a current in the wire while the magnet was in motion. Moving a conductor near a stationary permanent magnet caused a current to flow in the wire, too, as long as it was moving. Faraday visualized a magnetic field as composed of many lines of induction, along which a small magnetic compass would point. The aggregate of the lines intersecting a given area is called the magnetic flux. The electrical effects were thus attributed by Faraday to a changing magnetic flux. Some years later the Scottish physicist James Clerk Maxwell proposed that the fundamental effect of changing magnetic flux was the production of an electric field, not only in a conductor (where it could drive an electric charge) but also in space even in the absence of electric charges. Maxwell formulated the mathematical expression relating the change in magnetic flux to the induced electromotive force (E, or emf). This relationship, known as Faraday’s law of induction (to distinguish it from his laws of electrolysis), states that the magnitude of the emf induced in a circuit is proportional to the rate of change of the magnetic flux that cuts across the circuit. If the rate of change of magnetic flux is expressed in units of webers per second, the induced emf has units of volts. Faraday’s law is one of the four Maxwell equations that define electromagnetic theory. Quantitative Faraday's law of induction makes use of the magnetic flux ΦB through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be
moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral:
where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B·dA is a vector dot product (the infinitesimal amount of magnetic flux). In more visual terms, the magnetic flux through the wire loop is proportional to the number ofmagnetic flux lines that pass through the loop.
The wire loop (red) forms the boundary of a surface Σ (blue). The black arrows denote any vector field F(r, t) defined throughout space; in the case of Faraday's law, the relevant vector field is the magnetic flux density B, and it is integrated over the blue surface. The red arrow represents the fact that the wire loop may be moving and/or deforming.
The definition of surface integral relies on splitting the surface Σ into small surface elements. Each element is associated with a vector dA of magnitude equal to the area of the element and with direction normal to the element and pointing “outward” (with respect to the orientation of the surface). When the flux changes—because B changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an EMF
, defined as the energy available per unit charge that travels once
around the wire loop (the unit of EMF is the volt). Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads. According to the Lorentz force law (in SI units),
the EMF on a wire loop is:
where E is the electric field, B is the magnetic field (aka magnetic flux density, magnetic induction), dℓ is an infinitesimal arc length along the wire, and the line integral is evaluated along the wire (along the curve the conincident with the shape of the wire). The EMF is also given by the rate of change of the magnetic flux:
where
is the electromotive force (EMF) in volts and ΦB is the magnetic
flux in webers. The direction of the electromotive force is given by Lenz's law. For a tightly wound coil of wire, composed of N identical loops, each with the same ΦB, Faraday's law of induction states that
where N is the number of turns of wire and ΦB is the magnetic flux in webers through a single loop.
Maxwell–Faraday equation : The Maxwell–Faraday equation is a generalisation of Faraday's law that states that a time-varying magnetic field is always accompanied by a spatially-varying, non-conservative electric field, and vice-versa. The Maxwell–Faraday equation is
(in SI units) where
is the curl operator and again E(r, t) is the electric
field and B(r, t) is the magnetic field. These fields can generally be functions of position r and time t. The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in an integral form by the Kelvin-Stokes theorem:
where, as indicated in the figure: Σ is a surface bounded by the closed contour ∂Σ, E is the electric field, B is the magnetic field. dℓ is an infinitesimal vector element of the contour ∂Σ, dA is an infinitesimal vector element of surface Σ. If its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface.
Both dℓ and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin-Stokes theorem. For a planar surface Σ, a positive path element dℓ of curve ∂Σ is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal n to the surface Σ. The integral around ∂Σ is called a path integral or line integral. Notice that a nonzero path integral for E is different from the behavior of the electric field generated by charges. A charge-generated E-field can be expressed as the gradient of a scalar field that is a solution to Poisson's equation, and has a zero path integral. See gradient theorem. The integral equation is true for any path ∂Σ through space, and any surface Σ for which that path is a boundary. If the path Σ is not changing in time, the equation can be rewritten:
The surface integral at the right-hand side is the explicit expression for the magnetic flux ΦB through Σ.
Proof of Faraday's law from Maxwell's equations : The four Maxwell's equations (including the Maxwell–Faraday equation), along with
the Lorentz
force
law,
are
a
sufficient
foundation
to
derive everything in classical electromagnetism . Therefore it is possible to "prove" Faraday's law starting with these equations. Click "show" in the box below for an outline of this proof. (In an alternative approach, not shown here but equally valid, Faraday's law could be taken as the starting point and used to "prove" the Maxwell–Faraday equation and/or other laws.) Outline of proof of Faraday's law from Maxwell's equations and the Lorentz force law. Consider the time-derivative of flux through a possibly moving loop, with area
:
The integral can change over time for two reasons: The integrand can change, or the integration
region can change. These add linearly, therefore:
where t0 is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer EMF, the second to motional EMF (see above). The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation:
Area swept out by vector element dℓ of curve ∂Σ in timedt when moving with velocity v. Next, we analyze the second term on the right-hand side:
This is the most difficult part of the proof; more details and alternate approaches can be found in references. As the loop moves and/or deforms, it sweeps out a surface (see figure on right). The magnetic flux through this swept-out surface corresponds to the magnetic flux that is either entering or exiting the loop, and therefore this is the magnetic flux that contributes to the timederivative. (This step implicitly uses Gauss's law for magnetism: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.) As a small part of the loop vector
moves with velocity v for a short time
, it sweeps out a vector area
. Therefore, the change in magnetic flux through the loop here is
Therefore:
where v is the velocity of a point on the loop
.
Putting these together,
Meanwhile, EMF is defined as the energy available per unit charge that travels once around the wire loop. Therefore, by the Lorentz force law,
Combining these,
Examples of Faraday’s Law :
Faraday's disc electric generator. The disc rotates with angular rate ω, sweeping the conducting radius circularly in the static magnetic field B. The magnetic Lorentz force v × Bdrives the current along the conducting radius to the
conducting rim, and from there the circuit completes through the lower brush and the axle supporting the disc. Thus, current is generated from mechanical motion. •
A counterexample to Faraday's Law when over-broadly interpreted. A wire (solid red lines) connects to two touching metal plates (silver) to form a circuit. The whole system sits in a uniform magnetic field, normal to the page. If the word "circuit" is interpreted as "primary path of current flow" (marked in red), then the magnetic flux through the "circuit" changes dramatically as the plates are rotated, yet the EMF is almost zero, which contradicts Faraday's Law. Although Faraday's law is always true for loops of thin wire, it can give the wrong result if naively extrapolated to other contexts. One example is the homopolar generator (above left): A spinning circular metal disc in a homogeneous magnetic field generates a DC (constant in time) EMF. In Faraday's law, EMF is the time-derivative of flux, so a DC EMF is only possible if the magnetic flux is getting uniformly larger and larger perpetually. But in the generator, the magnetic field is constant and the disc stays in the same position, so no magnetic fluxes are growing larger and larger. So this example cannot be analyzed directly with Faraday's law. Another example, due to Feynman, has a dramatic change in flux through a circuit, even though the EMF is arbitrarily small. See figure and caption above right. In both these examples, the changes in the current path are different from the motion of the material making up the circuit. The electrons in a material tend to follow the motion of the atoms that make up the material, due to scattering in the
bulk and work function confinement at the edges. Therefore, motional EMF is generated when a material's atoms are moving through a magnetic field, dragging the electrons with them, thus subjecting the electrons to the Lorentz force. In the homopolar generator, the material's atoms are moving, even though the overall geometry of the circuit is staying the same. In the second example, the material's atoms are almost stationary, even though the overall geometry of the circuit is changing dramatically. On the other hand, Faraday's law always holds for thin wires, because there the geometry of the circuit always changes in a direct relationship to the motion of the material's atoms. Although Faraday's law does not apply to all situations, the Maxwell–Faraday equation and Lorentz force law are always correct and can always be used directly. Both of the above examples can be correctly worked by choosing the appropriate path of integration for Faraday's Law. Outside of context of thin wires, the path must never be chosen to go through the conductor in the shortest direct path.
Applications of faraday’s law : The principles of electromagnetic induction are applied in many devices and systems, including: •
Current clamp
•
Electrical generators
•
Electromagnetic forming
•
Graphics tablet
•
Hall effect meters
•
Induction cookers
•
Induction motors
•
Induction sealing
•
Induction welding
•
Inductive charging
•
Inductors
•
Magnetic flow meters
•
Mechanically powered flashlight
•
Pickups
•
Rowland ring
•
Transcranial magnetic stimulation
•
Transformers
•
Wireless energy transfer
The Maxwell Displacement Current Maxwell's Equations (ME) consist of two inhomogeneous partial differential equations and two homogeneous partial differential equations. At this point you should be familiar at least with the ``static'' versions of these equations by name and function:
in SI units, where
and
.
The astute reader will immediately notice two things. One is that these equations are not all, strictly speaking, static - Faraday's law contains a time derivative, and Ampere's law involves moving charges in the form of a current. The second is that they are almost symmetric. There is a divergence equation and a curl equation for each kind of field. The inhomogenous equations (which are connected to sources in the form of electric charge) involve the electric displacement and magnetic field, where the homogeneous equations suggest that there is no magnetic charge and consequently no screening of the magnetic induction or electric field due to magnetic charge. One asymmetry is therefore the
presence/existence of electric charge in contrast with the absence/nonexistence of magnetic charge. The other asymmetry is that Faraday's law connects the curl of the time derivative of the does not connect the curl of
field to the
field, but its apparent partner, Ampere's Law, to the time deriviative of
as one might
expect from symmetry alone. If one examines Ampere's law in its integral form, however:
one quickly concludes that the current through the open surface the closed curve
bounded by
is not invariant as one chooses different surfaces. Let us
analyze this and deduce an invariant form for the current (density), two ways.
Figure : Current flowing through a closed curve
and
.
bounded by two surfaces,
Consider a closed curve
that bounds two distinct open surfaces
that together form a closed surface (density) ``through'' the curve
and
. Now consider a current
, moving from left to right. Suppose that some of
this current accumulates inside the volume
bounded by
. The law of charge
conservation states that the flux of the current density out of the closed surface is equal to the rate that the total charge inside decreases. Expressed as an integral:
With this in mind, examine the figure above. If we rearrange the integrals on the
left and right so that the normal
points in to the volume (so we can compute
the current through the surface
moving from left to right) we can easily see
that charge conservation tells us that the current in through
out through
minus the current
must equal the rate at which the total charge inside this volume
increases. If we express this as integrals:
In this expression and figure, note well that
and
point through the loop in
the same sense (e.g. left to right) and note that the volume integral is over the
volume
bounded by the closed surface formed by
and
together.
Using Gauss's Law for the electric field, we can easily connect this volume integral of the charge to the flux of the electric field integrated over these two surfaces with outward directed normals:
Combining these two expressions, we get:
From this we see that the flux of the ``current density'' inside the brackets is invariant as we choose different surfaces bounded by the closed curve
.
In the original formulation of Ampere's Law we can clearly get a different answer on the right for the current ``through'' the closed curve depending on which surface we choose. This is clearly impossible. We therefore modify Ampere's Law to use the invariant current density:
where the flux of the second term is called the Maxwell displacement current (MDC). Ampere's Law becomes:
or
in terms of the magnetic field
and electric displacement
. The origin of the
term ``displacement current'' is obviously clear in this formulation. Using vector calculus on our old form of Ampere's Law allows us to arrive at this same conclusion much more simply. If we take the divergence of Ampere's Law we get:
If we apply the divergence theorem to the law of charge conservation expressed as a flux integral above, we get its differential form:
and conclude that in general we can not conclude that the divergence of
vanishes in general as this expression requires, as there is no guarantee that vanishes everywhere in space. It only vanishes for ``steady state currents'' on a background of uniform charge density, justifying our calling this form of Ampere's law a magnetostatic version. If we substitute in
(Gauss's Law) for
, we can see that it is true
that:
as an identity. A sufficient (but not necessary!) condition for this to be true is:
or
This expression is identical to the magnetostatic form in the cases where
is
constant in time but respects charge conservation when the associated (displacement) field is changing. We can now write the complete set of Maxwell's equations, including the Maxwell displacement current discovered by requiring formal invariance of the current and using charge conservation to deduce its form. Keep the latter in mind; it should not be surprising to us later when the law of charge conservation pops out of Maxwell's equations when we investigate their formal properties we can see that we deliberately encoded it into Ampere's Law as the MDC. Anyway, here they are. Learn them. They need to be second nature as we will spend a considerable amount of time using them repeatedly in many, many contexts as we investigate electromagnetic radiation.
(where I introduce and obvious and permanent abbreviations for each equation by name as used throughout the rest of this text). Aren't they pretty! The no-monopoles asymmetry is still present, but we now have two symmetric dynamic equations coupling the electric and magnetic fields and are ready to start studying electrodynamics instead of electrostatics. Note well that the two inhomogeneous equations use the in-media forms of the electric and magnetic field. These forms are already coarse-grain averaged over the microscopic distribution of point charges that make up bulk matter. In a truly microscopic description, where we consider only bare charges wandering around in free space, we should use the free space versions:
It is time to make these equations jump through some hoops.
Maxwell's Equations Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not
generally introduced in an introductory treatment of the subject, except perhaps as summary relationships Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe howelectric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the Scottish physicist and mathematician James Clerk Maxwell who published an early form of those equations between 1861 and 1862. The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current, including the complicated charges and currents in materials at the atomic scale; it has universal applicability, but may be unfeasible to calculate. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that describe large-scale behavior without having to consider these atomic scale details, but it requires the use of parameters characterizing the electromagnetic properties of the relevant materials. The term "Maxwell's equations" is often used for other forms of Maxwell's equations. For example, space-time formulations are commonly used in high energy and gravitational physics. These formulations defined on space-time, rather
than
space
and
time
separately
are manifestly[1] compatible
with specialand general relativity. In quantum mechanics, versions of Maxwell's equations based on the electric and magnetic potentials are preferred. Since the mid-20th century, it has been understood that Maxwell's equations are not exact laws of the universe, but are a classical approximation to the more accurate and fundamental theory of quantum electrodynamics. In most cases, though, quantum deviations from Maxwell's equations are immeasurably small. Exceptions occur when the particle nature of light is important or for very strong electric fields. Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields and how they affect
each other. (Mathematical descriptions are below). Using vector calculus there are four equations. Two describe how the fields vary in space due to sources (if any); electric fields emanating from electric charges in Gauss's law, and magnetic fields as closed field lines (not due to magnetic monopoles) in Gauss's law for magnetism. The other two describe how the fields 'circulate' around their respective sources; the magnetic field 'circulates' around electric currents and time varying electric fields in Ampère's law with Maxwell's correction, while the electric field 'circulates' around time varying magnetic fields inFaraday's law. Gauss's law Gauss's law describes the relationship between a static electric field and the electric charges that cause it: The static electric field points away from positive charges and towards negative charges. In the field line description, electric field lines begin only at positive electric charges and end only at negative electric charges. 'Counting' the number of field lines passing though a closed surface, therefore, yields the total charge (including bound charge due to polarization of material) enclosed by that surface divided by dielectricity of free space (the vacuum permittivity). More technically, it relates the electric flux through any hypothetical closed "Gaussian surface" to the enclosed electric charge.
Gauss's law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.
Gauss's law for magnetism Gauss's law for magnetism states that there are no "magnetic charges" (also called magnetic monopoles), analogous to electric charges. Instead, the magnetic field due to materials is generated by a configuration called a dipole. Magnetic dipoles are best represented as loops of current but resemble positive and negative 'magnetic charges', inseparably bound together, having no net 'magnetic charge'. In terms of field lines, this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back. In other words, any magnetic field line that enters a given volume must omewhere exit that volume. Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.
Faraday's law
In a geomagnetic storm, a surge in the flux of charged particles temporarily alters Earth's magnetic field, which induces electric fields in Earth's atmosphere, thus causing surges in electrical power grids. Artist's rendition; sizes are not to scale. Faraday's law describes how a time varying magnetic field creates ("induces") an electric field. This dynamically induced electric field has closed field lined just as the magnetic field, if NOT superposed by a static (charge induced) electric field. This aspect of electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field, which in turn generates an electric field in a nearby wire. (Note: there are two closely related equations which are called Faraday's law. The form used in Maxwell's equations is always valid but more restrictive than that originally formulated by Michael Faraday.)
Ampère's law with Maxwell's correction
An Wang's magnetic core memory (1954) is an application of Ampère's law. Each corestores one bit of data. Ampère's law with Maxwell's correction states that magnetic fields can be generated in two ways: byelectrical current (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's correction"). Maxwell's correction to Ampère's law is particularly important: it shows that not only does a changing magnetic field induce an electric field, but also a changing electric field induces a magnetic field. Therefore, these equations allow selfsustaining
"electromagnetic
waves"
to
travel
through
empty
space
(see electromagnetic wave equation). The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, exactly matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.
Maxwell's Equations and Electromagnetic Waves The Equations Maxwell’s four equations describe the electric and magnetic fields arising from varying distributions of electric charges and currents, and how those fields change in time. The equations were the mathematical distillation of decades of experimental observations of the electric and magnetic effects of charges and currents. Maxwell’s own contribution is just the last term of the last equation but realizing the necessity of that term had dramatic consequences. It made evident for the first time that varying electric and magnetic fields could feed off each other
these fields could propagate indefinitely through space, far from the
varying charges and currents where they originated. Previously the fields had been envisioned as tethered to the charges and currents giving rise to them. Maxwell’s new term (he called it the displacement current) freed them to move through space in a self-sustaining fashion, and even predicted their velocity
it
was the velocity of light! Here are the equations:
1. Gauss’ Law for electric fields:
(The integral of the
outgoing electric field over an area enclosing a volume equals the total charge inside, in appropriate units.) 2. The corresponding formula for magnetic fields:
(No magnetic
charge exists: no “monopoles”.) 3. Faraday’s Law of Magnetic Induction:
The
first term is integrated round a closed line, usually a wire, and gives the total voltage change around the circuit, which is generated by a varying magnetic field threading through the circuit.
4. Ampere’s
Law
plus
current:
Maxwell’s
displacement
This gives the total magnetic
force around a circuit in terms of the current through the circuit, plus any varying electric field through the circuit (that’s the “displacement current”). The purpose of this lecture is to review the first three equations and the original Ampere’s law fairly briefly, as they were already covered earlier in the course, then to demonstrate why the displacement current term must be added for consistency, and finally to show, without using differential equations, how measured values of static electrical and magnetic attraction are sufficient to determine the speed of light. Preliminaries: Definitions of µ0 and ε0, the Ampere and the Coulomb Ampere discovered that two long parallel wires carrying electric currents in the same direction attract each other magnetically, the force per unit length being proportional to the product of the currents (so oppositely directed currents repel) and decaying with distance as 1/r. In modern (SI) notation, his discovery is written (F in Newtons)
The modern convention is that the constant
appearing here is exactly 10-7,
this defines our present unit of current, the ampere. To repeat:
is not
something to measure experimentally, it's just a funny way of writing the number 10-7! That's not quite fair it has dimensions to ensure that both sides of the above equation have the same dimensionality. (Of course, there's a historical reason for this strange convention, as we shall see later). Anyway, if we bear in mind that dimensions have been taken care of, and just write the equation
it's clear that this defines the unit current one ampere as that current in a long straight wire which exerts a magnetic force of
newtons per meter of wire
on a parallel wire one meter away carrying the same current. However, after we have established our unit of current
the ampere
we have
also thereby defined our unit of charge, since current is a flow of charge, and the unit of charge must be the amount carried past a fixed point in unit time by unit current. Therefore, our unit of charge
the coulomb
is defined by stating
that a one amp current in a wire carries one coulomb per second past a fixed point. To be consistent, we must do electrostatics using this same unit of charge. Now, the electrostatic force between two charges is appearing here, now written turns out to be
The constant
, must be experimentally measured its value
.
To summarize: to find the value of
, two experiments have to be
performed. We must first establish the unit of charge from the unit of current by measuring the magnetic force between two current-carrying parallel wires. Second, we must find the electrostatic force between measured charges. (We could, alternatively, have defined some other unit of current from the start, then we would have had to find both
and
by experiments on magnetic and
electrostatic attraction. In fact, the ampere was originally defined as the current that deposited a definite weight of silver per hour in an electrolytic cell). Maxwell's Equations We established earlier in the course that the total flux of electric field out of a closed surface is just the total enclosed charge multiplied by
,
This is Maxwell’s first equation. It represents completely covering the surface with a large number of tiny patches having areas
. (The little areas are small
enough to be regarded as flat, the vector magnitude dA is just the value of the area, the direction of the vector is perpendicular to the area element, pointing outwards away from the enclosed volume.) Hence the dot product with the electric field selects the component of that field pointing perpendicularly outwards (it would count negatively if the field were pointing inwards)
this is
the only component of the field that contributes to actual electric flux across the surface. (Remember flux just means flow
the picture of the electric field in
this context is like a fluid flowing out from the charges, the field vector representing the direction and velocity of the flowing fluid.) The second Maxwell equation is the analogous one for the magnetic field, which has no sources or sinks (no magnetic monopoles, the field lines just flow around in closed curves). Again thinking of the force lines as representing a kind of fluid flow, the so-called "magnetic flux", we see that for a closed surface, as much magnetic flux flows into the surface as flows out
since there are no sources.
This can perhaps be visualized most clearly by taking a group of neighboring lines of force forming a slender tube
the "fluid" inside this tube flows round
and round, so as the tube goes into the closed surface then comes out again (maybe more than once) it is easy to see that what flows into the closed surface at one place flows out at another. Therefore the net flux out of the enclosed volume is zero, Maxwell’s second equation:
The first two Maxwell's equations, given above, are for integrals of the electric and magnetic fields over closed surfaces. Maxwell's other two equations, discussed below, are for integrals of electric and magnetic fields around closed curves (taking the component of the field pointing along the curve). These represent the work that would be needed to take a charge around a closed curve in
an electric field, and a magnetic monopole (if one existed!) around a closed curve in a magnetic field. The
simplest
version
of Maxwell's
third
equation is
for
the
special electrostatic case: The path integral
for electrostatics.
However, we know that this is only part of the truth, because from Faraday's Law of Induction, if a closed circuit has a changing magnetic flux through it, a circulating current will arise, which means there is a nonzero voltage around the circuit. The complete Maxwell's third equation is:
where the area integrated over on the right hand side spans the path (or circuit) on the left hand side, like a soap film on a loop of wire. (The best way to figure out the sign is to use Lenz’ law: the induced current will generate a magnetic field opposing the changing of the external field, so if an external upward field is decreasing, the current thereby generated around the loop will give an upward pointing field.) It may seem that the integral on the right hand side is not very clearly defined, because if the path or circuit lies in a plane, the natural choice of spanning surface (the "soap film") is flat, but how do you decide what surface to choose to do the integral over for a wire bent into a circuit that doesn’t lie in a plane? The answer is that it doesn’t matter what surface you choose, as long as the wire forms its boundary. Consider two different surfaces both having the wire as a boundary (just as both the northern hemisphere of the earth’s surface and the southern hemisphere have the equator as a boundary). If you add these two surfaces together, they form a single closed surface, and we know that for a closed surface
. This implies that
bounded by the path is equal to
for one of the two surfaces for the other one, so that the two will
add to zero for the whole closed surface. But don’t forget these integrals for the whole closed surface are defined with the little area vectors pointing outwards from the enclosed volume. By imagining two surfaces spanning the wire that are actually close to each other, it is clear that the integral over one of them is equal to the integral over the other if we take the
vectors to point in the same direction
for both of them, which in terms of the enclosed volume would be outwards for one surface, inwards for the other one. The bottom line of all this is that the surface integral
is the same for any surface spanning the path, so it
doesn’t matter which we choose. The equation analogous to the electrostatic version of the third equation given above, but for the magnetic field, is Ampere's law, for the magnetostatic case, where the currents counted are those threading through the path we're integrating around, so if there is a soap film spanning the path, these are the currents that punch through the film (of course, we have to agree on a direction, and subtract currents flowing in the opposite direction). We must now consider whether this equation, like the electrostatic one, has limited validity. In fact, it was not questioned for a generation after Ampere wrote it down: Maxwell's great contribution, in the 1860's, was to realize that it was not always valid. When Does Ampere's Law Go Wrong? A simple example to see that something must be wrong with Ampere's Law in the general case is given by Feynman in his Lectures in Physics (II, 18-3). Suppose we use a hypodermic needle to insert a spherically symmetric blob of charge in the middle of a large vat of solidified jello (which we assume conducts electricity). Because of electrostatic repulsion, the charge will dissipate, currents will flow outwards in a spherically symmetric way. Question: does this outward-
flowing current distribution generate a magnetic field? The answer must be no , because since we have a completely spherically symmetric situation, it could only generate a spherically symmetric magnetic field. But the only possible such fields are one pointing outwards everywhere and one pointing inwards everywhere, both corresponding to non-existent monopoles. So, there can be no magnetic field. However, imagine we now consider checking Ampere's law by taking as a path a horizontal circle with its center above the point where we injected the charge (think of a halo above someone’s head.) Obviously, the left hand side of Ampere's equation is zero, since there can be no magnetic field. On the other hand, the right hand side is most definitely not zero, since some of the outward flowing current is going to go through our circle. So the equation must be wrong. Ampere's law was established as the result of large numbers of careful experiments on all kinds of current distributions. So how could it be that something of the kind we describe above was overlooked? The reason is really similar to why electromagnetic induction was missed for so long. No-one thought about looking at changing fields, all the experiments were done on steady situations. With our ball of charge spreading outward in the jello, there is obviously a changing electric field. Imagine yourself in the jello near where the charge was injected: at first, you would feel a strong field from the nearby concentrated charge, but as the charge spreads out spherically, some of it going past you, the field will decrease with time. Maxwell's Example Maxwell himself gave a more practical example: consider Ampere's law for the usual infinitely long wire carrying a steady current I , but now break the wire at some point and put in two large circular metal plates, a capacitor, maintaining the steady current I in the wire everywhere else, so that charge is simply piling up on one of the plates and draining off the other. Looking now at the wire some distance away from the plates, the situation appears normal, and if we put the usual circular path around the wire, application of Ampere's law tells us that the magnetic field at distancer , from
is just
(Reminder on field direction: the right hand rule if you curl the fingers of your right hand around an imaginary wire, a current flowing in the direction indicated by your thumb will generate circular magnetic field lines in the direction indicated by your fingers.) Recall, however, that we defined the current threading the path in terms of current punching through a soap film spanning the path, and said this was independent of whether the soap film was flat, bulging out on one side, or whatever. With a single infinite wire, there was no escape no contortions of this covering surface could wriggle free of the wire going through it (actually, if you distort the surface enough, the wire could penetrate it several times, but you have to count the net flow across the surface, and the new penetrations would come in pairs with the current crossing the surface in opposite directions, so they would cancel). Once we bring in Maxwell's parallel plate capacitor, however, there is a way to distort the surface so that no current penetrates it at all: we can run it between the plates! The question then arises: can we rescue Ampere's law by adding another term just as the electrostatic version of the third equation was rescued by adding Faraday's induction term? The answer is of course yes: although there is no current crossing the surface if we put it between the capacitor plates, there is certainly a changing electric field , because the capacitor is charging up as the current I flows in. Assuming the plates are close together, we can take all the electric field lines from the charge q on one plate to flow across to the other plate, so the total electric flux across the surface between the plates,
Now, the current in the wire, I , is just the rate of change of charge on the plate,
Putting the above two equations together, we see that
Ampere's law can now be written in a way that is correct no matter where we put the surface spanning the path we integrate the magnetic field around:
This is Maxwell’s fourth equation. Notice that in the case of the wire, either the current in the wire, or the increasing electric field, contribute on the right hand side, depending on whether we have the surface simply cutting through the wire, or positioned between the plates. (Actually, more complicated situations are possible
we could imaging the
surface partly between the plates, then cutting through the plates to get out! In this case, we would have to figure out the current actually in the plate to get the right hand side, but the equation would still apply).
"Displacement Current" Maxwell referred to the second term on the right hand side, the changing electric field term, as the "displacement current". This was an analogy with a dielectric material. If a dielectric material is placed in an electric field, the molecules are distorted, their positive charges moving slightly to the right, say, the negative charges slightly to the left. Now consider what happens to a dielectric in an increasing electric field. The positive charges will be displaced to the right by a continuously increasing distance, so, as long as the electric field is increasing in strength, these charges are moving: there is actually a displacement current . (Meanwhile, the negative charges are moving the other way, but that is a current in the same direction, so adds to the effect of the positive charges' motion.) Maxwell's picture of the vacuum, the aether, was that it too had dielectric properties somehow, so he pictured a similar motion of charge in the vacuum to that we have just described in the dielectric. This is why the changing electric field term is often called the "displacement current", and in Ampere's law
(generalized) is just added to the real current, to give Maxwell's fourth and final equation. Another Angle on the Fourth Equation: the Link to Charge Conservation Going back for a moment to Ampere's law, we stated it as: for magnetostatics where the currents counted are those threading through the path we’re integrating around, so if there is a soap film spanning the path, these are the currents that punch through the film. Our mental picture here is usually of a few thin wires, maybe twisted in various ways, carrying currents. More generally, thinking of electrolytes, or even of fat wires, we should be envisioning a current density varying from point to point in space. In other words, we have a flux of current and the natural expression for the current threading our path is (analogous to the magnetic flux in the third equation) to write a surface integral of the current density
over a surface spanning the path, giving for magnetostatics
path integral
, (surface integral, over surface spanning path)
The question then arises as to whether the surface integral we have written on the right hand side above depends on which surface we choose spanning the path. From an argument exactly parallel to that for the magnetic flux in the third equation (see above), this will be true if and only if
for
a closed surface (with the path lying in the surface this closed surface is made up by combining two different surfaces spanning the path). Now,
taken over a closed surface is just the net current flow out of the
enclosed volume. Obviously, in a situation with steady currents flowing along wires or through conductors, with no charge piling up or draining away from anywhere, this is zero. However, if the total electric charge q , say, enclosed by the closed surface is changing as time goes on, then evidently
where we put in a minus sign because, with our convention,
is a little vector
pointing outwards, so the integral represents net flow of charge out from the surface, equal to the rate of decrease of the enclosed total charge. To summarize: if the local charge densities are changing in time, that is, if charge is piling up in or leaving some region, then around that region. That implies that will be different from
over a closed surface
over one surface spanning the wire
over another surface spanning the wire if these two
surfaces together make up a closed surface enclosing a region containing a changing amount of charge. The
key
to
fixing
this
up
is
to
realize
that
although it can be written as another surface integral over the same surface, using the first Maxwell equation, that is, the integral over a closed surface
where q is the total charge in the volume enclosed by the surface. By taking the time rate of change of both sides, we find
Putting this together with
gives:
for any closed surface, and consequently this is a surface integral that must be the same for any surface spanning the path or circuit! (Because two different surfaces spanning the same circuit add up to a closed surface. We’ll ignore the technically
trickier case where the two surfaces intersect each other, creating multiple volumes there one must treat each created volume separately to get the signs right.) Therefore, this is the way to generalize Ampere's law from the magnetostatic situation to the case where charge densities are varying with time, that is to say the path integral
and this gives the same result for any surface spanning the path. A Sheet of Current: A Simple Magnetic Field As a preliminary to looking at electromagnetic waves, we consider the magnetic field configuration from a sheet of uniform current of large extent. Think of the sheet as perpendicular to this sheet of paper, the current running vertically down into the paper. It might be helpful to visualize the sheet as many equal parallel fine wires uniformly spaced close together, carrying equal (small) currents. The magnetic field from this current sheet can be found using Ampere's law applied to a rectangular contour in the plane of the paper, with the current sheet itself bisecting the rectangle, so the rectangle's top and bottom are equidistant from the current sheet in opposite directions.
Applying Ampere’s law to the above rectangular contour, there are contributions to
(taken clockwise) only from the top and bottom, and they add to
give 2BL if the rectangle has side L. The total current enclosed by the rectangle is IL, taking the current density of the sheet to be I amperes per meter (how many little wires per meter multiplied by the current in each wire). Thus,
immediately gives:
B = µ0I/2 a magnetic field strength independent of distance d from the sheet. (This is the magnetostatic analog of the electrostatic result that the electric field from an infinite sheet of charge is independent of distance from the sheet.) In real life, where there are no infinite sheets of anything, these results are good approximations for distances from the sheet small compared with the extent of the sheet. Switching on the Sheet: How Fast Does the Field Build Up? Consider now how the magnetic field develops if the current in the sheet is suddenly switched on at time t = 0. We will assume that sufficiently close to the
sheet, the magnetic field pattern found above using Ampere's law is rather rapidly established. In fact, we will assume further that the magnetic field spreads out from the sheet like a tidal wave, moving in both directions at some speed v , so that after time t the field within distance vt of the sheet is the same as that found above for the magnetostatic case, but beyond vt there is at that instant no magnetic field present. Let us now apply Maxwell's equations to this guess to see if it can make sense. Certainly Ampere's law doesn't work by itself, because if we take a rectangular path as we did in the previous section, for d < vteverything works as before, but for a rectangle extending beyond the spreading magnetic field, d > vt , there will be no magnetic field contribution from the top and bottom of the rectangle, and hence
but there is definitely enclosed current! We are forced to conclude that for Maxwell's fourth equation to be correct, there must also be a changing electric field through the rectangular contour. Let us now try to nail down what this electric field through the contour must look like. First, it must be through the contour, that is, have a component perpendicular to the plane of the contour, in other words, perpendicular to the magnetic field. In fact, electric field components in other directions won't affect the fourth equation we are trying to satisfy, so we shall ignore them. Notice first that for a rectangular contour with d < vt, Ampere's law works, so we don't want a changing electric field through such a contour (but a constant electric field would be ok). Now apply Maxwell's fourth equation to a rectangular contour with d > vt,
It is: path integral
(over surface spanning path).
For the rectangle shown above, the integral on the left hand side is zero because is perpendicular to
along the sides, so the dot product is zero, and
is zero at
the top and bottom, because the outward moving "wave" of magnetic field hasn’t gotten there yet. Therefore, the right hand side of the equation must also be zero. We know
, so we must have:
Finding the Speed of the Outgoing Field Front: the Connection with Light So, as long as the outward moving front of magnetic field, travelling at v , hasn't reached the top and bottom of the rectangular contour, the electric field through the contour increases linearly with time, but the increase drops to zero (because Ampere's law is satisfied) the moment the front reaches the top and bottom of the rectangle. The simplest way to get this behavior is to have an electric field of strength E, perpendicular to the magnetic field, everywhere there is a magnetic
field, so the electric field also spreads outwards at speed v. (Note that, unlike the magnetic field, the electric field must point the same way on both sides of the current sheet, otherwise its net flux through the rectangle would be zero.)
After time t , then, the electric field flux through the rectangular contour (in the yz-plane in the diagram above) will be just field x area = E.2.vtL , and the rate of change will be 2EvL . (It's spreading both ways, hence the 2). Therefore ε0E.2.vL = -LI , the electric field is downwards and of strength E = I/ (2ε0v ). Since B = µ0I/2, this implies: B = µ0ε0vE. But we have another equation linking the field strengths of the electric and magnetic fields, Maxwell's third equation:
We can apply this equation to a rectangular contour with sides parallel to the E field, one side being within vt of the current sheet, the other more distant, so the only contribution to the integral is EL from the first side, which we take to have length L. (This contour is all on one side of the current sheet.) The area of the rectangle the magnetic flux is passing through will be increasing at a rate Lv (square meters per second) as the magnetic field spreads outwards.
It follows that E = vB. Putting this together with the result of the fourth equation, B = µ0ε0vE, we deduce v2 = 1/µ0ε0 Substituting the defined value of µ0, and the experimentally measured value of ε0, we find that the electric and magnetic fields spread outwards from the switchedon current sheet at a speed of 3 x 108 meters per second. To understand how this relates to wave propagation, imagine now that shortly after the current is switched on, the value of the current is suddenly doubled. Repeating the argument above for this more complicated situation, we find the following scenario:
We could have ramped up the field in a series of steps and the profile of the magnetic and electric fields would, effectively, be a graph of how the current built up over time. The next step is to imagine an electric current in the sheet that’s oscillating like a sine wave as a function of time: the magnetic and electric fields will evidently be sine waves too! In fact, this is how electromagnetic waves are generated. Of course, there’s no such thing as an infinite current sheet, an antenna has an
oscillating current going up and down a wire. But the mechanism is essentially the same: the only difference is that the geometry of the waves is complicated. Far away, they’ll look like expanding spheres, a three-dimensional version of the ripples on a pond when a stone falls in, instead of propagating planes. But at large distances a small fraction of these expanding spheres w, and that looks like a series of planes. The picture above of how the electric and magnetic fields relate to each other and to the direction of propagation of the wave is correct. Formu lation
"Microscopic" Name (material
equations "Macroscopic"
contributes
charge and current!)
to (easy
free
equations
charge
and
current in material)
Gauss' s law Gauss 's law for
Same as microscopic
magn etism Integr Max al
well– Farad ay equat
Same
ion
microscopic;
(Fara day's law of induc tion)
as
Ampè re's circui tal law ( with Max well's corre ction) Gauss' s law Gauss 's law for
Same as microscopic
magn etism Max well– Differe ntial
Farad ay equat ion (Fara day's law of induc tion) Ampè re's circui
Same as microscopic
tal law (with Max well's corre ction)
This is how Maxwell discovered a speed equal to the speed of light from a purely theoretical argument based on experimental determinations of forces between currents in wires and forces between electrostatic charges. This of course led to the realization that light is an electromagnetic wave, and that there must be other such waves with different wavelengths. Hertz detected other waves, of much longer wavelengths, experimentally, and this led directly to radio, t v, radar, cell phones, etc. Maxwell's equations – free space and linear isotropic media : (i) In Free Space: In absence of any free charges or currents or in the empty space i.e.,when q or p and i or J are zero. Then Maxwell’s equations take the form
(ii) In Isotropic Medium: In an isotropic dielectric (or non-conducting isotropic medium)
D =ε E , µ H ,J =σ E =0 and p=0
The Magnetic Vector Potential : Electric fields generated by stationary charges obey
This immediately allows us to write
since the curl of a gradient is automatically zero. In fact, whenever we come across an irrotational vector field in physics we can always write it as the gradient of some scalar field. This is clearly a useful thing to do, since it enables us to replace a vector field by a much simpler scalar field. The quantity
in the above
equation is known as the electric scalar potential. Magnetic fields generated by steady currents (and unsteady currents, for that matter) satisfy This immediately allows us to write
since the divergence of a curl is automatically zero. In fact, whenever we come across a solenoidal vector field in physics we can always write it as the curl of some other vector field. This is not an obviously useful thing to do, however, since it only allows us to replace one vector field by another. Nevertheless, Eq. is one of the most useful equations we shall come across in this lecture course. The quantity
is known as the magnetic vector potential.
We know from Helmholtz's theorem that a vector field is fully specified by its divergence and its curl. The curl of the vector potential gives us the magnetic field via Eq. . However, the divergence of are completely free to choose
has no physical significance. In fact, we to be whatever we like. Note that,
according to Eq. , the magnetic field is invariant under the transformation
In other words, the vector potential is undetermined to the gradient of a scalar field. This is just another way of saying that we are free to choose
. Recall
that the electric scalar potential is undetermined to an arbitrary additive constant, since the transformation
leaves the electric field invariant in Eq. . The transformations
are examples of
what mathematicians call gauge transformations. The choice of a particular
function
or a particular constant
is referred to as a choice of the gauge. We
are free to fix the gauge to be whatever we like. The most sensible choice is the one which makes our equations as simple as possible. The usual gauge for the scalar potential
is such that
at infinity. The usual gauge for
is such
that This particular choice is known as the Coulomb gauge. It is obvious that we can always add a constant to
so as to make it zero at
infinity. But it is not at all obvious that we can always perform a gauge transformation such as to make vector field
zero. Suppose that we have found some
whose curl gives the magnetic field but whose divergence in non-
zero. Let
The question is, can we find a scalar field
such that after we perform the gauge
transformation we are left with
. Taking the divergence of Eq. it is
clear that we need to find a function
which satisfies
But this is just Poisson's equation. We know that we can always find a unique solution of this equation This proves that, in practice, we can always set the divergence of
equal to zero.
Let us again consider an infinite straight wire directed along the carrying a current
-axis and
. The magnetic field generated by such a wire is written
We wish to find a vector potential
whose curl is equal to the above magnetic
field, and whose divergence is zero. It is not difficult to see that
fits the bill. Note that the vector potential is parallel to the direction of the current. This would seem to suggest that there is a more direct relationship between the vector potential and the current than there is between the magnetic field and the current. The potential is not very well-behaved on the
-axis, but this is just
because we are dealing with an infinitely thin current. Let us take the curl of Eq. We find that
where use has been made of the Coulomb gauge condition . We can combine the above relation with the field equation to give
Writing this in component form, we obtain
But, this is just Poisson's equation three times over. We can immediately write the unique solutions to the above equations:
These solutions can be recombined to form a single vector solution
Of course, we have seen a equation like this before:
Equations are the unique solutions (given the arbitrary choice of gauge) to the field equations : they specify the magnetic vector and electric scalar potentials
generated by a set of stationary charges, of charge density
steady currents, of current density Eq. satisfies the gauge condition (with currents.
and
, and a set of
. Incidentally, we can prove that above by repeating the analysis of Eqs.
), and using the fact that
for steady
Scalar potential Scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.
gravitational potential well of an increasing mass where A scalar potential is a fundamental concept in vector analysis and physics (the adjective scalar is frequently omitted if there is no danger of confusion with vector potential). The scalar potential is an example of a scalar field. Given a vector field F, the scalar potential P is defined such that:
where ∇P is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinates x,y,z. In some cases, mathematicians may use a positive sign in front of the gradient to define the potential. Because of this definition of P in terms of the gradient, the direction of F at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length. In order for F to be described in terms of a scalar potential only, the following have to be true:
1.
, where the integration is over a Jordan arc passing from location a to location b and P(b) is P evaluated at location b .
2.
, where the integral is over any simple closed path, otherwise known as a Jordan curve.
3. The first of these conditions represents the fundamental theorem of the gradient and is true for any vector field that is a gradient of a differentiable single valued scalar field P. The second condition is a requirement of F so that it can be expressed as the gradient of a scalar function. The third condition re-expresses the second condition in terms of the curl of F using the fundamental theorem of the curl. A vector field F that satisfies these conditions is said to be irrotational (Conservative). Scalar potentials play a prominent role in many areas of physics and engineering. The gravity potential is the scalar potential associated with the gravity per unit mass, i.e., the acceleration due to the field, as a function of position. The gravity potential is the gravitational potential energy per unit mass. In electrostatics the electric potential is the scalar potential associated with the electric field, i.e., with the electrostatic force per unit charge. The electric potential is in this case the electrostatic potential energy per unit charge. In fluid dynamics, irrotational lamellar fields have a scalar potential only in the special case when it is a Laplacian field. Certain aspects of the nuclear force can be described by a Yukawa potential. The potential play a prominent role in theLagrangian and Hamiltonian formulations of classical mechanics. Further, the scalar potential is the fundamental quantity in quantum mechanics. Not every vector field has a scalar potential. Those that do are called conservative, corresponding to the notion of conservative force in physics. Examples of non-conservative forces include frictional forces,
magnetic forces, and in fluid mechanics a solenoidal field velocity field. By the Helmholtz decomposition theorem however, all vector fields can be describable in terms of a scalar potential and corresponding vector potential. In electrodynamics the electromagnetic scalar and vector potentials are known together as the electromagnetic four-potential.
Integrablity Condition : If F is a conservative vector field (also called irrotational, curl-free, or potential), and its components have continuous partial derivatives, the potential of F with respect to a reference point
is defined in terms of the line integral:
where C is a parametrized path from
to
The fact that the line integral depends on the path C only through its terminal points
and
is, in essence, the path independence property of a conservative
vector field. The fundamental theorem of calculus for line integrals implies that if V is defined in this way, then
so that V is a scalar potential of the
conservative vector field F. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference point The Vector and Scalar Potentials Problem: Given Maxwell's four equations, demonstrate the existence of a vector magnetic potential and a scalar electric potential. Derive field equations for these potentials. Solution: Maxwell's four equations read: xH= xE=. H=0
0 0
E/ t + j H/ t
1. 2. 3.
.
E=
/
4.
0
where H is the magnetic field (A/m), E is the electric field (V/m), j is the vector current density (A/m2), space,
0
=4
0
= 8.8542 x 10-12 F/m is the permittivity of free
x 10-7H/m is the permeability of free space, and
is the scalar
charge density (C/m3). Existence Of The Vector And Scalar Potentials Let us begin with eq. 3. By a theorem of vector calculus,
.
H = 0 if and only if
there exists a vector field A such that 5.
H= xA Call A the vector magnetic potential.
Let us now take eq. 2. We will find a characteristic solution Ec and a particular solution Ep. The total electric field will then be the sum Ec + Ep. a. Characteristic solution:
x Ec = 0. By another theorem of vector calculus, this
expression can be true if and only if there exists a scalar field -
. Call
the scalar electric potential.
b. Particular solution: x Ep = -
such that Ec =
0
(
x Ep = -
x A)/ t =
0
x (-
H/ t, and H = 0
x A allows us to write
A/ t)
We may obtain a particular solution Ep by simply choosing Ep = -
0
A/ t.
c. Total electric field: E = Ec + Ep = - 0 A/ t Relationships Between The Vector And Scalar Potentials
6.
Let us now substitute the expressions derived above intoeqs. 4 and 1. From eq. 4, we obtain .
(- 0 A/ t) = and from eq. 1, we obtain
/
0
7.
8. x ( x A) = 0 (- 0 A/ t)/ t + j By still another theorem of vector calculus, we have the identity x(
x A) =
(
.
A) -
2
A
so that eq. 8 becomes ( . A) - 2A = 0 (or, after some simplification .
(
2
A) -
Since
A=-
(
0
-
0
A/ t)/
/
t) -
0
0
8a.
t+j 2
A/ t2+ j 9.
x grad = 0, eq. 9 may be manipulated by taking the curl of both sides.
The terms involving the gradient then vanish leaving us with the identity 2
x(
A) =
x(
0
0
2
A/ t2 + j)
from which we may infer, without loss of generality, that 2
A=
0
or
10. 10a.
2
0
A/ t2 + j 2 A/ t2 = j 0 0
2
A-
Eq. 10a is one of the field equations we sought. We may simplify eq. 10a somewhat if we recognize that now introduce a new operator 2
2
=
- (1/c2)
2
2
0
= 1/c2 where c is the speed of light. We
0
defined by
/ t2
so that eq. 10a becomes 2
11.
A=j Using eq. 10 in eq. 9, we obtain .
(
A) = -
(
/ t)
0
from which we may infer, again without loss of generality, that 12.
.
A=- 0 / t We must next rewrite eq. 7 by distributing the operator
. 7.
.
(- 0 A/ t) = / 0 becomes - 2 - 0 ( .A)/ t = Substituting eq. 12 into eq. 7a gives 2
-
-
0
(-
0
/ t)/
t=
/
/
7a.
0
0
which may be simplified: 2
- 0 0 2 or, recalling that 2
/ 0
t2 = - / 0 2 0 = 1/c , and using the operator
=- / 0 Eq. 13a is the other field equation that we sought.
13. 2
13a.
To summarize: 1. The existence of a vector magnetic potential A and a scalar electric potential and
was demonstrated. The respective field equations for A
were 2
and
=-
found /
to
be
2
A
=
j
0
1. where j is the vector current density (A/m 2), potential (C/m3), and
0
is the scalar charge
= 8.8542 x 10-12 F/m is the permittivity of free
space. Dimensional analysis on the above equations shows that A has units of electric current (amperes) and
has units of electric potential (volts).
The Vector and Scalar Potentials Problem: Given Maxwell's four equations, demonstrate the existence of a vector magnetic potential and a scalar electric potential. Derive field equations for these potentials. Solution: Maxwell's four equations read: xH= xE=. H=0 .
E=
0
/
1. 2. 3. 4.
E/ t + j H/ t
0
0
where H is the magnetic field (A/m), E is the electric field (V/m), j is the vector current density (A/m2), space,
0
=4
0
= 8.8542 x 10-12 F/m is the permittivity of free
x 10-7H/m is the permeability of free space, and
charge density (C/m3). Existence Of The Vector And Scalar Potentials
is the scalar
Let us begin with eq. 3. By a theorem of vector calculus,
.
H = 0 if and only if
there exists a vector field A such that 5.
H= xA Call A the vector magnetic potential.
Let us now take eq. 2. We will find a characteristic solution E c and a particular solution Ep. The total electric field will then be the sum Ec + Ep. a. Characteristic solution:
x Ec = 0. By another theorem of vector calculus, this
expression can be true if and only if there exists a scalar field -
. Call
the scalar electric potential.
b. Particular solution: x Ep = -
such that Ec =
(
0
x Ep = -
H/ t, and H =
0
x A)/ t =
x (-
x A allows us to write
A/ t)
0
We may obtain a particular solution Ep by simply choosing Ep = -
0
A/ t.
c. Total electric field: E = Ec + Ep = - 0 A/ t Relationships Between The Vector And Scalar Potentials
6.
Let us now substitute the expressions derived above intoeqs. 4 and 1. From eq. 4, we obtain .
(- 0 A/ t) = and from eq. 1, we obtain
/
7.
0
8. x ( x A) = 0 (- 0 A/ t)/ t + j By still another theorem of vector calculus, we have the identity x(
x A) =
(
.
2
A) -
A
so that eq. 8 becomes ( . A) - 2A = 0 (or, after some simplification (
.
A) -
Since
2
A=-
(
/
0
0
t) -
A/ t)/ t + j 0
0
2
A/ t2+ j 9.
x grad = 0, eq. 9 may be manipulated by taking the curl of both sides.
The terms involving the gradient then vanish leaving us with the identity x(
8a.
2
A) =
x(
0
0
2
A/ t2 + j)
from which we may infer, without loss of generality, that
2
A=
0
or
10. 10a.
2
0
A/ t2 + j 2 A/ t2 = j 0 0
2
A-
Eq. 10a is one of the field equations we sought. We may simplify eq. 10a somewhat if we recognize that now introduce a new operator 2
2
=
- (1/c2)
2
2
0
0
= 1/c2 where c is the speed of light. We
defined by
/ t2
so that eq. 10a becomes 2
11.
A=j Using eq. 10 in eq. 9, we obtain .
(
A) = -
(
/ t)
0
from which we may infer, again without loss of generality, that 12.
.
A=- 0 / t We must next rewrite eq. 7 by distributing the operator
. 7.
.
(- 0 A/ t) = / 0 becomes - 2 - 0 ( .A)/ Substituting eq. 12 into eq. 7a gives 2
-
-
0
(-
/ t)/
0
t=
t= /
/
7a.
0
0
which may be simplified: 2
- 0 0 2 or, recalling that
/ t2 = - / 0 2 0 0 = 1/c , and using the operator
13. 2
13a.
2
=- / 0 Eq. 13a is the other field equation that we sought. To summarize:
2. The existence of a vector magnetic potential A and a scalar electric potential and and
was demonstrated. The respective field equations for A
were 2
=-
found /
to
be
A
=
j
0
2. where j is the vector current density (A/m2), potential (C/m3), and space.
2
0
is the scalar charge
= 8.8542 x 10-12 F/m is the permittivity of free
3. Dimensional analysis on the above equations shows that A has units of electric current (amperes) and
Gauge Theory In physics, a gauge
theory is
has units of electric potential (volts).
a
type
of field
theory in
which
the Lagrangian is invariant under a continuous group of local transformations. The term gauge refers to redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group which is referred to as thesymmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding vector field called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, the gauge theory is referred to as nonabelian, the usual example being the Yang–Mills theory. Many powerful theories in physics are described by Lagrangians which are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the space in which the physical processes occur, they are said to have a global symmetry. The requirement of local symmetry, the cornerstone of gauge theories, is a stricter constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in space-time. Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1)and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1)×SU(2)×SU(3)and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.
Gauge theories are also important in explaining gravitation in the theory of general relativity. Its case is somewhat unique in that the gauge field is a tensor, the Lanczos tensor. Theories ofquantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton. Gauge symmetries can be viewed as analogues of the principle of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of general covariance with a true gauge principle with new gauge fields. Historically, these ideas were first stated in the context of classical electromagnetism and later in general relativity. However, the modern importance of
gauge
symmetries
appeared
first
in
therelativistic
quantum
mechanics of electrons – quantum electrodynamics, elaborated on below. Today, gauge
theories
are
useful
in condensed
matter, nuclear and high
energy
physics among other subfields. Gauge fields The "gauge covariant" version of a gauge theory accounts for this effect by introducing a gauge field (in mathematical language, an Ehresmann connection) and formulating all rates of change in terms of the covariant derivative with respect to this connection. The gauge field becomes an essential part of the description of a mathematical configuration. A configuration in which the gauge field can be eliminated by a gauge transformation has the property that its field strength (in mathematical language, its curvature) is zero everywhere; a gauge theory is not limited to these configurations. In other words, the distinguishing characteristic of a gauge theory is that the gauge field does not merely compensate for a poor choice of coordinate system; there is generally no gauge transformation that makes the gauge field vanish.
When analyzing the dynamics of a gauge theory, the gauge field must be treated as a dynamical variable, similarly to other objects in the description of a physical situation. In addition to itsinteraction with other objects via the covariant derivative, the gauge field typically contributes energy in the form of a "selfenergy" term. One can obtain the equations for the gauge theory by: •
starting from a naïve ansatz without the gauge field (in which the derivatives appear in a "bare" form);
•
listing those global symmetries of the theory that can be characterized by a continuous parameter (generally an abstract equivalent of a rotation angle);
•
computing the correction terms that result from allowing the symmetry parameter to vary from place to place; and
•
reinterpreting these correction terms as couplings to one or more gauge fields, and giving these fields appropriate self-energy terms and dynamical behavior.
This is the sense in which a gauge theory "extends" a global symmetry to a local symmetry, and closely resembles the historical development of the gauge theory of gravity known as general relativity. Physical experiments Gauge theories are used to model the results of physical experiments, essentially by: •
limiting the universe of possible configurations to those consistent with the information used to set up the experiment, and then
•
computing the probability distribution of the possible outcomes that the experiment is designed to measure.
The mathematical descriptions of the "setup information" and the "possible measurement outcomes" (loosely speaking, the "boundary conditions" of the experiment) are generally not expressible without reference to a particular coordinate system, including a choice of gauge. (If nothing else, one assumes that the experiment has been adequately isolated from "external" influence, which is itself a gauge-dependent statement.) Mishandling gauge dependence in boundary conditions is a frequent source of anomalies in gauge theory calculations, and
gauge theories can be broadly classified by their approaches to anomaly avoidance. Continuum theories The two gauge theories mentioned above (continuum electrodynamics and general relativity) are examples of continuum field theories. The techniques of calculation in a continuum theory implicitly assume that: given a completely fixed choice of gauge, the boundary conditions of an
•
individual configuration can in principle be completely described; given a completely fixed gauge and a complete set of boundary conditions,
•
the principle of least action determines a unique mathematical configuration (and therefore a unique physical situation) consistent with these bounds; the likelihood of possible measurement outcomes can be determined by:
•
establishing a probability distribution over all physical situations
•
determined by boundary conditions that are consistent with the setup information, establishing a probability distribution of measurement outcomes
•
for each possible physical situation, and convolving these two probability distributions to get a distribution
•
of possible measurement outcomes consistent with the setup information; and •
fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.
These assumptions are close enough to be valid across a wide range of energy scales and experimental conditions, to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life, from light, heat, and electricity to eclipses and spaceflight. They fail only at the smallest and largest scales (due to omissions in the theories themselves) and when the mathematical techniques themselves break down (most notably in the case of turbulence and other chaotic phenomena). Gauge invariance
The electric and magnetic fields are obtained from the vector and scalar potentials according to the prescription
These fields are important because they determine the electromagnetic forces exerted on charged particles. Note that the above prescription does not uniquely determine the two potentials. It is possible to make the following transformation, known as a gauge transformation, which leaves the fields unaltered:
where
is a general scalar field. It is necessary to adopt some form of
convention, generally known as a gauge condition, to fully specify the two potentials. In fact, there is only one gauge condition which is consistent with Eqs. This is the Lorentz gauge condition,
Note that this condition can be written in the Lorentz invariant form
This implies that if the Lorentz gauge holds in one particular inertial frame then it automatically holds in all other inertial frames. A general gauge transformation can be written
Note that even after the Lorentz gauge has been adopted the potentials are undetermined to a gauge transformation using a scalar field
which satisfies the
sourceless wave equation
However, if we adopt ``sensible'' boundary conditions in both space and time then the only solution to the above equation is Plane Wave Solution D = 2, Supersymmetric Yang-Mills Quantum Mechanics Supersymmetric Yang-Mills quantum mechanics can be obtained by a dimensional reduction of a supersymmetric, D = d + 1 dimensional Yang-Mills quantum field theory to one point in d-dimensional space. The initial local gauge symmetry of field theory is thus reduced to a global symmetry of the quantum mechanical system. The simplest ones among such systems are those obtained by reduction of the N = 1 Yang-Mills gauge theory in two dimensions They are described by a scalar field φA and a complex fermion λA, where A labels the generators of the gauge group. The Hamiltonian is1
with the algebra of operators given by
The generator of the gauge transformations is
The supersymmetry charges are
And
The fermionic term of the Hamiltonian is proportional to the generator of the gauge symmetry and hence it is supposed to vanish on any physical (gauge invariant) state due to the Gauss law. Hence, the Hamiltonian is simply
Subsequently, we introduce bosonic creation and annihilation operators,
and express them, as well as the fermionic creation and annihilation operators, in the matrix notation,
where T Aij are the generators of the SU(N) group in the fundamental representation. Hence, all operators become operator valued N × N matrices. Such a notation has a very practical feature, namely, any gauge invariant operator can be simply written as a trace of a product of appropriate operator valued matri- ces . In situations when it is self-evident we will use a simplified notation for the trace of any such matrix, namely, tr(O) ≡ (O). We get
and
Therefore, in the case of the SU(3) group we have,
Obviously, the Hamiltonian eq.(11) conserves the fermionic occupation number. Hence, we can analyze its spectrum separately in each subspace of the Hilbert
space with fixed fermionic occupation number. There are 9 such fermionic sectors because of the Pauli exclusion principle. This Hamiltonian possess also another symmetry, namely, the particle-hole symmetry , which can be thought of as a quantum mechanical precursor of the charge conjugation sym- metry in quantum field theory. This symmetry can be observed as a perfect matching of the energy eigenvalues in the sectors with nF and 8−nF fermionic quanta2. Thus, it is sufficient to consider only the sectors with nF = 0, . . . , 4 fermionic quanta. The cut Fock basis method The cut Fock space approach turned out to be very useful in the studies of gauge systems for several reasons. First of all, it is a fully non-perturbative tool. Second, it treats bosons and fermions on an equal footing, therefore calculations can be
extended to all fermionic sectors without difficulties. Finally, it can be generalized, to handle SU(N) gauge groups with N ≥ 2, as well as systems defined in spaces of various dimensionality. The analytic approach used in this work is inspired by this numerical treat- ment. Hence, we now briefly summarize the numerical approach in order to introduce the basic notions needed in the following parts. For a more extensive discussion see The basic ingredient of the numerical approach is a systematic construction of the Fock basis using the concepts of elementary bosonic bricks and composite fermionic bricks (see tables 1 and 2). Elementary bricks are linearly independent single trace operators, composed uniquely of creation operators, which cannot be reduced by the Cayley-Hamilton theorem. The set of states obtained by acting with all possible products of powers of bosonic elementary bricks on the Fock vacuum spans the bosonic sector of the Hilbert space . As far as the fermionic sectors are concerned, apart of the bosonic bricks one has to use fermionic bricks which need not to be single traces operators . The spectra of the Hamiltonian are obtained by diagonalizing numerically the Hamiltonian matrix evaluated in the cut basis. Therefore, one introduces a cut-off, Ncut, which limits the maximal number of quanta contained in the basis states. Once the cut-off is set the Hamiltonian matrix becomes finite and it is possible to evaluate all its elements. Subsequently, we can diagonalize numerically such matrix, as well as any other matrices corresponding to other physical observables. The physically reliable results are those obtained in the infinite cut-off limit, called also the continuum limit . The procedure requires to perform several calculations with increasing cut-off and to extrapolate the re- sults to the continuum limit. It was shown that the eigenenergies corresponding to localized and nonlocalized states behave differently . The former ones converge rapidly to their physical limit, whereas the latter ones slowly fall to zero.
SU(3) fermionic bricks in the sectors with nF = 1, . . . , 4 fermions. Our analytic approach uses the cut Fock basis since it provides a system-atic control of the Hilbert space. However, instead of evaluating the matrix elements of the Hamiltonian operator, one transforms the eigenvalue problem into a problem of finding a solution of some recursion relation. The analytic formulae derived in this article are particularly useful since they describe both the approximate numerical results and the exact continuum solutions. This is possible because they are parameterized by the cut-off Ncut. In the limit of infinite Ncut our formulae give the exact solutions, whereas for any finite cut-off they can be directly compared with numerical results. Gauge fixing : In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailedlocal field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.
Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration by aparticular detailed configuration (or even a weighted distribution of them). Judicious gauge fixing can simplify calculations immensely, but becomes progressively harder as the physical model becomes m
Coulomb gauge : The Coulomb gauge (also known as the transverse gauge) is much used in quantum chemistry and condensed matter physics and is defined by the gauge condition (more precisely, gauge fixing condition)
It is particularly useful for "semi-classical" calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb interaction is not. The Coulomb gauge has a number of properties: 1. The potentials can be expressed in terms of instantaneous values of the fields and densities (in SI units)
where ρ(r, t) is the electric charge density, R = |r - r'| (where r is any position vector in space and r' a point in the charge or current distribution), the del operates on r and d3r is the volume element at r'. The instantaneous nature of these potentials appears, at first sight, to violate causality, since motions of electric charge or magnetic field appear everywhere instantaneously as changes to the potentials. This is justified by noting that the scalar and vector potentials themselves do not affect the motions of charges, only the combinations of their derivatives that form the electromagnetic field strength. Although one can
compute the field strengths explicitly in the Coulomb gauge and demonstrate that changes in them propagate at the speed of light, it is much simpler to observe that the field strengths are unchanged under gauge transformations and to demonstrate causality in the manifestly Lorentz covariant Lorenz gauge described below. Another expression for the vector potential, in terms of the time-retarded electric current density J(r, t), has been obtained to be:[1]
. 2. Further gauge transformations that retain the Coulomb gauge
condition might be made with gauge functions that satisfy ∇2 ψ = 0, but as the only solution to this equation that vanishes at infinity (where all fields are required to vanish) is ψ(r, t) = 0, no gauge arbitrariness remains. Because of this, the Coulomb gauge is said to be a complete gauge, in contrast to gauges where some gauge arbitrariness remains, like the Lorenz gauge below. 3. The Coulomb gauge is a minimal gauge in the sense that the integral of A2 over all space is minimal for this gauge: all other gauges give a larger integral.[2] The minimum value given by the Coulomb gauge is
. 4. In regions far from electric charge the scalar potential becomes zero.
This is known as the radiation gauge. Electromagnetic radiation was first quantized in this gauge. 5. The Coulomb gauge is not Lorentz covariant. If a Lorentz transformation to a new inertial frame is carried out, a further gauge transformation has to be made to retain the Coulomb gauge condition. Because of this, the Coulomb gauge is not used in covariant perturbation theory, which has become standard for the treatment of relativistic quantum field theories such as quantum electrodynamics. Lorentz covariant gauges such as the Lorenz gauge are used in these theories.
6. For a uniform and constant magnetic field B the vector potential in the Coulomb gauge is
which can be confirmed by calculating the div and curl of A. The divergence of A at infinity is a consequence of the unphysical assumption that the magnetic field is uniform throughout the whole of space. Although this vector potential is unrealistic in general it can provide a good approximation to the potential in a finite volume of space in which the magnetic field is uniform. 7. As a consequence of the considerations above, the electromagnetic potentials may be expressed in their most general forms in terms of the electromagnetic fields as
where ψ(r, t) is an arbitrary scalar field called the gauge function. The fields that are the derivatives of the gauge function are known as pure gauge fields and the arbitrariness associated with the gauge function is known as gauge freedom. In a calculation that is carried out correctly the pure gauge terms have no effect on any physical observable. A quantity or expression that does not depend on the gauge function is said to be gauge invariant: all physical observables are required to be gauge invariant. A gauge transformation from the Coulomb gauge to another gauge is made by taking the gauge function to be the sum of a specific function which will give the desired gauge transformation and the arbitrary function. If the arbitrary function is then set to zero, the gauge is said to be fixed. Calculations may be carried out in a fixed gauge but must be done in a way that is gauge invariant.
Lorenz Gauge : The Lorenz gauge is given, in SI units, by:
and in Gaussian units by:
This may be rewritten as:
where Aμ =
(φ/c,
−A)
is
the electromagnetic
four-potential,
∂μ the 4-
gradient [using the metric signature (+−−−)]. It is unique among the constraint gauges in retaining manifest Lorentz invariance. Note, however, that this gauge was originally named after the Danish physicist Ludvig Lorenz and not afterHendrik Lorentz; it is often misspelled "Lorentz gauge". (Neither was the first to use it in calculations; it was introduced in 1888 by George F. FitzGerald.) The Lorenz gauge leads to the following inhomogeneous wave equations for the potentials:
It can be seen from these equations that, in the absence of current and charge, the solutions are potentials which propagate at the speed of light. The Lorenz gauge is incomplete in the sense that there remains a subspace of gauge transformations which preserve the constraint. These remaining degrees of freedom correspond to gauge functions which satisfy the wave equation
These remaining gauge degrees of freedom propagate at the speed of light. To obtain a fully fixed gauge, one must add boundary conditions along the light cone of the experimental region. Maxwell's equations in the Lorenz gauge simplify to where jν = (ρc, −j) is the four-current. Two solutions of these equations for the same current configuration differ by a solution of the vacuum wave equation . In this form it is clear that the components of the potential separately satisfy the Klein-Gordon equation, and hence that the Lorenz gauge condition allows transversely, longitudinally, and "time-like" polarized waves in the four-potential. The transverse polarizations correspond to classical radiation, i. e., transversely polarized waves in the field strength. To suppress the "unphysical" longitudinal and time-like polarization states, which are not observed in experiments at classical distance scales, one must also employ auxiliary constraints known as Ward identities. Classically, these identities are equivalent to the continuity equation . Many of the differences between classical and quantum electrodynamics can be accounted for by the role that the longitudinal and time-like polarizations play in interactions between charged particles at microscopic distances. Gauge transformations Electric and magnetic fields can be written in terms of scalar and vector potentials, as follows:
However, this prescription is not unique. There are many different potentials which can generate the same fields. We have come across this problem before. It
is called gauge invariance. The most general transformation which leaves the and
fields unchanged in Eqs.
This is clearly a generalization of the gauge transformation which we found earlier for static fields:
where
is a constant. In fact, if
then Eqs. becomes
We are free to choose the gauge so as to make our equations as simple as possible. As before, the most sensible gauge for the scalar potential is to make it go to zero at infinity:
For steady fields, we found that the optimum gauge for the vector potential was the so-called Coulomb gauge: We can still use this gauge for non-steady fields , that it is always possible to transform away the divergence of a vector potential, remains valid. One of the nice features of the Coulomb gauge is that when we write the electric field,
we find that the part which is generated by charges (i.e., the first term on the right-hand side) is conservative, and the part induced by magnetic fields (i.e., the second term on the right-hand side) is purely solenoidal. Earlier on, we proved mathematically that a general vector field can be written as the sum of a conservative field and a solenoidal field . Now we are finding that when we split up the electric field in this manner the two fields have different physical origins:
the conservative part of the field emanates from electric charges, whereas the solenoidal part is induced by magnetic fields. Equation can be combined with the field equation
(which remains valid for non-steady fields) to give
With the Coulomb gauge condition,
, the above expression reduces
to
which is just Poisson's equation. Thus, we can immediately write down an expression for the scalar potential generated by non-steady fields. It is exactly the same as our previous expression for the scalar potential generated by steady fields, namely
However, this apparently simple result is extremely deceptive. Equation is a typical action at a distance law. If the charge density changes suddenly at the potential at
then
responds immediately. However, we shall see later that the full
time-dependent Maxwell's equations only allow information to propagate at the speed of light (i.e., they do not violate relativity). How can these two statements be reconciled? The crucial point is that the scalar potential cannot be measured directly, it can only be inferred from the electric field. In the time-dependent case, there are two parts to the electric field: that part which comes from the scalar potential, and that part which comes from the vector potential . So, if the scalar potential responds immediately to some distance rearrangement of charge density it does not necessarily follow that the electric field also has an immediate response. What actually happens is that the change in the part of the electric field which comes from the scalar potential is balanced by an equal and opposite change in the part which comes from the vector potential, so that the overall
electric field remains unchanged. This state of affairs persists at least until sufficient time has elapsed for a light signal to travel from the distant charges to the region in question. Thus, relativity is not violated, since it is the electric field, and not the scalar potential, which carries physically accessible information. It is clear that the apparent action at a distance nature of Eq. is highly misleading. This suggests, very strongly, that the Coulomb gauge is not the optimum gauge in the time-dependent case. A more sensible choice is the so called Lorentz gauge:
It can be shown, by analogy with earlier arguments , that it is always possible to make a gauge transformation, at a given instance in time, such that the above equation is satisfied. Substituting the Lorentz gauge condition into Eq., we obtain
It turns out that this is a three-dimensional wave equation in which information propagates at the speed of light. But, more of this later. Note that the magnetically
induced part of the electric field (i.e.,
) is not purely solenoidal in the
Lorentz gauge. This is a slight disadvantage of the Lorentz gauge with respect to the Coulomb gauge. However, this disadvantage is more than offset by other advantages which will become apparent presently. Incidentally, the fact that the part of the electric field which we ascribe to magnetic induction changes when we change the gauge suggests that the separation of the field into magnetically induced and charge induced components is not unique in the general time-varying case (i.e., it is a convention). Poynting's Theorem : In electrodynamics, Poynting's theorem is a statement of energy conservation for the electromagnetic field, in the form of a partial differential equation, due to the British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic
field to the work done on a charge distribution (i.e. an electrically charged object), through energy flux Poynting's Theorem, Work and Energy Recall from elementary physics that the rate at which work is done on an electric charge by an electromagnetic field is:
If one follows the usual method of constructing a current density made up of many charges, it is easy to show that this generalizes to:
for the rate at which an electric field does work on a current density throughout a volume. The magnetic field, of course, does no work because the force it creates is always perpendicular to
or
.
If we use AL to eliminate
and integrate over a volume of space to compute the rate the electromagnetic field is doing work within that volume:
Using:
(which can be easily shown to be true as an identity by distributing the
derivatives) and then use FL to eliminate
It is easy to see that:
, one gets:
from which we see that these terms are the time derivative of the electromagnetic field energy density:
Moving the sign to the other side of the power equation above, we get:
as the rate at which power flows out of the volume
(which is arbitrary).
Equating the terms under the integral:
where we introduce the Poynting vector
This has the precise appearance of conservation law. If we apply the divergence theorem to the integral form to change the volume integral of the divergence of
where
into a surface integral of its flux:
is the closed surface that bounds the volume
. Either the differential
or integral forms constitute the Poynting Theorem. In words, the sum of the work done by all fields on charges in the volume, plus the changes in the field energy within the volume, plus the energy that flows out of the volume carried by the field must balance - this is a version of the workenergy theorem, but one expressed in terms of the fields. In this interpretation, we see that
must be the vector intensity of the
electromagnetic field - the energy per unit area per unit time - since the flux of the
Poynting vector through the surface is the power passing through it. It's magnitude is the intensity proper, but it also tells us the direction of energy flow. With this said, there is at least one assumption in the equations above that is not strictly justified, as we are assuming that the medium is dispersionless and has no resistance. We do not allow for energy to appear as heat, in other words, which surely would happen if we drive currents with the electric field. We also used the macroscopic field equations and energy densities, which involve a coarsegrained average over the microscopic particles that matter is actually made up of it is their random motion that is the missing heat. It seems, then, that Poynting's theorem is likely to be applicable in a microscopic description of particles moving in a vacuum, where their individual energies can be tracked and tallied:
but not necessarily so useful in macroscopic media with dynamical dispersion that we do not yet understand. There we can identify the
term as the rate at
which the mechanical energy of the charged particles that make up
changes
and write:
(where
is, recall, an outward directed normal) so that this says that the rate at
which energy flows into the volume carried by the electromagnetic field equals the rate at which the total mechanical plus field energy in the volume increases. This is a marvelous result!
Momentum can similarly be considered, again in a microscopic description. There we start with Newton's second law and the Lorentz force law:
summing with coarse graining into an integral as usual:
As before, we eliminate sources using the inhomogeneous MEs (this time starting from the beginning with the vacuum forms):
or
Again, we distribute:
or
substitute it in above, and add
Finally, substituting in FL:
:
Reassembling and rearranging:
The quantity under the integral on the left has units of momentum density. We define:
to be the field momentum density. Proving that the right hand side of this interpretation is consistent with this is actually amazingly difficult. It is simpler to just define the Maxwell Stress Tensor:
In terms of this, with a little work one can show that:
That is, for each component, the time rate of change of the total momentum (field plus mechanical) within the volume equals the flux of the field momentum through the closed surface that contains the volume. Example Plane Parallel Capacitor The plane parallel capacitor of Fig. is familiar from Example . The circular electrodes are perfectly conducting, while the region between the electrodes is free space. The system is driven by a voltage source distributed around the edges of the electrodes. Between the electrodes, the electric field is simply the voltage divided by the plate spacing
while the magnetic field that follows from the integral form of Ampère's law is
Figure Plane parallel circular electrodes are driven by a distributed voltage source. Poynting flux through surface denoted by dashed lines accounts for rate of change of electric energy stored in the enclosed volume. Consider the application of the integral version of (8) to the surface S enclosing the region between the electrodes in Fig. First we determine the power flowing into the volume through this surface by evaluating the left-hand side of (8). The density of power flow follows from (11) and (12).
The top and bottom surfaces have normals perpendicular to this vector, so the only contribution comes from the surface at r = b. Because S is constant on that surface, the integration amounts to a multiplication.
where
Here the expression has been written as the rate of change of the energy stored in the capacitor. With Eagain given by (11), we double-check the expression for the time rate of change of energy storage.
From the field viewpoint, power flows into the volume through the surface at r = b and is stored in the form of electrical energy in the volume between the plates. In the quasistatic approximation used to evaluate the electric field, the magnetic
energy storage is neglected at the outset because it is small compared to the electric energy storage. As a check on the implications of this approximation, consider the total magnetic energy storage. From (12),
Comparison of this expression with the electric energy storage found in (15) shows that the EQS approximation is valid provided that
For a sinusoidal excitation of frequency , this gives
where c is the free space velocity of light Example . Long Solenoidal Inductor The perfectly conducting one-turn solenoid of Fig. is familiar from Example . In terms of the terminal current i = Kd, the magnetic field intensity inside is ,
while the electric field is the sum of the particular and conservative homogeneous parts [ for the particular part and Eh for the conservative part].
Figure One-turn solenoid surrounding volume enclosed by surface S denoted by dashed lines. Poynting flux through this surface accounts for the rate of change of magnetic energy stored in the enclosed volume. Consider how the power flow through the surface S of the volume enclosed by the coil is accounted for by the time rate of change of the energy stored. The Poynting flux implied by (19) and (20) is
This Poynting vector has no component normal to the top and bottom surfaces of the volume. On the surface at r = a, the first term in brackets is constant, so the integration on S amounts to a multiplication by the area. Because Eh is irrotational, the integral of Eh
ds = E
h
rd around a contour at r = a must be
zero. For this reason, there is no net contribution of Eh to the surface integral.
where
Here the result shows that the power flow is accounted for by the rate of change of the stored magnetic energy. Evaluation of the right hand side of (8), ignoring the electric energy storage, indeed gives the same result.
The validity of the quasistatic approximation is examined by comparing the magnetic energy storage to the neglected electric energy storage. Because we are only interested in an order of magnitude comparison and we know that the homogeneous solution is proportional to the particular solution , the latter can be approximated by the first term in (20).
We conclude that the MQS approximation is valid provided that the angular frequency
is small compared to the time required for an electromagnetic wave
to propagate the radius a of the solenoid and that this is equivalent to having an electric energy storage that is negligible compared to the magnetic energy storage.
A note of caution is in order. If the gap between the "sheet" terminals is made very small, the electric energy storage of the homogeneous part of the E field can become large. If it becomes comparable to the magnetic energy storage, the structure approaches the condition of resonance of the circuit consisting of the gap capacitance and solenoid inductance. In this limit, the MQS approximation breaks down. In practice, the electric energy stored in the gap would be dominated by that in the connecting plates, and the resonance could be described as the coupling of MQS and EQS systems The Lorentz force In physics, particularly electromagnetism, the Lorentz force is the force on a point charge due to electromagnetic fields. If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force
Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a particle which might be traveling near the speed of light (relativistic form of the Lorentz force). The first derivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889, although other historians suggest an earlier origin in an 1865 paper by James Clerk Maxwell. Lorentz derived it a few years after Heaviside. The Lorentz force The flow of an electric current down a conducting wire is ultimately due to the motion of electrically charged particles (in most cases, electrons) through the conducting medium. It seems reasonable, therefore, that the force exerted on the
wire when it is placed in a magnetic field is really the resultant of the forces exerted on these moving charges. Let us suppose that this is the case. Let
be the (uniform) cross-sectional area of the wire, and let
be the number
density of mobile charges in the conductor. Suppose that the mobile charges each have charge
and velocity
. We must assume that the conductor also contains
stationary charges, of charge
and number density
(say), so that the net
charge density in the wire is zero. In most conductors, the mobile charges are electrons and the stationary charges are atomic nuclei. The magnitude of the electric current flowing through the wire is simply the number of coulombs per second which flow past a given point. In one second, a mobile charge moves a distance
, so all of the charges contained in a cylinder of cross-sectional area
and length
is
flow past a given point. Thus, the magnitude of the current
. The direction of the current is the same as the direction of motion of
the charges, so the vector current is
. According to Eq. , the force
per unit length acting on the wire is
However, a unit length of the wire contains
moving charges. So, assuming
that each charge is subject to an equal force from the magnetic field (we have no reason to suppose otherwise), the force acting on an individual charge is
We can combine this with Eq. to give the force acting on a charge with velocity
in an electric field
and a magnetic field
moving
:
This is called the Lorentz force law, after the Dutch physicist Hendrik Antoon Lorentz who first formulated it. The electric force on a charged particle is parallel
to the local electric field. The magnetic force, however, is perpendicular to both the local magnetic field and the particle's direction of motion. No magnetic force is exerted on a stationary charged particle. The equation of motion of a free particle of charge
and mass
moving in
electric and magnetic fields is
according to the Lorentz force law. This equation of motion was first verified in a famous experiment carried out by the Cambridge physicist J.J. Thompson in 1897. Thompson was investigating cathode rays, a then mysterious form of radiation emitted by a heated metal element held at a large negative voltage (i.e., a cathode) with respect to another metal element (i.e., an anode) in an evacuated tube. German physicists held that cathode rays were a form of electromagnetic radiation, whilst British and French physicists suspected that they were, in reality, a stream of charged particles. Thompson was able to demonstrate that the latter view was correct. In Thompson's experiment, the cathode rays passed though a region of ``crossed'' electric and magnetic fields (still in vacuum). The fields were perpendicular to the original trajectory of the rays, and were also mutually perpendicular. Let us analyze Thompson's experiment. Suppose that the rays are originally traveling in the the
-direction, and are subject to a uniform electric field
-direction and a uniform magnetic field
in the
in
-direction. Let us
assume, as Thompson did, that cathode rays are a stream of particles of mass and charge
where
. The equation of motion of the particles in the
is the velocity of the particles in the
-direction is
-direction. Thompson started off
his experiment by only turning on the electric field in his apparatus, and
measuring the deflection distance
of the ray in the
through the electric field. It is clear from the equation of motion that
where the ``time of flight''
if
-direction after it had traveled a
is replaced by
. This formula is only valid
, which is assumed to be the case. Next, Thompson turned on the
magnetic field in his apparatus, and adjusted it so that the cathode ray was no longer deflected. The lack of deflection implies that the net force on the particles in the
-direction was zero. In other words, the electric and magnetic forces
balanced exactly. It follows from Eq. that with a properly adjusted magnetic field strength
Thus, Eqs. and can be combined and rearranged to give the charge to mass ratio of the particles in terms of measured quantities:
Using this method, Thompson inferred that cathode rays were made up of negatively charged particles (the sign of the charge is obvious from the direction of the deflection in the electric field) with a charge to mass ratio
of
C/kg. A decade later, in 1908, the American Robert Millikan
performed his famous ``oil drop'' experiment, and discovered that mobile electric
charges are quantized in units of
C. Assuming that mobile
electric charges and the particles which make up cathode rays are one and the same thing, Thompson's and Millikan's experiments imply that the mass of these
particles is
kg. Of course, this is the mass of an electron (the
modern value is
kg), and
C is the charge of an
electron. Thus, cathode rays are, in fact, streams of electrons which are emitted from a heated cathode, and then accelerated because of the large voltage difference between the cathode and anode. Consider, now, a particle of mass magnetic field,
and charge
moving in a uniform
. According, to Eq , the particle's equation of motion
can be written:
This reduces to
Here, be solved to give
and
is called the cyclotron frequency. The above equations can
According to these equations, the particle trajectory is a spiral whose axis is
parallel to the magnetic field. The radius of the spiral is
, where
is the particle's constant speed in the plane perpendicular to the magnetic field. The particle drifts parallel to the magnetic field at a constant velocity,
.
Finally, the particle gyrates in the plane perpendicular to the magnetic field at the cyclotron frequency. Finally, if a particle is subject to a force interval
and moves a distance
in a time
, then the work done on the particle by the force is
The power input to the particle from the force field is
where
is the particle's velocity. It follows from the Lorentz force law, Eq. that
the power input to a particle moving in electric and magnetic fields is
Note that a charged particle can gain (or lose) energy from an electric field, but not from a magnetic field. This is because the magnetic force is always perpendicular to the particle's direction of motion, and, therefore, does no work on the particle [see Eq. ]. Thus, in particle accelerators, magnetic fields are often used to guide particle motion (e.g., in a circle) but the actual acceleration is performed by electric fields.
Conservation laws for a system of charges and electromagnetic fields : conservation laws, in physics, basic laws that together determine which processes can or cannot occur in nature; each law maintains that the total value of the quantity governed by that law, e.g., mass or energy, remains unchanged during physical processes. Conservation laws have the broadest possible
application of all laws in physics and are thus considered by many scientists to be the most fundamental laws in nature. Conservation of classical processes : Most conservation laws are exact, or absolute, i.e., they apply to all possible processes; a few conservation laws are only partial, holding for some types of processes but not for others. By the beginning of the 20th cent. physics had established conservation laws governing the following quantities: energy, mass (or matter), linear momentum, angular momentum, and electric charge. When the theory of relativity showed (1905) that mass was a form of energy, the two laws governing these quantities were combined into a single law conserving the total of mass and energy. Conservation of elementary particle properties : With the rapid development of the physics of elementary particles during the 1950s, new conservation laws were discovered that have meaning only on this subatomic level. Laws relating to the creation or annihilation of particles belonging to the baryon and lepton classes of particles have been put forward. According to these conservation laws, particles of a given group cannot be created or destroyed except in pairs, where one of the pair is an ordinary particle and the other is an antiparticle belonging to the same group. Recent work has raised the possibility that the proton, which is a type of baryon, may in fact be unstable and decay into lighter products; the postulated methods of decay would violate the conservation of baryon number. To date, however, no such decay has been observed, and it has been determined that the proton has a lifetime of at least 1031 years. Two partial conservation laws, governing the quantities known as strangeness and isotopic spin, have been discovered for elementary particles. Strangeness is conserved during the so-called strong interactions and the electromagnetic interactions, but not during the weak interactions associated with particle decay; isotopic spin is conserved only during the strong interactions. Conservation of natural symmetries :
One very important discovery has been the link between conservation laws and basic symmetries in nature. For example, empty space possesses the symmetries that it is the same at every location (homogeneity) and in every direction (isotropy); these symmetries in turn lead to the invariance principles that the laws of physics should be the same regardless of changes of position or of orientation in space. The first invariance principle implies the law of conservation of linear momentum, while the second implies conservation of angular momentum. The symmetry known as the homogeneity of time leads to the invariance principle that the laws of physics remain the same at all times, which in turn implies the law of conservation of energy. The symmetries and invariance principles underlying the other conservation laws are more complex, and some are not yet understood. Three special conservation laws have been defined with respect to symmetries and invariance principles associated with inversion or reversal of space, time, and charge. Space inversion yields a mirror-image world where the "handedness" of particles and processes is reversed; the conserved quantity corresponding to this symmetry is called space parity, or simply parity, P. Similarly, the symmetries leading to invariance with respect to time reversal and charge conjugation (changing particles into their antiparticles) result in conservation of time parity, T, and charge parity, C. Although these three conservation laws do not hold individually for all possible processes, the combination of all three is thought to be an absolute conservation law, known as the CPT theorem, according to which if a given process occurs, then a corresponding process must also be possible in which particles are replaced by their antiparticles, the handedness of each particle is reversed, and the process proceeds in the opposite direction in time. Thus, conservation laws provide one of the keys to our understanding of the universe and its material basis. Conservation laws (physics) Principles which state that the total values of specified quantities remain constant in time for an isolated system. Conservation laws occupy enormously important positions both at the foundations of physics and in its applications.
Realization in classical mechanics There are three great conservation laws of mechanics: the conservation of linear momentum, often referred to simply as the conservation of momentum; the conservation of angular momentum; and the conservation of energy. The linear momentum, or simply momentum, of a particle is equal to the product of its mass and velocity. It is a vector quantity. The total momentum of a system of particles is simply the sum of the momenta of each particle considered separately. The law of conservation of momentum states that this total momentum does not change in time. See Conservation of momentum, Momentum The angular momentum of a particle is more complicated. It is defined by the vector product of the position and momentum vectors. The law of conservation of angular momentum states that the total angular momentum of an isolated system is constant in time. See Angular momentum The conservation of energy is perhaps the most important law of all. Energy is a scalar quantity, and takes two forms: kinetic and potential. The kinetic energy of a particle is defined to be one-half the product of its mass and the square of its velocity. The potential energy is loosely defined as the ability to do work. The total energy is the sum of the kinetic and potential energies, and according to the conservation law it remains constant in time for an isolated system. The essential difficulty in applying the conservation of energy law can be appreciated by considering the problem of two colliding bodies. In general, the bodies emerge from the collision moving more slowly than when they entered. This phenomenon seems to violate the conservation of energy, until it is recognized that the bodies involved may consist of smaller particles. Their random small-scale motions will require kinetic energy, which robs kinetic energy from the overall coherent large-scale motion of the bodies that are observed directly. One of the greatest achievements of nineteenth-century physics was the recognition that small-scale motion within macroscopic bodies could be identified with the perceived property of heat. See Conservation of energy, Energy, Kinetic theory of matter
Position in modern physics As physics has evolved, the great conservation laws have likewise evolved in both form and content, but have never ceased to be important guiding principles. In order to account for the phenomena of electromagnetism, it was necessary to go beyond the notion of point particles, to postulate the existence of continuous electric and magnetic fields filling all space. To obtain valid conservation laws, energy, momentum, and angular momentum must be ascribed to the electromagnetic
fields. See Electromagnetic
radiation, Maxwell's
equations, Poynting's vector In the special theory of relativity, energy and momentum are not independent concepts. Einstein discovered perhaps the most important consequence of special relativity, that is, the equivalence of mass and energy, as a consequence of the conservation laws. The “law” of conservation of mass is understood as an approximate consequence of the conservation of energy. See Conservation of mass, Relativity A remarkable, beautiful, and very fruitful connection has been established between symmetries and conservation laws. Thus the law of conservation of linear momentum is understood as a consequence of the homogeneity of space, the conservation of angular momentum as a consequence of the isotropy of space, and the conservation of energy as a consequence of the homogeneity of time. See Symmetry laws (physics) The development of general relativity, the modern theory of gravitation, necessitates attention to a fundamental question for the conservation laws: The laws refer to an “isolated system,” but it is not clear that any system is truly isolated. This is a particularly acute problem for gravitational forces, which are long range and add up over cosmological distances. It turns out that the symmetry of physical laws is actually a more fundamental property than the conservation laws themselves, for the symmetries remain valid while the conservation laws, strictly speaking, fail. In quantum theory, the great conservation laws remain valid in a very strong sense. Generally, the formalism of quantum mechanics does not allow prediction
of the outcome of individual experiments, but only the relative probability of different possible outcomes. One might therefore entertain the possibility that the conservation laws were valid only on the average. However, momentum, angular momentum, and energy are conserved in every experiment. See Quantum mechanics, Quantum theory of measurement Conservation laws of particle type There is another important class of conservation laws, associated not with the motion of particles but with their type. Perhaps the most practically important of these laws is the conservation of chemical elements. From a modern viewpoint, this principle results from the fact that the small amount of energy involved in chemical transformations is inadequate to disrupt the nuclei deep within atoms. It is not an absolute law, because some nuclei decay spontaneously, and at sufficiently high energies it is grossly violated. See Radioactivity Several conservation laws in particle physics are of the same character: They are useful even though they are not exact because, while known processes violate them, such processes are either unusually slow or require extremely high energy
UNIT IV : Electromagnetic Waves Plane waves in non-conducting media - Linear and circular polarization, reflection and refraction at a plane interface Fresnel’s law, interference, coherence and diffraction -Waves in a conducting medium - Propagation of waves in a rectangular wave guide - Inhomogeneous wave equation and retarded potentials Radiation from a localized source - Oscillating electric dipole.
Plane waves in non-conducting media : Propagation in LIH nonconducting media Propagation velocity and refractive index The refractive index n of a given material is defined as the speed of light in vacuum divided by that in the material. Hence Because always has a value greater than one, the speed of light in a material is always less than in a vacuum. As both Pr and n may vary strongly with frequency, discrepancies may arise when comparing values of Pf (often the static, DC value is used) with n 2 (generally the value appropriate to optical frequencies). Relationship between E and B In vacuum E=cB. In a material this is modified to E=cmB=cB/n and because H=BXV
Reflection and refraction at the interface between two different materials The aim is to establish the properties of electromagnetic waves when they encounter a plane interface separating two different non-conducting materials (generally two different dielectrics). In particular equations will be derived which give the fractions of the incident wave which are reflected and transmitted at the interface Mathematical description of a plane wave not propagating along one of the principal axes So far we have only considered waves which propagate along one of the principal axes. For example a plane wave propagating along the x-axis is described by an equation of the formE=E0sin(kx-t), one propagating along the y-axis by E=E0sin(ky-t) etc.
In the following we will need to consider plane waves which do not propagate along one of these principal axes. However we can always resolve the direction of the wave onto two or more of the principal axes. For example in the figure below the wave lies in the x/y-plane and propagates in a direction which makes an angle to the y-axis. There are hence components cos along the y-axis and sin along the x-axis. The equation for this plane wave is hence of the form E=E0sin(k(xsin$+ycos$)-wt) The wave vector, k, in the above equations is given by 2/ where is the wavelength of the wave. In addition if the wave propagates in a material with a velocity cm then k=nw/c Boundary conditions for E, D, B and H at an interface In the lectures on dielectrics and magnetic materials it was shown that, in the absence of free surface charge and conduction currents, the following boundary conditions existed for E,D, B and H D and B: Normal components continuous E and H: Tangential components continuous These boundary conditions will be used below for electromagnetic waves incident at the boundary between two materials. Frequency and direction of reflected and refracted waves In the figure below a plane electromagnetic wave travelling in a material of refractive index n1 is incident on the plane boundary with a second material of refractive index n2. The incident wave propagates at an angle i to the normal of the boundary and has E- and H-fields of magnitude Ei and Hi. The E-field of the incident wave lies in the plane of incidence (the x/y-plane), this is known as the E parallel configuration.
For this configuration the B- and H-fields must be normal to the plane of incidence. In general there will be, in addition to the incident wave, a reflected wave at an angle r and a refracted or transmitted wave at an angle t. Each of these waves will
have E- and H-fields
as
shown
with
amplitudes
related
by
the
expression H=nE/0c where n is the refractive index of the appropriate material. The E-fields of the three waves are given by the following expressions Ei=Ei0sin(k1(xsini-ycosi)-1t)
(A)
Er=Er0sin(k1(xsinr+ycosr)-1t)
(B)
Et=Et0sin(k2(xsint-ycost)-2t)
(C)
Where Ei0, Er0 and Et0 are
the
amplitudes
of
the E-fields
and
k1 and
k2 and 1 and 2 are the wavevectors and angular frequencies of the waves in the two materials. The boundary condition for E requires that the tangential component (the component parallel to the boundary) be continuous. Hence if we evaluate this component on both sides of the boundary we must obtain the same result. The tangential component of each E-field is given by the magnitude of the E-field multiplied by cos, where is the appropriate angle. On the side of the boundary in material 1 the total tangential component of the Efield is the difference of the components due to the incident and reflected waves (see above figure). In material 2 there is only the transmitted wave. Hence the requirement that the tangential component of the E-field be continuous can be written Eicosiy=0 - Ercosry=0=Etcosty=0
(D)
Where y=0 indicates that the preceding expression is evaluated at the boundary y=0. Substituting in (D) the expressions for Ei, Er and Et given by (A), (B) and (C) and evaluated for y=0 Ei0cosisin(k1xsini-1t)-Er0cosrsin(k1xsinr-1t)=Et0costsin(k2xsint2t)
(E)
This equation must hold at all times and for all values of x. This is only possible if all the coefficients of x and all the coefficients of t are equal. This requires 1=2 and k1sini=k1sinr=k2sint Hence
The above results would have been obtained if E were polarized normal to the plane of incidence (H polarised parallel to the plane of incidence). These results hence apply to any incident wave. Amplitudes of the reflected and refracted waves The above procedure provided information on the frequencies and directions of the incident, reflected and transmitted waves. The next step is to derive expressions which give their relative amplitudes. Returning to equation (E) above, which is valid when E is parallel to the plane of incidence, the spatial and time dependent components have been shown to be equal. Hence (E) reduces to Ei0cosi - Er0cosi=Et0cost
(F)
Where r has been replaced by i. In addition the tangential component of the H-field must be continuous at the boundary between the two materials. For the present case H is normal to the incident plane so that the tangential component of H is simply H. For the tangential component of H to be continuous we must therefore have Hi0+Hr0=Ht0 But H=nE/0c so the above can be written as n1Ei0+n1Er0=n2Et0
(G)
Equations (F) and (G) can now be used to eliminate either Er0 or Et0. Eliminating Et0: From (G) Et0=(Ei0+Er0)n1/n2 Now eliminating Er0. From (G) Er0=(n2Et0-n1Ei0)/n1
Substituting into (F) Ei0cosi-(n2Et0-n1Ei0)/n1cosi=Et0cost Ei0n1cosi-(n2Et0-n1Ei0)cosi=Et0n1cost 2Ei0n1cosi=Et0(n1cost+n2cosi) r// and t//, which relate the amplitudes of the reflected and transmitted E-fields to that of the incident E-field, are known as the reflection and transmission coefficients for E parallel to the plane of incidence. The polarisation of E can also be aligned normal to the plane of incidence (H is now parallel to the plane of incidence). The boundary conditions that the tangential components of E and H are continuous now require: Ei0+Er0=Et0 Properties of the Fresnel Equations The properties can be more easily seen if we consider a special case where one of the materials is air (n=1) and the other has a refractive index n. Consider the case where the wave is incident from air. Hence n1=1 and n2=n. The Fresnel equations become with sini/sint=n The above figure (plotted for n=2) shows a number of points: The reflection coefficients tend to 1 (total reflection) and the magnitude of the transmission coefficients tend to zero (zero transmission). For a certain angle B, known as the Brewster angle, r// becomes zero whereas r remains non-zero. For this angle light can only be reflected with E perpendicular to the plane of incidence. If unpolarised light is incident at the Brewster angle then the reflected light will be polarised. The Brewster angle is found by setting r//=0. and hence Waves propagating from a material into air We now have n1=n and n2=1. Fresnel’s equations have the form and nsini=sint
In this case the equations for r// and r and for t// and t are interchanged compared to those for when the wave is incident from air. However because nsini=sint t>i. For i equal to a certain value, the critical angle c, sint becomes equal to unity and hence t=90. At this point both reflection coefficients become equal to one and the waves are totally reflected. As the value of sint can not exceed unity, for i>c the reflection coefficients remain equal to unity. Hence for i>c all the incident light is reflected. This is known as ‘total internal reflection’. Power reflection and transmission coefficients The reflection and transmission coefficients derived above give the amplitudes of the electric fields associated with the waves. However in general it is the power of the wave which is measured experimentally. the power density of an electromagnetic wave is given by the Poynting vector N=ExH. This has a magnitude EH = nE2/(c0), using H=nE/(c0). Hence the energy density nE2 The coefficients which give the fraction of energy reflected or transmitted at a boundary between two materials equal the appropriate values of r 2 or t2 with the inclusion of the appropriate value(s) of n. If R is the power reflection coefficient at an interface between a material of refractive index n1 and one of n2 then where r is the appropriate reflection coefficient given by the Fresnel relationships. Similarly if T is the power transmission coefficient because energy must always be conserved R+T=1
Power reflection and transmission coefficients Linear and circular polarization : Circular Polarization : In electrodynamics, circular
polarization of
an electromagnetic
wave is
a polarization in which the electric field of the passing wave does not change strength but only changes direction in a rotary manner.
The electric field vectors of a traveling circularly polarized electromagnetic wave. In electrodynamics the strength and direction of an electric field is defined by what is called an electric field vector. In the case of a circularly polarized wave, as seen in the accompanying animation, the tip of the electric field vector, at a given point in space, describes a circle as time progresses. If the wave is frozen in time, the electric field vector of the wave describes a helix along the direction of propagation. Circular polarization is a limiting case of the more general condition of elliptical polarization. The other special case is the easier-to-understand linear polarization. The phenomenon of polarization arises as a consequence of the fact that light behaves as a two-dimensional transverse wave. On the right is an illustration of the electric field vectors of a circularly polarized electromagnetic wave.The electric field vectors have a constant magnitude but
their direction changes in a rotary manner. Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis. Specifically, given that this is a circularly polarized plane wave, these vectors indicate that the electric field, from plane to plane, has a constant strength while its direction steadily rotates. It is considered to be right-hand, clockwise circularly polarized if viewed by the receiver. Since this is anelectromagnetic wave each electric field vector has a corresponding, but not illustrated, magnetic field vector that is at a right angle to the electric field vector and proportional in magnitude to it. As a result, the magnetic field vectors would trace out a second helix if displayed. Circular polarization is often encountered in the field of optics and in this section, the electromagnetic wave will be simply referred to as light. The nature of circular polarization and its relationship to other polarizations is often understood by thinking of the electric field as being divided into two components which are at right angles to each other. Refer to the second illustration on the right. The vertical component and its corresponding plane are illustrated in blue while the horizontal component and its corresponding plane are illustrated in green. Notice that the rightward (relative to the direction of travel) horizontal component leads the vertical component by one quarter of a wavelength. It is thisquadrature phase relationship which creates the helix and causes the points of maximum magnitude of the vertical component to correspond with the points of zero magnitude of the horizontal component, and vice versa. The result of this alignment is that there are select vectors, corresponding to the helix, which exactly match the maxima of the vertical and horizontal components. (To minimize visual clutter these are the only helix vectors displayed.) To appreciate how this quadrature phase shift corresponds to an electric field that rotates while maintaining a constant magnitude, imagine a dot traveling clockwise in a circle. Consider how the vertical and horizontaldisplacements of the dot, relative to the center of the circle, vary sinusoidally in time and are out of phase by one quarter of a cycle. The displacements are said to be out of phase by one quarter of a cycle because the horizontal maximum displacement (toward the left)
is reached one quarter of a cycle before the vertical maximum displacement is reached. Now referring again to the illustration, imagine the center of the circle just described, traveling along the axis from the front to the back. The circling dot will trace out a helix with the displacement toward our viewing left, leading the vertical displacement. Just as the horizontal and vertical displacements of the rotating dot are out of phase by one quarter of a cycle in time, the magnitude of the horizontal and vertical components of the electric field are out of phase by one quarter
of
a
wavelength.
Left-handed/counter-clockwise circularly polarized light displayed with and without the use of components. This would be considered right-handed/clockwise
circularly polarized if defined from the point of view of the source rather than the receiver. The next pair of illustrations is that of left-handed, counter-clockwise circularly polarized light when viewed by the receiver. Because it is left-handed, the rightward (relative to the direction of travel) horizontal component is nowlagging the vertical component by one quarter of a wavelength rather than leading it. One should appreciate that our choice to focus on the horizontal and vertical components was arbitrary. Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them. To convert a given handedness of polarized light to the other handedness one can use a half-wave plate. A half-wave plate shifts a given component of light one half of a wavelength relative to the component to which it is orthogonal. The handedness of polarized light is also reversed when it is reflected off of a mirror. Initially, as a result of the interaction of the electromagnetic field with the conducting surface of the mirror, both orthogonal components are effectively shifted by one half of a wavelength. However as a result of the change in direction, a mirror image of the wave is created and the two components' phase relationship is reversed. For a better appreciation of the nature of circularly polarized light one may find it useful to read how circularly polarized light is converted to and from linearly polarized light in the circular polarizer article. Left / Right hardness convention : Circular polarization may be referred to as right-handed or left-handed, and clockwise or counter-clockwise, depending on the direction in which the electric field vector rotates. Unfortunately, two opposing historical conventions exist. From the point of view of the source Using this convention, polarization is defined from the point of view of the source. When using this convention, left or right handedness is determined by
pointing one's left or right thumb away from the source, in the same direction that the wave is propagating, and matching the curling of one's fingers to the direction of the temporal rotation of the field at a given point in space. When determining if the wave is clockwise or counter-clockwise circularly polarized, one again takes the point of view of the source, and while looking away from the source and in the same direction of the wave’s propagation, one observes the direction of the field’s temporal rotation. Using this convention, the electric field vector of a right handed circularly polarized
wave
is
as
follows: As a specific example, refer to the circularly polarized wave in the first animation. Using this convention that wave is defined as right-handed because when one points one's right thumb in the same direction of the wave’s propagation, the fingers of that hand curl in the same direction of the field’s temporal rotation. It is considered clockwise circularly polarized because from the point of view of the source, looking in the same direction of the wave’s propagation, the field rotates in the clockwise direction. The second animation is that of left-handed or counterclockwise light using this same convention. This convention is in conformity with the Institute of Electrical and Electronics Engineers (IEEE) standard and as a result it is generally used in the engineering community. Quantum physicists also use this convention of handedness because it is consistent with their convention of handedness for a particle’s spin. In quantum mechanics the direction of spin of a photon is tied to the handedness of the circularly polarized light and the spin of a beam of photons is similar to the spin of a beam of particles, such as electrons. Many radio astronomers also use this convention. From the point of view of the receiver In this alternative convention, polarization is defined from the point of view of the receiver. Using this convention, left or right handedness is determined by pointing
one’s left or right thumb toward the source, against the direction of propagation, and then matching the curling of one's fingers to the temporal rotation of the field. When using this convention, in contrast to the other convention, the defined handedness of the wave matches the handedness of the screw type nature of the field in space. Specifically, if one freezes a right-handed wave in time, when one curls the fingers of one’s right hand around the helix, the thumb will point in the direction which the helix progresses given that sense of rotation. Note that it is the nature of all screws and helices that it does not matter in which direction you point your thumb when determining its handedness. When determining if the wave is clockwise or counter-clockwise circularly polarized, one again takes the point of view of the receiver and, while looking toward the source, against the direction of propagation, one observes the direction of the field’s temporal rotation. Many optics textbooks use this second convention. Uses of the two conventions As stated earlier, there is significant confusion with regards to these two conventions. As a general rule the engineering and quantum physics community use the first convention where the wave is observed from the point of view of the source.In many physics textbooks dealing with optics the second convention is used where the light is observed from the point of view of receiver. To avoid confusion, it is good practice to specify “as defined from the point of view of the source” or "as defined from the point of view of the receiver" when discussing polarization matters. The archive of the US Federal Standard 1037C proposes two contradictory conventions of handedness Circularly polarized luminance : Circularly polarized luminescence (CPL) can occur when either a luminophore or an ensemble of luminophores is chiral. The extent to which emissions are polarized is quantified in the same way it is for circular dichroism, in terms of the dissymmetry , also sometimes referred to as the anisotropy factor. This value is given by
where light, and
corresponds to the quantum yield of left-handed circularly polarized to that of right-handed light. The maximum absolute value of gem,
corresponding to purely left- or right-handed circular polarization, is therefore 2. Meanwhile the smallest absolute value that gem can achieve, corresponding to linearly polarized or unpolarized light, is zero.
Mathamatical Discription : The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is
where k is the wavenumber, is the angular frequency of the wave,
is an orthogonal
whose columns span the transverse x-y plane and
is the speed of light.
matrix
Here is the amplitude of the field and
is the Jones vector in the x-y plane. If
is rotated by
radians with respect to
and the x amplitude equals the
y amplitude the wave is circularly polarized. The Jones vector is
where the plus sign indicates left circular polarization and the minus sign indicates right circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane. If basis vectors are defined such that
and
then the polarization state can be written in the "R-L basis" as where
and
Linear Polarization : In electrodynamics, linear polarization or plane polarization of electromagnetic radiation is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. See polarization for more information. The orientation of a linearly polarized electromagnetic wave is defined by the direction of the electric field vector. For example, if the electric field vector is vertical (alternately up and down as the wave travels) the radiation is said to be vertically polarized. Mathematical description of linear polarization :
The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is (cgs units)
for the magnetic field, where k is the wavenumber, is the angular frequency of the wave, and
is the speed of light.
Here is the amplitude of the field and
is the Jones vector in the x-y plane. The wave is linearly polarized when the phase angles
are equal,
. This represents a wave polarized at an angle
with respect to the x axis. In that
case the Jones vector can be written
. The state vectors for linear polarization in x or y are special cases of this state vector. If unit vectors are defined such that
and
then the polarization state can written in the "x-y basis" as
Light in the form of a plane wave in space is said to be linearly polarized. Light is a transverse electromagnetic wave, but natural light is generally unpolarized, all planes of propagation being equally probable. If light is composed of two plane waves of equal amplitude by differing in phase by 90°, then the light is said to be circularly polarized. If two plane waves of differing amplitude are related in phase by 90°, or if the relative phase is other than 90° then the light is said to be elliptically polarized.
Reflection and Refraction at a Plane Interface : The reflection and refraction of a beam of light is treated. Approximate solutions of Maxwell’s equations are used to describe the electromagnetic field of the beam being limited in the transverse direction. The point of departure is the classical paper by Schaefer and Pich. The laws of reflection and refraction are derived. Fresnel’s formulas and their corrections are presented for both polarizations. The case of total reflection is investigated for E polarization in greater detail. The electromagnetic fields and the time-average Poynting vectors are explicitly derived for both the optically dense and less-dense media. The flow of energy at total reflection is studied extensively. It is shown that, due to the flow of energy in the less-dense medium, the center of gravity of the reflected beam is displaced, as was suggested by v. Fragstein. This leads to a shift of the totally reflected beam with respect to a geometrically reflected beam . .
Suppose a plane wave is incident upon a plane surface that is an interface between
two materials, one with
and the other with
.
Figure : Geometry for reflection and refraction at a plane interface between two media, one
with permittivity/permeability
, one with permittivity/permeability
.
In order to derive an algebraic relationship between the intensities of the incoming wave, the reflected wave, and the refracted wave, we must begin by defining the algebraic form of each of these waves in terms of the wave numbers. The reflected wave and incident wave do not leave the first medium and hence retain
speed
,
wave changes to speed
,
and
.
,
,
The
refracted
.
Note that the frequency of the waves is the same in both media as a kinematic constraint! Why is that? This yields the following forms for the various waves: Incident Wave
Refracted Wave
Reflected Wave
Our goal is to completely understand how to compute the reflected and refracted wave from the incident wave. This is done by matching the wave across the boundary interface. There are two aspects of this matching - astatic or kinematic matching of the waveform itself and a dynamic matching associated with the (changing) polarization in the medium. These two kinds of matching lead to two distinct and well-known results.
Fresnel’s Equations : This article is about the Fresnel equations describing reflection and refraction of light at uniform planar interfaces. For the diffraction of light through an aperture, see Fresnel diffraction. For the thin lens and mirror technology, see Fresnel lens.
Partial transmission and reflection amplitudes of a wave travelling from a low to high refractive index medium.
The Fresnel equations (or Fresnel conditions), deduced by Augustin-Jean Fresnel , describe the behaviour of light when moving between media of differing refractive indices. The reflection of light that the equations predict is known as Fresnel reflection. Overview : When light moves from a medium of a given refractive index n1 into a second medium with refractive index n2, both reflection and refraction of the light may occur. The Fresnel equations describe what fraction of the light is reflected and what fraction is refracted (i.e., transmitted). They also describe the phase shift of the reflected light. The equations assume the interface is flat, planar, and homogeneous, and that the light is a plane wave. Definitions and power equations
Variables used in the Fresnel equations. In the diagram on the right, an incident light ray IO strikes the interface between two media of refractive indices n1 and n2 at point O. Part of the ray is reflected as ray OR and part refracted as ray OT. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θi, θr and θt, respectively. The relationship between these angles is given by the law of reflection: and Snell's law:
The fraction of the incident power that is reflected from the interface is given by the reflectance R and the fraction that is refracted is given by the transmittance T. The media are assumed to be non-magnetic. The calculations of R and T depend on polarisation of the incident ray. Two cases are analyzed: •
The incident light is s-polarized. That means its electric field is in the plane of the interface (perpendicular to the plane of the diagram above).
•
The incident light is p-polarized. That means its electric field is in a perpendicular direction to s-polarized above (in the plane of the diagram above).
For the s-polarized light, the reflection coefficient is given by
, where the second form is derived from the first by eliminating θt using Snell's law and trigonometric identities. For the p-polarized light, the R is given by
. As a consequence of the conservation of energy, the transmission coefficients are given by
and
These relationships hold only for power coefficients, not for amplitude coefficients as defined below. If the incident light is unpolarised (containing an equal mix of s- and ppolarisations), the reflection coefficient is
For common glass, the reflection coefficient is about 4%. Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides. The combined reflection coefficient for this case is 2R/(1 + R), when interference can be neglected (see below). The discussion given here assumes that the permeability μ is equal to the vacuum permeability μ0 in
both
media.
This
is
approximately
true
for
most dielectric materials, but not for some other types of material. The completely general Fresnel equations are more complicated.
Special angles At one particular angle for a given n1 and n2, the value of Rp goes to zero and a ppolarised incident ray is purely refracted. This angle is known as Brewster's angle, and is around 56° for a glass medium in air or vacuum. Note that this statement is only true when the refractive indices of both materials are real numbers, as is the
case for materials like air and glass. For materials that absorb light, like metals and semiconductors, n is complex, and Rp does not generally go to zero. When moving from a denser medium into a less dense one (i.e., n1 > n2), above an incidence angle known as the critical angle, all light is reflected and Rs = Rp = 1. This phenomenon is known as total internal reflection. The critical angle is approximately 41° for glass in air. Amplitude equations Equations for coefficients corresponding to ratios of the electric field complexvalued amplitudes of the waves (not necessarily real-valued magnitudes) are also called "Fresnel equations". These take several different forms, depending on the choice of formalism and sign convention used. The amplitude coefficients are usually represented by lower case r and t.
Amplitude ratios: air to glass
Amplitude ratios: glass to air
Conventions used here In
this
treatment,
the
coefficient r is
the
ratio
of
the
reflected
wave's complex electric field amplitude to that of the incident wave. The coefficient t is the ratio of the transmitted wave's electric field amplitude to that of the incident wave. The light is split into s and p polarizations as defined above. (In the figures to the right, s polarization is denoted " " and p is denoted " ".) For s-polarization, a positive r or t means that the electric fields of the incoming and reflected or transmitted wave are parallel, while negative means anti-parallel. For p-polarization, a positive r or t means that the magnetic fields of the waves are parallel, while negative means anti-parallel. Formulas Using the conventions above,
Notice that
but
.
Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the amplitude reflection coefficient is related to the reflectance R by . The transmittance T is generally not equal to |t|2, since the light travels with different direction and speed in the two media. The transmittance is related to t by. .
The factor of n2/n1 occurs from the ratio of intensities (closely related to irradiance). The factor of cos θt/cos θi represents the change in area m of the pencil of rays, needed since T, the ratio of powers, is equal to the ratio of (intensity × area). In terms of the ratio of refractive indices, , and of the magnification m of the beam cross section occurring at the interface, . Multiple surfaces When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's coherence length, which for ordinary white light is few micrometers; it can be much larger for light from a laser. An example of interference between reflections is the iridescent colours seen in a soap bubble or in thin oil films on water. Applications include Fabry–Pérot interferometers, antireflection coatings, and optical filters. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference. The transfer-matrix method, or the recursive Rouard method can be used to solve multiple-surface problems.
Interference of Waves Interference (wave propagation), in physics (optics), the superposition of
•
two or more waves resulting in a new wave pattern •
Thin-film interference. a special kind of optical interference Interference (communication), anything which alters, modifies, or disrupts
•
a message as it travels along a channel •
Electromagnetic interference (EMI)
•
Co-channel interference (CCI), also known as crosstalk
•
Adjacent-channel interference (ACI), interference caused by extraneous power from a signal in an adjacent channel
•
Intersymbol interference (ISI), distortion of a signal in which one symbol interferes with subsequent symbols
•
Inter-carrier
interference (ICI),
caused
by
doppler
shift
in OFDM modulation •
Vaccine interference, may occur when two or more vaccines are mixed in the same formulation
•
Interference fit, two items attempting to occupy the same space
•
Interference engine, an engine whose pistons can strike and damage the piston valves when the engine is cranked while the timing belt is broken
•
Interference theory in psychology
•
Statistical interference, a method of determining when one distribution of values exceeds another
•
Interference
(genetic),
a
phenomenon
by
which
a chromosomal
crossover in one interval decreases the probability that additional crossovers will occur nearby •
RNA interference, a process within living cells that moderates the activity of their genes
What happens when two waves meet while they travel through the same medium? What effect will the meeting of the waves have upon the appearance of the medium? Will the two waves bounce off each other upon meeting (much like two billiard balls would) or will the two waves pass through each other? These questions involving the meeting of two or more waves along the same medium pertain to the topic of wave interference. Wave interference is the phenomenon that occurs when two waves meet while traveling along the same medium. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium. To begin our exploration of wave interference,
consider two pulses of the same amplitude traveling in different directions along the same medium. Let's suppose that each displaced upward 1 unit at its crest and has the shape of a sine wave. As the sine pulses move towards each other, there will eventually be a moment in time when they are completely overlapped. At that moment, the resulting shape of the medium would be an upward displaced sine pulse with an amplitude of 2 units. The diagrams below depict the before and during interference snapshots of the medium for two such pulses. The individual sine pulses are drawn in red and blue and the resulting displacement of the medium is drawn in green.
This
type
of
interference
is
sometimes
called
constructive
interference. Constructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the same direction. In this case, both waves have an upward displacement; consequently, the medium has an upward displacement that is greater than the displacement of the two interfering pulses. Constructive interference is observed at any location where the two interfering waves are displaced upward. But it is also observed when both interfering waves are displaced downward. This is shown in the diagram below for two downward displaced pulses.
In this case, a sine pulse with a maximum displacement of -1 unit (negative means a downward displacement) interferes with a sine pulse with a maximum displacement of -1 unit. These two pulses are drawn in red and blue. The resulting shape of the medium is a sine pulse with a maximum displacement of -2 units. Destructive interference is a type of interference that occurs at any location along the medium where the two interfering waves have a displacement in the
opposite direction. For instance, when a sine pulse with a maximum displacement of +1 unit meets a sine pulse with a maximum displacement of -1 unit, destructive interference occurs. This is depicted in the diagram below.
In the diagram above, the interfering pulses have the same maximum displacement but in opposite directions. The result is that the two pulses completely destroy each other when they are completely overlapped. At the instant of complete overlap, there is no resulting displacement of the particles of the medium. This "destruction" is not a permanent condition. In fact, to say that the two waves destroy each other can be partially misleading. When it is said that the two pulses destroy each other, what is meant is that when overlapped, the effect of one of the pulses on the displacement of a given particle of the medium is destroyed or canceled by the effect of the other pulse. Recall fromLesson 1 that waves transport energy through a medium by means of each individual particle pulling upon its nearest neighbor. When two pulses with opposite displacements (i.e., one pulse displaced up and the other down) meet at a given location, the upward pull of one pulse is balanced (canceled or destroyed) by the downward pull of the other pulse. Once the two pulses pass through each other, there is still an upward displaced pulse and a downward displaced pulse heading in the same direction that they were heading before the interference. Destructive interference leads to only a momentary condition in which the medium's displacement is less than the displacement of the largest-amplitude wave. The two interfering waves do not need to have equal amplitudes in opposite directions for destructive interference to occur. For example, a pulse with a maximum displacement of +1 unit could meet a pulse with a maximum displacement of -2 units. The resulting displacement of the medium during complete overlap is -1 unit.
This is still destructive interference since the two interfering pulses have opposite displacements. In this case, the destructive nature of the interference does not lead to complete cancellation. Interestingly, the meeting of two waves along a medium does not alter the individual waves or even deviate them from their path. This only becomes an astounding behavior when it is compared to what happens when two billiard balls meet or two football players meet. Billiard balls might crash and bounce off each other and football players might crash and come to a stop. Yet two waves will meet, produce a net resulting shape of the medium, and then continue on doing what they were doing before the interference.
The task of determining the shape of the resultant demands that the principle of superposition is applied. Theprinciple of superposition is sometimes stated as follows: When two waves interfere, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that same location. In the cases above, the summing the individual displacements for locations of complete overlap was made out to be an easy task - as easy as simple arithmetic: Displacement of Pulse 1 +1 -1 +1 +1
Displacement of Pulse 2 +1 -1 -1 -2
=Resulting Displacement =+2 =-2 =0 =-1
In actuality, the task of determining the complete shape of the entire medium during interference demands that the principle of superposition be applied for every point (or nearly every point) along the medium. As an example of the complexity of this task, consider the two interfering waves at the right. A snapshot of the shape of each individual wave at a particular instant in time is shown. To determine the precise shape of the medium at this given instant in time, the principle of superposition must be applied to several locations along the medium. A short cut involves measuring the displacement from equilibrium at a few strategic locations. Thus, approximately 20 locations have been picked and labeled as A, B, C, D, etc. The actual displacement of each individual wave can be counted by measuring from the equilibrium position up to the particular wave. At position A, there is no displacement for either individual wave; thus, the resulting displacement of the medium at position will be 0 units. At position B, the smaller wave has a displacement of approximately 1.4 units (indicated by the red dot); the larger wave has a displacement of approximately 2 units (indicated by the blue dot). Thus, the resulting displacement of the medium will be approximately 3.4 units. At position C, the smaller wave has a displacement of approximately 2 units; the larger wave has a displacement of approximately 4 units; thus, the resulting displacement of the medium will be approximately 6 units. At position D, the smaller wave has a displacement of approximately 1.4 units; the larger wave has a displacement of approximately 2 units; thus, the resulting displacement of the medium will be approximately 3.4 units. This process can be repeated for every position. When finished, a dot (done in green below) can be marked on the graph to note the displacement of the medium at each given location. The actual shape of the medium can then be sketched by estimating the position between the various marked points and sketching the wave. This is shown as the green line in the diagram below.
Coherence : In physics, coherence is an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference. It contains several distinct concepts, which are limit cases that never occur in reality but allow an understanding of the physics of waves, and has become a very important concept in quantum physics. More generally, coherence describes all properties of the correlation between physical quantities of a single wave, or between several waves or wave packets. Interference is nothing more than the addition, in the mathematical sense, of wave functions. In quantum mechanics, a single wave can interfere with itself, but this is due to its quantum behavior and is still an addition of two waves (see Young's slits experiment). This implies that constructive or destructive interferences are limit cases, and that waves can always interfere, even if the result of the addition is complicated or not remarkable. When interfering, two waves can add together to create a wave of greater amplitude than either one (constructive interference) or subtract from each other to create a wave of lesser amplitude than either one (destructive interference), depending on their relative phase. Two waves are said to be coherent if they have a constant relative phase. The degree of coherence is measured by the interference visibility, a measure of how perfectly the waves can cancel due to destructive interference.
Spatial coherence describes the correlation between waves at different points in space. Temporal coherence describes the correlation or predictable relationship between waves observed at different moments in time. Both are observed in the Michelson–Morley experiment and Young's interference experiment. Once the fringes are obtained in the Michelson–Morley experiment, when one of the mirrors is moved away gradually, the time for the beam to travel increases and the infringes become dull and finally are lost, showing Temporal Coherence. Similarly, if in Young's double slit experiment the space between the two slits is increased, the coherence dies gradually and finally the infringes disappear, showing spatial coherence Introduction Coherence was originally conceived in connection with Thomas Young's doubleslit experiment in optics but is now used in any field that involves waves, such as acoustics, electrical engineering,neuroscience, and quantum mechanics. The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and
telescope
interferometers
(astronomical
optical
interferometers and radio telescopes). Coherence and correlation The coherence of two waves follows from how well correlated the waves are as quantified by the cross-correlation function. The cross-correlation quantifies the ability to predict the value of the second wave by knowing the value of the first. As an example, consider two waves perfectly correlated for all times. At any time, if the first wave changes, the second will change in the same way. If combined they can exhibit complete constructive interference/superposition at all times, then it follows that they are perfectly coherent. As will be discussed below, the second wave need not be a separate entity. It could be the first wave at a different time or position. In this case, the measure of correlation is the autocorrelation function (sometimes called self-coherence). Degree of correlation involves correlation functions.
Examples of wave-like states These states are unified by the fact that their behavior is described by a wave equation or some generalization thereof. •
Waves in a rope (up and down) or slinky (compression and expansion)
•
Surface waves in a liquid
•
Electric signals (fields) in transmission cables
•
Sound
•
Radio waves and Microwaves
•
Light waves (optics)
•
Electrons, atoms and any other object (such as a baseball, as described by quantum physics)
In most of these systems, one can measure the wave directly. Consequently, its correlation with another wave can simply be calculated. However, in optics one cannot measure the electric fielddirectly as it oscillates much faster than any detector’s time resolution. Instead, we measure the intensity of the light. Most of the concepts involving coherence which will be introduced below were developed in the field of optics and then used in other fields. Therefore, many of the standard measurements of coherence are indirect measurements, even in fields where the wave can be measured directly. Temporal coherence
Figure : The amplitude of a single frequency wave as a function of time t (red) and a copy of the same wave delayed by τ(green). The coherence time of the wave is infinite since it is perfectly correlated with itself for all delays τ.
Figure : The amplitude of a wave whose phase drifts significantly in time τ c as a function of time t (red) and a copy of the same wave delayed by 2τ c(green). At any particular time t the wave can interfere perfectly with its delayed copy. But, since half the time the red and green waves are in phase and half the time out of phase, when averaged over t any interference disappears at this delay. Temporal coherence is the measure of the average correlation between the value of a wave and itself delayed by τ, at any pair of times. Temporal coherence tells us how monochromatic a source is. In other words, it characterizes how well a wave can interfere with itself at a different time. The delay over which the phase or amplitude wanders by a significant amount (and hence the correlation decreases by significant amount) is defined as the coherence time τc. At τ=0 the degree of coherence is perfect whereas it drops significantly by delay τc. The coherence length Lc is defined as the distance the wave travels in time τc. One should be careful not to confuse the coherence time with the time duration of the signal, nor the coherence length with the coherence area (see below). The relationship between coherence time and bandwidth It can be shown that the faster a wave decorrelates (and hence the smaller τ c is) the larger the range of frequencies Δf the wave contains. Thus there is a tradeoff: . Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum (the intensity of each frequency) to its autocorrelation. Examples of temporal coherence We consider four examples of temporal coherence.
•
A wave containing only a single frequency (monochromatic) is perfectly correlated at all times according to the above relation. (See Figure 1)
•
Conversely, a wave whose phase drifts quickly will have a short coherence time. (See Figure 2)
•
Similarly, pulses (wave packets) of waves, which naturally have a broad range of frequencies, also have a short coherence time since the amplitude of the wave changes quickly. (See Figure 3)
•
Finally, white light, which has a very broad range of frequencies, is a wave which varies quickly in both amplitude and phase. Since it consequently has a very short coherence time (just 10 periods or so), it is often called incoherent.
The most monochromatic sources are usually lasers; such high monochromaticity implies long coherence lengths (up to hundreds of meters). For example, a stabilized helium-neon laser can produce light with coherence lengths in excess of 5 m . Not all lasers are monochromatic, however (e.g. for a mode-locked Tisapphire laser, Δλ ≈ 2 nm - 70 nm). LEDs are characterized by Δλ ≈ 50 nm, and tungsten filament lights exhibit Δλ ≈ 600 nm, so these sources have shorter coherence times than the most monochromatic lasers. Holography requires light with a long coherence time. In contrast, Optical coherence tomography uses light with a short coherence time. Measurement of temporal coherence
Figure
: The amplitude of a wavepacket whose amplitude changes
significantly in time τc (red) and a copy of the same wave delayed by 2τc(green) plotted as a function of time t. At any particular time the red and
green waves are uncorrelated; one oscillates while the other is constant and so there will be no interference at this delay. Another way of looking at this is the wavepackets are not overlapped in time and so at any particular time there is only one nonzero field so no interference can occur.
Figure : The time-averaged intensity (blue) detected at the output of an interferometer plotted as a function of delay τ for the example waves in Figures . As the delay is changed by half a period, the interference switches between constructive and destructive. The black lines indicate the interference envelope, which gives the degree of coherence. Although the waves in Figures have different time durations, they have the same coherence time. In optics, temporal coherence is measured in an interferometer such as the Michelson interferometer or Mach–Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time τ. A detector measures the time-averaged intensity of the light exiting the interferometer. The resulting interference visibility (e.g. see Figure ) gives the temporal coherence at delay τ. Since for most natural light sources, the coherence time is much shorter than the time resolution of any detector, the detector itself does the time averaging. Consider the example shown in Figure . At a fixed delay, here 2τ c, an infinitely fast detector would measure an intensity that fluctuates significantly over a time tequal to τc. In this case, to find the temporal coherence at 2τc, one would manually time-average the intensity. Spatial coherence
In some systems, such as water waves or optics, wave-like states can extend over one or two dimensions. Spatial coherence describes the ability for two points in space, x1 and x2, in the extent of a wave to interfere, when averaged over
time.
More
precisely,
the
spatial
coherence
is
the cross-
correlation between two points in a wave for all times. If a wave has only 1 value of amplitude over an infinite length, it is perfectly spatially coherent. The range of separation between the two points over which there is significant interference is called the coherence area, Ac. This is the relevant type of coherence for the Young’s double-slit interferometer. It is also used in optical imaging systems and particularly in various types of astronomy telescopes. Sometimes people also use “spatial coherence” to refer to the visibility when a wave-like state is combined with a spatially shifted copy of itself. Examples of spatial coherence •
Spatial coherence
•
Figure 5: A plane wave with an infinite coherence length. •
Figure 6: A wave with a varying profile (wavefront) and infinite coherence length.
•
Figure 7: A wave with a varying profile (wavefront) and finite coherence length. •
Figure 8: A wave with finite coherence area is incident on a pinhole (small aperture). The wave will diffract out of the pinhole. Far from the pinhole the emerging spherical wavefronts are approximately flat. The coherence area is now infinite while the coherence length is unchanged. •
Figure 9: A wave with infinite coherence area is combined with a spatially shifted copy of itself. Some sections in the wave interfere constructively and some will interfere destructively. Averaging over these sections, a detector with length D will measure reducedinterference visibility. For example a misaligned Mach–Zehnder interferometer will do this. Consider a tungsten light-bulb filament. Different points in the filament emit light independently and have no fixed phase-relationship. In detail, at any point in time the profile of the emitted light is going to be distorted. The profile will change randomly over the coherence time light source such as a light-bulb
. Since for a white-
is small, the filament is considered a
spatially incoherent source. In contrast, a radio antenna array, has large spatial coherence because antennas at opposite ends of the array emit with a fixed
phase-relationship. Light waves produced by a laser often have high temporal and spatial coherence (though the degree of coherence depends strongly on the exact properties of the laser). Spatial coherence of laser beams also manifests itself as speckle patterns and diffraction fringes seen at the edges of shadow. Holography
requires
temporally
and
spatially
coherent
light.
Its
inventor, Dennis Gabor, produced successful holograms more than ten years before lasers were invented. To produce coherent light he passed the monochromatic light from an emission line of a mercury-vapor lamp through a pinhole spatial filter. In February 2011, Dr Andrew Truscott, leader of a research team at the ARC Centre of Excellence for Quantum-Atom Optics at Australian National University in Canberra, that helium atoms
Australian
cooled
to
Capital near absolute
Territory,
showed
zero / Bose-Einstein
condensate state, can be made to flow and behave as a coherent beam as occurs in a laser. Spectral coherence
Figure : Waves of different frequencies (i.e. colors) interfere to form a pulse if they are coherent.
Figure : Spectrally incoherent light interferes to form continuous light with a randomly varying phase and amplitude Waves of different frequencies (in light these are different colours) can interfere to form a pulse if they have a fixed relative phase-relationship (see Fourier transform). Conversely, if waves of different frequencies are not coherent, then, when combined, they create a wave that is continuous in time (e.g. white light or white noise). The temporal duration of the pulse bandwidth of the light
is limited by the spectral
according to:
, which follows from the properties of the Fourier transform and results in Küpfmüller's uncertainty principle (for quantum particles it also results in the Heisenberg uncertainty principle). If the phase depends linearly on the frequency (i.e.
) then the pulse
will have the minimum time duration for its bandwidth (a transformlimited pulse), otherwise it is chirped (see dispersion).
Measurement of spectral coherence Measurement of the spectral coherence of light requires a nonlinear optical interferometer, such as an intensity optical correlator,frequency-resolved optical gating (FROG), or Spectral phase interferometry for direct electric-field reconstruction (SPIDER). Polarization coherence Light also has a polarization, which is the direction in which the electric field oscillates. Unpolarized light is composed of incoherent light waves with random polarization angles. The electric field of the unpolarized light wanders in every direction and changes in phase over the coherence time of the two light waves. An absorbing polarizer rotated to any angle will always transmit half the incident intensity when averaged over time. If the electric field wanders by a smaller amount the light will be partially polarized so that at some angle, the polarizer will transmit more than half the intensity. If a wave is combined with an orthogonally polarized copy of itself delayed by less than the coherence time, partially polarized light is created. The polarization of a light beam is represented by a vector in the Poincare sphere. For polarized light the end of the vector lies on the surface of the sphere, whereas the vector has zero length for unpolarized light. The vector for partially polarized light lies within the sphere
Applications Holography Coherent superpositions of optical wave fields include holography. Holographic objects are used frequently in daily life in bank notes and credit cards. Non-optical wave fields Further applications concern the coherent superposition of non-optical wave fields. In quantum mechanics for example one considers a probability field, which is related to the wave function
(interpretation: density of the probability
amplitude). Here the applications concern, among others, the future technologies
of quantum computing and the already available technology of quantum cryptography. Additionally the problems of the following subchapter are treated. Quantum coherence In quantum mechanics, all objects have wave-like properties (see de Broglie waves). For instance, in Young's Double-slit experiment electrons can be used in the place of light waves. Each electron's wave-function goes through both slits, and hence has two separate split-beams that contribute to the intensity pattern on a screen. According to standard wave theory [Fresnel, Huygens] these two contributions give rise to an intensity pattern of bright bands due to constructive interference, interlaced with dark bands due to destructive interference, on a downstream screen. (Each split-beam, by itself, generates a diffraction pattern with less noticeable, more widely spaced dark and light bands.) This ability to interfere and diffract is related to coherence (classical or quantum) of the wave. The association of an electron with a wave is unique to quantum theory. When the incident beam is represented by a quantum pure state, the split beams downstream of the two slits are represented as a superposition of the pure states representing each split beam. (This has nothing to do with two particles or Bell's inequalities relevant to an entangled state: a 2-body state, a kind of coherence between two 1-body states.) The quantum description of imperfectly coherent paths is called a mixed state. A perfectly coherent state has a density matrix (also called the "statistical operator") that is a projection onto the pure coherent state, while a mixed state is described by a classical probability distribution for the pure states that make up the mixture. Large-scale (macroscopic) quantum coherence leads to novel phenomena, the socalled macroscopic
quantum
phenomena.
For
instance,
the laser, superconductivity, and superfluidity are examples of highly coherent quantum systems, whose effects are evident at the macroscopic scale. The macroscopic quantum coherence (Off-Diagonal Long-Range Order, ODLRO) [Penrose & Onsager (1957), C. N. Yang (1962)] for laser light, and superfluidity, is related to first-order (1-body) coherence/ODLRO, while superconductivity is related to second-order coherence/ODLRO. (For fermions, such as electrons, only
even orders of coherence/ODLRO are possible.) Superfluidity in liquid He4 is related to a partial Bose–Einstein condensate. Here, the condensate portion is described by a multiply occupied single-particle state. [e.g., Cummings & Johnston (1966)] On the other hand, the Schrödinger's cat thought experiment highlights the fact that quantum coherence cannot be arbitrarily applied to macroscopic situations. In order to have a quantum superposition of dead and alive cat, one needs to have pure states associated with aliveness and pure states associated with death, which are then superposed. Given the problem of defining death (absence of EEG, heart beat, ...) it is hard to imagine a set of quantum parameters that could be used in constructing such superposition. In any case, this is not a good topic for a description of quantum coherence. Regarding the occurrence of quantum coherence at a macroscopic level, it is interesting to note that the classical electromagnetic field exhibits macroscopic quantum coherence. The most obvious example is carrier signals for radio and TV. They satisfy Glauber's quantum description of coherence.
Diffraction : Diffraction refers to various phenomena which occur when a wave encounters an obstacle. In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects occur when a light wave travels through a medium with a varying refractive index, or a sound wave travels through one with varying acoustic
impedance.
Diffraction
occurs
with
all
waves,
including sound waves, water waves, and electromagnetic waves such as visible light, X-rays and radio waves. As physical objects have wave-like properties (at the atomic level), diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. [N]o-one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them.
He suggested that when there are only a few sources, say two, we call it interference, as in Young's slits, but with a large number of sources, the process be labelled diffraction. While diffraction occurs whenever propagating waves encounter such changes, its effects are generally most pronounced for waves whose wavelength is roughly similar to the dimensions of the diffracting objects. If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result. This is due to the superposition, or interference, of different parts of a wave that travels to the observer by different paths (see diffraction grating). The formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the expanding profile of a laser beam, the beam shape of a radar antenna and the field of view of an ultrasonic transducer can all be analysed using diffraction equations. Examples
Solar glory at the steam from hot springs. A glory is an optical phenomenon produced
by
light
backscattered
(a
combination
of
diffraction, reflection and refraction) towards its source by a cloud of uniformly sized water droplets. The effects of diffraction are often seen in everyday life. The most striking examples of diffraction are those involving light; for example, the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern seen when looking at a disk. This principle can be extended to engineer a
grating with a structure such that it will produce any diffraction pattern desired; the hologram on a credit card is an example.Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. A shadow of a solid object, using light from a compact source, shows small fringes near its edges. The speckle pattern which is observed when laser light falls on an optically rough surface is also a diffraction phenomenon. All these effects are a consequence of the fact that light propagates as a wave. Diffraction can occur with any kind of wave. Ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, which is why one can still hear someone calling even when hiding behind a tree. Diffraction can also be a concern in some technical applications; it sets afundamental limit to the resolution of a camera, telescope, or microscope. Mechanism
Photograph of single-slit diffraction in a circular ripple tank
Diffraction arises because of the way in which waves propagate; this is described by the Huygens–Fresnel principle and the principle of superposition of waves. The propagation of a wave can be visualized by considering every point on a wavefront as a point source for a secondary spherical wave. The wave displacement at any subsequent point is the sum of these secondary waves. When waves are added together, their sum is determined by the relative phases as well as the amplitudes of the individual waves so that the summed amplitude of the waves can have any value between zero and the sum of the individual amplitudes. Hence, diffraction patterns usually have a series of maxima and minima.
There are various analytical models which allow the diffracted field to be calculated, including the Kirchhoff-Fresnel diffraction equation which is derived from wave equation, the Fraunhofer diffraction approximation of the Kirchhoff equation which applies to the far field and the Fresnel diffraction approximation which applies to the near field. Most configurations cannot be solved analytically, but
can
yield
numerical
solutions
through finite
element and boundary
element methods. It is possible to obtain a qualitative understanding of many diffraction phenomena by considering how the relative phases of the individual secondary wave sources vary, and in particular, the conditions in which the phase difference equals half a cycle in which case waves will cancel one another out. The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For water waves, this is already the case; water waves propagate only on the surface of the water. For light, we can often neglect one direction if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three dimensional nature of the problem. Diffraction of light Some examples of diffraction of light are considered below. Single-slit diffraction
Numerical approximation of diffraction pattern from a slit of width equal to wavelength of an incident plane wave in 3D spectrum visualization
Numerical approximation of diffraction pattern from a slit of width equal to five times the wavelength of an incident plane wave in 3D spectrum visualization
Diffraction of red laser beam on the hole
Numerical approximation of diffraction pattern from a slit of width four wavelengths with an incident plane wave. The main central beam, nulls, and phase reversals are apparent.
Graph and image of single-slit diffraction. A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wavefront which emerges from the slit is a cylindrical wave of uniform intensity. A slit which is wider than a wavelength produces interference effects in the space downstream of the slit. These can be explained by assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit. The analysis of this system is simplified if we consider light of a single wavelength. If the incident light is coherent, these sources all have the same phase. Light incident at a given point in the space downstream of the slit is made up of contributions from each of these point sources and if the relative phases of these contributions vary by 2π or more, we may expect to find minima and maxima in the diffracted light. Such phase differences are caused by differences in the path lengths over which contributing rays reach the point from the slit. We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path
difference is given by
so that the minimum intensity occurs at an
angle θmin given by where •
d is the width of the slit, is the angle of incidence at which the minimum intensity occurs, and
•
is the wavelength of the light
•
A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles θn given by where •
n is an integer other than zero.
There is no such simple argument to enable us to find the maxima of the diffraction pattern. The intensity profile can be calculated using the Fraunhofer diffraction equation as
where is the intensity at a given angle,
• • •
is the original intensity, and the sinc function is given by sinc(x) = sin(πx)/(πx) if x ≠ 0, and sinc(0) = 1
This analysis applies only to the far field, that is, at a distance much larger than the width of the slit.
2-slit (top) and 5-slit diffraction of red laser light
Diffraction of a red laser using a diffraction grating.
A diffraction pattern of a 633 nm laser through a grid of 150 slits Diffraction grating A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation where •
θi is the angle at which the light is incident,
•
d is the separation of grating elements, and
•
m is an integer which can be positive or negative.
The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns. The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.
A computer-generated image of an Airy disk.
Computer generated light diffraction pattern from a circular aperture of diameter 0.5 micrometre at a wavelength of 0.6 micrometre (red-light) at distances of 0.1 cm – 1 cm in steps of 0.1 cm. One can see the image moving from the Fresnel region into the Fraunhofer region where the Airy pattern is seen. Circular aperture The far-field diffraction of a plane wave incident on a circular aperture is often referred to as the Airy Disk. The variation in intensity with angle is given by
, where a is the radius of the circular aperture, k is equal to 2π/λ and J1 is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams.
General aperture The wave that emerges from a point source has amplitude
at location r that is
given by the solution of thefrequency domain wave equation for a point source (The Helmholtz Equation),
where
is the 3-dimensional delta function. The delta function has only radial
dependence, so the Laplace operator (aka scalar Laplacian) in thespherical coordinate system simplifies to (see del in cylindrical and spherical coordinates)
By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system(and using the physics time convention
) is:
This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector
and the
field point is located at the point , then we may represent the scalar Green's function (for arbitrary source location) as:
Therefore, if an electric field, Einc(x,y) is incident on the aperture, the field produced by this aperture distribution is given by the surface integral:
On the calculation of Fraunhofer region fields where the source point in the aperture is given by the vector
In the far field, wherein the parallel rays approximation can be employed, the Green's function,
simplifies to
as can be seen in the figure to the right (click to enlarge). The expression for the far-zone (Fraunhofer region) field becomes
Now, since
and the expression for the Fraunhofer region field from a planar aperture now becomes,
Letting, and the Fraunhofer region field of the planar aperture assumes the form of a Fourier transform
In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture distribution. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see Fourier optics). Propagation of a laser beam The way in which the profile of a laser beam changes as it propagates is determined by diffraction. The output mirror of the laser is an aperture, and the subsequent beam shape is determined by that aperture. Hence, the smaller the output beam, the quicker it diverges. Paradoxically, it is possible to reduce the divergence of a laser beam by first expanding it with one convex lens, and then collimating it with a second convex lens whose focal point is coincident with that of the first lens. The resulting beam has a larger aperture, and hence a lower divergence. Diffraction-limited imaging Main article: Diffraction-limited system
The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this lucky image of the binary star zeta Boötis. The ability of an imaging system to resolve detail is ultimately limited by diffraction. This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane with radius to first null of where λ is the wavelength of the light and N is the f-number (focal length divided by diameter) of the imaging optics. In object space, the correspondingangular resolution is
where D is the diameter of the entrance pupil of the imaging lens (e.g., of a telescope's main mirror). Two point sources will each produce an Airy pattern – see the photo of a binary star. As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image. The Rayleigh criterion specifies that two point sources can be considered to be resolvable if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.
Thus, the larger the aperture of the lens, and the smaller the wavelength, the finer the resolution of an imaging system. This is why telescopes have very large lenses or mirrors, and why optical microscopes are limited in the detail which they can see. Speckle patterns The speckle pattern which is seen when using a laser pointer is another diffraction phenomenon. It is a result of the superpostion of many waves with different phases, which are produced when a laser beam illuminates a rough surface. They add together to give a resultant wave whose amplitude, and therefore intensity varies randomly. Patterns
The upper half of this image shows a diffraction pattern of He-Ne laser beam on an elliptic aperture. The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture. Several qualitative observations can be made of diffraction in general: •
The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: The smaller the diffracting object, the
'wider' the resulting diffraction pattern, and vice versa. (More precisely, this is true of the sines of the angles.) •
The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.
•
When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The third figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing, between the center of one slit and the next.
Particle diffraction Quantum theory tells us that every particle exhibits wave properties. In particular, massive particles can interfere and therefore diffract. Diffraction of electrons and neutrons stood as one of the powerful arguments in favor of quantum mechanics. The wavelength associated with a particle is the de Broglie wavelength
where h is Planck's constant and p is the momentum of the particle (mass × velocity for slow-moving particles). For most macroscopic objects, this wavelength is so short that it is not meaningful to assign a wavelength to them. A sodium atom traveling at about 30,000 m/s would have a De Broglie wavelength of about 50 pico meters. Because the wavelength for even the smallest of macroscopic objects is extremely small, diffraction of matter waves is only visible for small particles, like electrons, neutrons, atoms and small molecules. The short wavelength of these matter waves makes them ideally suited to study the atomic crystal structure of solids and large molecules like proteins. Relatively larger molecules like buckyballs were also shown to diffract
Propagation of waves in a rectangular wave guide :
In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning is a hollow metal pipe used to carry radio waves. This type of waveguide is used as atransmission line mostly
at microwave frequencies,
microwave transmitters and receivers to
for
such
purposes
their antennas,
in
as
connecting
equipment
such
as microwave ovens, radar sets, satellite communications, and microwave radio links. A dielectric waveguide employs a solid dielectric rod rather than a hollow pipe. An optical
fibre is
a
frequencies.Transmission
dielectric
guide
lines such
designed
to
work
at
as microstrip,
optical coplanar
waveguide, stripline or coaxial may also be considered to be waveguides. The electromagnetic waves in (metal-pipe) waveguide may be imagined as travelling down the guide in a zig-zag path, being repeatedly reflected between opposite walls of the guide. For the particular case of rectangular waveguide, it is possible to base an exact analysis on this view. Propagation in dielectric waveguide may be viewed in the same way, with the waves confined to the dielectric by total internal reflection at its surface. Some structures, such as Nonradiative dielectric waveguide and the Goubau line, use both metal walls and dielectric surfaces to confine the wave.
Short length of rectangular waveguide (WG17 with UBR120 connectionflanges) Rectangular Waveguides Rectangular waveguides are important for two reasons. First of all, the Laplacian operator separates nicely in Cartesian coordinates, so that the boundary value
problem that must be solved is both familiar and straightforward. Second, they are extremely common in actual application in physics laboratories for piping e.g. microwaves around as experimental probes. In Cartesian coordinates, the wave equation becomes:
This wave equation separates and solutions are products of sin, cos or exponential functions in each variable separately. To determine which combination to use it suffices to look at the BC's being satisfied. For TM waves, one solves
for
where
subject to
and
, which is automatically true if:
are the dimensions of the
rectangle and where in principle
and
sides of the boundary
.
However, the wavenumber of any given mode (given the frequency) is determined from:
where
for a ``wave'' to exist to propagate at all. If either index
or
is zero, there is no wave, so the first mode that can propagate has a dispersion relation of:
so that:
Each combination of permitted
and
is associated with a cutoff of this sort -
waves with frequencies greater than or equal to the cutoff can support propogation in all the modes with lower cutoff frequencies. If we repeat the argument above for TE waves (as is done in Jackson, which is why I did TM here so you could see them both) you will be led by nearly identical arguments to the conclusion that the lowest frequency mode cutoff occurs
for
,
and
to produce the
solution
to the wave equation above. The cutoff in this case is:
There exists, therefore, a range of frequencies in between where only one TE mode is supported with dispersion:
Note well that this mode and cutoff corresponds to exactly one-half a free-space wavelength across the long dimension of the waveguide. The wave solution for the right-propagating TE mode is:
We used
second of these, and
and
) to get the last one.
to get the
There is a lot more one can study in Jackson associated with waveguides, but we must move on at this time to a brief look at resonant cavities (another important topic) and multipoles.
Inhomogeneous electromagnetic wave equation Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a inhomogeneous electromagnetic wave equation (or often "non homogeneous electromagnetic wave equation") with sources. The addition of sources
to
the
wave
equations
makes
the partial
differential
equations inhomogeneous. SI units Maxwell's equations in a vacuum with charge
and current
written in terms of the vector and scalar potentials as
where
and . If the Lorenz gauge condition is assumed
then the nonhomogeneous wave equations become
sources can be
. CGS and Lorentz–Heaviside units In cgs units these equations become
with
and the Lorenz gauge condition . For Lorentz–Heaviside units, sometimes used in high dimensional relativistic calculations, the charge and current densities in cgs units translate as
. Covariant form of the inhomogeneous wave equation
Time dilation in transversal motion. The requirement that the speed of light is constant in every inertial reference frame leads to the theory of relativity The relativistic needed]
Maxwell's
equations can
be
written
form as
where J is the four-current ,
is the 4-gradient and the electromagnetic four-potential is
with the Lorenz gauge condition . Here
in covariant[disambiguation
is the d'Alembert operator. Curved spacetime The electromagnetic wave equation is modified in two ways in curved spacetime, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears (SI units).
where
is the Ricci curvature tensor. Here the semicolon indicates covariant differentiation. To obtain the equation in cgs units, replace the permeability with
.
Generalization of the Lorenz gauge condition in curved spacetime is assumed . Solutions to the inhomogeneous electromagnetic wave equation
Retarded spherical wave. The source of the wave occurs at time t'. The wavefront moves away from the source as time increases for t>t'. For advanced solutions, the wavefront moves backwards in time from the source t
and
where
is a Dirac delta function. For SI units
. For Lorentz–Heaviside units,
. These solutions are known as the retarded Lorenz gauge potentials. They represent a superposition of spherical light waves traveling outward from the sources of the waves, from the present into the future. There are also advanced solutions (cgs units)
and
.
These represent a superposition of spherical waves travelling from the future into the present.
Solution of the inhomogeneous wave equation Equations all have the general form
Can we find a unique solution to the above equation? Let us assume that the source function
can be expressed as a Fourier integral
The inverse transform is
Similarly, we may write the general potential
with the corresponding inverse
Fourier transformation of Eq. yields
where
.
as a Fourier integral
The above equation, which reduces to Poisson's equation in the limit
, and
is called Helmholtz's equation, is linear, so we may attempt a Green's function method of solution. Let us try to find a function
such that
The general solution is then
The
``sensible''
that
spatial
boundary
as
conditions
which
we
impose
are
. In other words, the field goes to zero a long
way from the source. Since the system we are solving is spherically symmetric about the point
it is plausible that the Green's function itself is spherically
symmetric. It follows that
where in the region
and
. The most general solution to the above equation
is1
We know that in the limit
the Green's function for Helmholtz's equation
must tend towards that for Poisson's equation, which is
This is only the case if
.
Reconstructing
from Eqs. , we obtain
It follows from Eq. that
Now, the real space Green's function for the inhomogeneous wave equation satisfies
Hence, the most general solution of this equation takes the form
Comparing Eqs. we obtain
where
and
.
The real space Green's function specifies the response of the system to a point source at position
which appears momentarily at time
the retarded Green's function centred on position
. According to
the response consists of a spherical wave,
, which propagates forward in time. In order for the wave to reach at time
the retarded
time
it must have been emitted from the source at .
According
to
the advanced
at
Green's
function
the response consists of a spherical wave, centred on
, which
propagates backward in time. Clearly, the advanced potential is not consistent with our ideas about causality, which demand that an effect can never precede its cause in time. Thus, the Green's function which is consistent with our experience is
We are able to find solutions of the inhomogeneous wave equation
which
propagate backward in time because this equation is time symmetric (i.e., it is invariant under the transformation
).
In conclusion, the most general solution of the inhomogeneous wave equation which satisfies sensible boundary conditions at infinity and is consistent with causality is
This expression is sometimes written
where the rectangular bracket symbol
denotes that the terms inside the bracket
are to be evaluated at the retarded time Eq. that if there is no source (i.e., (i.e.,
. Note, in particular, from ) then there is no field
). But, is the above solution really unique? Unfortunately, there
is a weak link in our derivation, between Eqs. (2.110) and (2.111), where we assume that the Green's function for the Helmholtz equation subject to the
boundary condition
as
is spherically symmetric. Let
us try to fix this problem. With the benefit of hindsight, we can see that the Green's function
corresponds to the retarded solution in real space and is, therefore, the correct physical Green's function. The Green's function
corresponds to the advanced solution in real space and must, therefore, be rejected. We can select the retarded Green's function by imposing the following boundary condition at infinity
This is called the Sommerfeld radiation condition; it basically ensures that sources radiate
waves
instead
of
absorbing
them.
But,
does
this
condition uniquely select the spherically symmetric Green's function
boundary as the
solution of
Here,
are spherical polar coordinates. If it does then we can be sure that
Eq. represents the unique solution of the wave equation which is consistent with causality. Let us suppose that there are two solutions of Eq. which satisfy the boundary condition and revert to the unique Green's function for Poisson's equation in the limit
. Let us call these solutions
and
, and let us form the
difference
. Consider a surface
which is a sphere of arbitrarily
small radius centred on the origin. Consider a second surface sphere of arbitrarily large radius centred on the origin. Let enclosed by these surfaces. The difference function
which is a
denote the volume
satisfies the homogeneous
Helmholtz equation,
throughout
where
. According to Green's theorem
denotes a derivative normal to the surface in question. It is clear
from Eq. that the volume integral is zero. It is also clear that the first surface integral is zero, since both Poisson's equation in the limit
and
must revert to the Green's function for . Thus,
Equation can be written
where
is the spherical harmonic operator
The most general solution of Eq. takes the form
Here, the
and
are arbitrary
coefficients,
the
are
spherical
harmonics,2 and
where
are Hankel functions of the first and second kind. It can be
demonstrated that4
where
and The large
. Note that the summations in Eqs. terminate after behaviour of the
terms.
is clearly inconsistent with the Sommerfeld
radiation condition . It follows that all of the
in Eq. are zero. The most
general solution can now be expressed in the form
where the
are various weighted sums of the spherical harmonics.
Substitution of this solution into the differential equation yields
Replacing the index of summation
in the first term of the parentheses by
we obtain
which gives us the recursion relation
It follows that if
then all of the
are equal to zero.
Let us now consider the surface integral . Since we are interested in the limit
where that
we can replace
by the first term of its expansion in , so
is a unit of solid angle. It is clear that and, hence, that
. This implies
. Thus, there is only one solution of
Eq. which is consistent with the Sommerfeld radiation condition, and this is given by Eq. We can now be sure that Eq. is a unique solution of Eq. subject to the boundary condition . This boundary condition basically says that infinity is an absorber of radiation but not an emitter, which seems entirely reasonable.
Retarded potentials Equations have the same form as the inhomogeneous wave equation , so we can immediately write the solutions to these equations as
Moreover, we can be sure that these solutions are unique, subject to the reasonable proviso that infinity is an absorber of radiation but not an emitter. This is a crucially important point. Whenever the above solutions are presented in physics textbooks there is a tacit assumption that they are unique. After all, if they were not unique why should we choose to study them instead of one of the other possible solutions? The uniqueness of the above solutions has a physical interpretation. It is clear from Eqs. that in the absence of any charges and currents there are no electromagnetic fields. In other words, if we observe an electromagnetic field we can be certain that if we were to trace it backward in time we would eventually discover that it was emitted by a charge or a current. In proving that the solutions of Maxwell's equations are unique, and then finding a solution in which all waves are emitted by sources, we have effectively ruled out the possibility that the vacuum can be ``unstable'' to the production of electromagnetic waves without the need for any sources. Equations can be combined to form the solution of the 4-vector wave equation ,
Here, the components of the 4-potential are evaluated at some event time,
is the distance of the volume element
from
in space-
, and the square
brackets indicate that the 4-current is to be evaluated at the retarded time; i.e., at a time
before
.
But, does the right-hand side of Eq. really transform as a contravariant 4-vector? This is not a trivial question since volume integrals in 3-space are not, in general, Lorentz invariant due to the length contraction effect. However, the integral in Eq. is not a straightforward volume integral because the integrand is evaluated at the retarded time. In fact, the integral is best regarded as an integral over events in space-time. The events which enter the integral are those which intersect a spherical light wave launched from the event
and evolved backwards in time.
In other words, the events occur before the event respect to
and have zero interval with
. It is clear that observers in all inertial frames will, at least, agree on
which events are to be included in the integral, since both the interval between events and the absolute order in which events occur are invariant under a general Lorentz transformation. We shall now demonstrate that all observers obtain the same value of each elementary contribution to the integral. Suppose that
and
for are two
inertial frames in the standard configuration. Let unprimed and primed symbols denote corresponding quantities in S and S’, respectively. Let us assign coordinates (o,o,o,o,) to r and (x,y,z,d) to the retarded event Q for which r and dv are evaluated. Using the standard Lorentz transformation , the fact that the interval between events P and Q is zero, and the fact that both
where
and
are negative, we obtain
is the relative velocity between frames
factor, and
where
and
We might be tempted to set
is the Lorentz
, etc. It follows that
is the angle (in 3-space) subtended between the line
We now know the transformation for
,
and the
. What about the transformation for
-axis. ?
, according to the usual length
contraction rule. However, this is wrong. The contraction by a factor
only
applies if the whole of the volume is measured at the same time, which is not the case in the present problem. Now, the dimensions of axes are the same in both
and
and
, according to Eqs. (2.19). For the
dimension these equations give measured at times differing by
along the
. The extremities of
are
, where5
Thus,
giving
It follows from Eqs. that even when Thus,
and
. This result will clearly remain valid
are not in the standard configuration.
is an invariant and, therefore,
is a contravariant 4-vector.
For linear transformations, such as a general Lorentz transformation, the result of adding 4-tensors evaluated at different 4-points is itself a 4-tensor. It follows that the right-hand side of Eq. is a contravariant 4-vector. Thus, this 4-vector equation can be properly regarded as the solution to the 4-vector wave equation .
Radiation from a localized source : While wave propagation in gyromagnetic media has been investigated extensively in the technical literature, little attention has been given to the excitation of such waves by a prescribed source. This problem is treated in the present paper. To render the analysis tractable, the source function is taken as a time‐harmonic line
source of electric currents with a rapidly varying phase, embedded in an infinite homogeneous
gyromagnetic
medium
whose
axis
of
magnetization
is
perpendicular to the source direction. For further simplification, the operating frequency is taken to be near gyromagnetic resonance. The resulting model incorporates relevant anomalies of the radiation process, and also serves as a prototype for more practical thin slab configurations. The analysis is performed for permeability tensors in which spin exchange effects are ignored and included. In the former case, the refractive index surface may have open branches and gives rise to an infinity in the total radiated power (infinity catastrophe); this anomaly is removed when spin exchange is accounted for. Detailed study of the radiation fields and radiated power densities shows that except for certain initial directions, the outward power flow is almost entirely in the electromagnetic waves, and the total radiated power in the electromagnetic waves far exceeds that in the spin exchange waves, even when the source dimensions tend to zero. These results imply that a localized electromagnetic current source does not strongly excite the spin exchange waves. 1 General Results If a localized current is given as a function of space and time, the vector potential in the Lorentz gauge is
( ,t) = d 3r' . The retardation is often simpler if the current is Fourier transformed in time,
( )=
dt
( ,t)e i
t
so that
( ,t) = ( )e - i t . Since the current is real, the complex conjugate of the last equation shows that ( )= alone as
*
-
( ) . The current can be written in terms of positive frequencies
( ,t) = Re ( )e - i t which therefore simply looks like a sum (i.e. the integral) of terms each with an e i
t
time dependence.
Plugging this form in, the vector potential is
( ,t) = Re e-i t . To extract the radiation fields, the distance from the source to observation point is expanded assuming the source is localized and the observation point goes to r . The vector potential is
( ,t) = Re d 3r' ( )e - i Notice that the current integral is just the ``on energy shell'' space-time Fourier transform of the current. That is defining the Fourier transform as
(
,
)=
d 3rdt
( ,t)e - i
+i
t
the vector potential in the radiation zone is
( ,t) = Re There are two cases of interest. The first is where the current (or at least the radiating part) is localized in time. In that case the total radiated energy in a small frequency range is finite. This of course is the only physical situation. However, in many cases, the motion is periodic for a very long time, and it is the energy radiated per cycle or equivalently the power radiated that is desired. If the motion continues indefinitely, the total energy radiated is infinite, so the mathematics needs to be slightly modified as shown in the spectral resolution notes.
The magnetic field in the radiation zone is given by taking the curl of
, and to
give a r - 1 dependence, the gradient must operate on the exponent. Similarly, the electric field is given for a harmonic time dependence as
x =i The fields are therefore
.
= ( ,t) Re = ( ,t) Re The
x
(
) is the transform of the transverse component of the current.
x
The Poynting vector is |2/4
given by c|
, and with direction along
x
and magnitude
.
As shown in the spectral resolution notes, if the time integral of the product of two functions of time f (t)
=
g(t)
=
Re
(
)
Re
(
)
is called I , the differential I per frequency range d
=
Re
(
)
(
is
).
so that the energy radiated in frequency interval d around
is
=
2
in solid angle increment d
where
is the detected polarization which must be perpendicular to
.
Summing over polarizations will gives the total energy radiated in the frequency interval and solid angle interval. Compare this result to Jackson Eq. Again applying results of the spectral resolution notes to the case where the
motion is periodic with period T = frequency n
, the power radiated per solid angle at the
is
=
2
where
+ in t ( )= dt d 3rJ( ,t)e - i . Essentially, if you imagine that you have a signal that repeats with fundamental
frequency
, but in each period is localized to time much smaller than the
period, then the intensity of radiation for one period would be given by d 2I/d
d
where the current integration would be over just one period. Since the motion repeats, the power radiated in a frequency n frequency
multiplied by
=n
is the energy radiated at that
divided by the period 2 /
to convert from d
to get the power, and
to dn . The overall factor is
/2
which
is the difference in the prefactors of the two expressions above. 2. Long Wavelength Approximations If the current is localized in a region or a set regions, each which has spatial extent much smaller than a wavelength,
= 2 c/
, the origin of integration
can be translated to the approximate center of a region, and the exponential
expanded in a power series. The terms in the power series are roughly powers of the extent of the region of nonzero current divided by the wavelength. The exponential becomes
e -i =1-i + ... and the current integrals needed to this order are I (0)k
= d 3r'J
I (1)jk
(
k
)
=
d 3r'x'jJ k( ) . Just as in magnetostatics, some integrations by parts using the divergence theorem are useful, (xk
(xjxk
= xk = i xk = xjxk = i xjxk
)
)
+J +J
k
k
+ [xjJ + [xjJ
k
k
+ xkJ
+ xkJ
j
].
The symmetry in j and k of the last term suggests writing I (1) as
I (1)jk =
and the integrals become I (0)k = -i I
(1) jk
=
.
d 3r'x'k
(
)=-i d
k
]
j
The I (0)k integral is proportional to the k component of the electric dipole moment. The I (1)jk has two terms. The second looks like the magnetic dipole moment
d 3r'[x'jJ k( ) - x'jJ k( )] = 2c m . Noting that in the field equation, one component is dotted with is crossed with
, while the other
, a constant diagonal element can be added to it without
changing the result. That is
rjri
=0.
so that the replacement
d 3r'x'jx'k ( ) d 3r' ( ) = Q jk does not change the fields. The latter expression is the quadrupole moment Qjk . The three terms are then identified as the electric dipole, magnetic dipole and quadrupole terms. Explicitly we can make the replacements = -i
+i
x
-
+ ...
or = n
- in
n
+ in
x
n
-
n
+ ...
where the subscript n indicates the Fourier integral over one period of
of e in
t
.
As a check against Jackson's results, look at the case where only the diagonal quadrupole terms contribute with Q33 = Q0 , and Q11 = Q22 = - Q0/2 each oscillating as cos(
t) , so that only the n = 1 term survives, and the time integration gives
= The current transform is then 1
.
= The power radiated per solid angle is 1
= 2 =
=
which agrees with Jackson Eq. after unit conversion. Radiation from an Oscillating Electric Dipole
For a charge to radiate electromagentic waves in free space, it must be either accelerated or decelerated. Charges undergoing oscillatory motion are continuously accelerated and decelerated and if the velocity of charges is nonrelativistic, electromagentic waves are radiated at the oscillation frequency. Radiation from antennas can be analyzed by superposing elementary radiation fields emitted by individual electrons .
Electric field lines of static dipole. Animation shows the electric field lines due to an electric dipole oscillating vertically at the origin. Near the dipole, the field lines are essentially those of a static dipole leaving a positive charge and ending up at a negative charge. However, at a distance of the order of half wavelength (lambda = 2Pi is assumed here) or greater, the field lines are completely detached from the dipole. This detachment characterizes radiation fields which propagate freely (without being attached to charges) in free space at the speed c.
