Electromagnetic Field Theory Fundamentals

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/246041673

Electromagnetic Field Theory Fundamentals Article · January 2004 DOI: 10.1017/CBO9781139165297

CITATIONS

READS

161

9,198

2 authors, including:

Huseyin Hiziroglu Hiziroglu Kettering University University 48 PUBLICATIONS 329 CITATIONS SEE PROFILE

Some of the authors of this publication are also working on these related projects:

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Cambridge University Press 0521830168 - Electromagnetic Field Theory Fundamentals, Second Edition Bhag Singh Guru and Huseyin R. Hiziroglu Excerpt More information

1 Electromagnetic field theory

1.1 Introduction .................................

What is a field? Is it a scalar field or a vector field? What is the na of a field? Is it a continuous or a rotational field? How is the magn field produced by a current-carrying coil? How does a capacitor s energy? How does a piece of wire (antenna) radiate or receive sign How do electromagnetic fields propagate in space? What really hap You're Reading a Preview when electromagnetic energy travels from one end of a hollow p (waveguide) to the other? The primary purpose of this text is to ans Unlock full access with a free trial. some of these questions pertaining to electromagnetic fields. In this chapter we With intend to show Download Free Trial that the study of electromagn field theory is vital to understanding many phenomena that take p in electrical engineering. To do so we make use of some of the conc and equations of other areas of electrical engineering. We aim to s light on theoriginof these concepts and equations using electroma field theory. Before we proceed any further, however, we mention that the de opment of science depends upon some quantities that cannot be defi precisely. We refer to these as fundamental quantities; they are mass length (), time (t ), charge (q ), and temperature (T ). For exam what is time? When did time begin? Likewise, what is temperatu  What is hot or cold? We do have feelings about th Read Free Foron 30this Days Signsome up to intuitive vote title  quantities but lack precise definitions. To measure and express eac Not useful  Useful  Cancel anytime. Special offer for students: Only $4.99/month. these quantities, we need to define a system of units. In the International System of Units (SI for short), we have adop

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Table 1.1. Derived units for some electromagnetic quantities Symbol Y ω C ρ G σ W F f Z L F

µ  P R

Quantity

Unit

admittance angular frequency capacitance charge density conductance conductivity energy force frequency impedance inductance magnetomotive force permeability permittivity power reluctance

siemen radian/second farad coulomb/meter3 siemen siemen/meter joule newton hertz ohm henry ampere-turn henry/meter farad/meter watt henry−1

You're Reading a Preview

Ab

Unlock full access with a free trial.

Table 1.2. Unit conversion factors From

Download With Free Trial Multiply by

gilbert ampere-turn/cm ampere-turn/inch oersted line (maxwells) gauss (lines/cm2 ) line/inch2 gauss inch foot meter square inch ounce pound


0.79577 2.54 39.37 79.577 1 × 10−8 6.4516 0.155 × 10−4 10−4 2.54 30.48 100 Read Free Foron 30this Days Sign up to vote title 6.4516  Useful  Not useful 28.35 Cancel anytime. 0.4536

To obtai

ampereampereampereampereweber (Wb line/inch Wb/m2 (tes Wb/m2 centimet centimet  centimet  square c gram kilogram

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1.2 Field concept

Table 1.3. A partial list of field quantities Variable

Definition

Type

Un

A B D E F I J q S u V

magnetic vector potential magnetic flux density electric flux density electric field intensity Lorentz force electric current volume current density free charge Poynting vector velocity of free charge electric potential

vector vector vector vector vector scalar vector scalar vector vector scalar

Wb Wb C/m V/m N A A/m C W/ m/s V

















Table 1.4. A partial list of relationships between various field quantities D B J F ∇·D ∇·B 

=



=



=



=



=



=

∇·J = 

permittivity ( ) E permeability (µ) µH You're Reading a Preview conductivity (σ ), Ohm’s law σ E Unlock (E +full access free trial. force equation q u× B) with aLorentz Gauss’s law (Maxwell’s equation) ρ 0Download With Free Gauss’s Trial law (Maxwell’s equation) ∂ρ continuity equation − ∂t 









∂B





∇×E = − 

∂t ∂D

Faraday’s law (Maxwell’s equation)



∇×H= 

J + 

∂t

Ampère’s law (Maxwell’s equation)

1.2 Field concept .................................

Prior to undertaking electromagnetic fields we must de Master your semester with Scribd the study ofRead  Free For 30this Days up tothe vote on title the concept of a field. When weSign define behavior of a quantity  & The New York Timesgiven region in terms of a set of Usefulone useful values, forNot each point in that re Cancel anytime.

Special offer for students: Only $4.99/month. we refer to this behavior of the quantity as a field. The value at e

point of a field can be either measured experimentally or predicted

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From the equations listed in Table 1.4, Maxwell was able to p that electromagnetic fields propagate in a vacuum with the sp light. That is, c = (µ0 0 )−1/2 m/s

1.3 Vector analysis .................................

Vector analysis is the language used in the study of electroma fields. Without the use of vectors, the field equations would be unwieldy to write and onerous to remember. For example, the product of two vectors A and B can be simply written as 



A×B = C 





where C is another vector. When expressed in scalar form, this yields a set of three scalar equtions. In addition, the appearance of scalar equations depends upon the coordinate system. In the rec coordinate system, the aprevious You're Reading Preview equation is a concise version following three equations: 

Unlock full access with a free trial.

A y B z − A z B y = C x

With Free Trial A z B x −Download A x B z = C y

A x B y − A y B x = C z

You can easily see that the vector equation conveys the sens cross product better than its three scalar counterparts. Moreove vector representation is independent of the coordinate system. vector analysis helps us to simplify and unify field equations. By the time a student is required to take the first course in elec netic theory, he/she has had a very limited exposure to vector an The student may be competent to perform such vector operations gradient, divergence, and curl, but may not be able to describe th  nificance of each operation. The ofthis each Read Free Foron 30 Days Signknowledge up to vote titlevector opera  essential to appreciate the development of electromagnetic field t Not useful  Useful  Cancel anytime. Quite often, a student does not know that (a) the unit vecto transforms a scalar surface to a vector surface is always normal


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1.4 Differential and integral formulations

proficiency in all vector operations prior to exploring electromagn field theory. We prefer the latter approach and for this reason h devoted Chapter 2 to the study of vectors.

1.4 Differential and integral formulations .................................

Quite often a student does not understand why we present the s idea in two different forms: the differential form and the integral fo It must be pointed out that the integral form is useful to explain significance of an equation, whereas the differential form is conven for performing mathematical operations. For example, we express equation of continuity of current in the differential form as ∇ · J = −

∂ρ



∂ t

where J is the volume current density and ρ is the volume ch density. This equation states that the divergence of current density You're Reading Preview point is equal to the rate atawhich the charge density is changing at point. TheUnlock usefulness ofwith thisa equation full access free trial. lies in the fact that we can u to calculate the rate at which the charge density is changing at a p when the current density known Download WithisFree Trialat that point. However, to highl the physical significance of this equation, we have to enclose the in a volume v and perform volume integration. In other words, we h to express (1.3) as 



∇ · J d v = − 

v

 v

∂ρ ∂ t

d v

We can now apply the divergence theorem to transform the volu integral on the left-hand side into a closed surface integral. We also interchange the operations of integration and differentiation on right-hand side of equation (1.4). We can now obtain

 Scribd  Master your semester with ∂ & The New York Times J · d s = − ∂ t ρ d v 

s

Special offer for students: Only $4.99/month.



v

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This equation is an integral formulation of (1.3). The integral on

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The details of the preceding development are given in Chapter used this example at this time just to show that (1.3) and (1.5) a same and that they embody the same basic idea.

1.5 Static fields .................................

Once again we face the dilemma of how to begin the presentat electromagnetic field theory. Some authors believe in starting w presentation of Maxwell’s equations as a basic set of postulates summarizing the results of many years of experimental observati electromagnetic effects. We, however, think that the field theory always be developed by making maximum possible use of the co previously discussed in earlier courses in physics. For this reas first discuss static fields. In the study of electrostatics, or static electric fields, we assum (a) all charges are fixed in space, (b) all charge densities are cons time, and (c) the charge is the source of the electric field. Our inte You're Reading a Preview to determine (a) the electric field intensity at any point, (b) the po Unlock(c) full the access with aexerted free trial. by the charges on other charge distribution, forces (d) the electric energy distribution in the region. We will also e Download Withenergy. Free Trial how a capacitor stores To do so, we will begin our disc with Coulomb’s law and Gauss’s law and formulate such well-k equations as Poisson’s equation and Laplace’s equation in terms tential functions. We will show that the electric field at any p perpendicular to an equipotential surface and emphasize its ram tions. Some of the equations pertaining to electrostatic fields are in Table 1.5 (see below). Table 1.5. Electrostatic field equations Coulomb’s law: Master your semester with Scribd & The New York Times Electric field: Special offer for students: Only $4.99/month.

Gauss’s law:

F = qE  Read Free 30this Days Sign up to vote title 1 QFor a Ron =  or E= E 2 useful  Useful  Not 4π R 4π  Cancel anytime. 







 

∇·D = 

ρ

or

s



D · d s = 



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1.6 Time-varying fields

Table 1.6. Magnetostatic field equations Force equation:

F = qu × B µ I d  ×a r d B = 4π r 2

or d F = I d

∇×H=

or









Biot–Savart law: Ampère’s law:









J





 H d   

·



c

Gauss’s law:

∇·B = 

0

or

B · d s 



s



Magnetic vector potential:

B = ∇×A

Magnetic flux:

=





B · d s 



s

wm = 12 B · H

Poisson’s equation:

∇

Constitutive relationship:

B = µH







or  =

µ

 4π

A 

c

Magnetic energy:

2

or A =



A = −µJ 





We already know that a charge in motion creates a current. If movement of the charge is restricted in such a way that the resul Reading a Preview current is You're constant in time, the field thus created is called a magn field. Since the current is constant in time, the magnetic field is Unlock full access with a free trial. constant in time. The branch of science relating to constant magn fields is called magnetostatics, static magnetic fields. In this case Download With Freeor Trial are interested in the determination of (a) magnetic field intensity, magnetic flux density, (c) magnetic flux, and (d) the energy stored magnetic field. To this end we will begin our discussion with the B Savart law and Ampère’s law and develop all the essential equati From time to time we will also stress the correlation between the st electric and magnetic fields. Some of the important equations tha will either state or formulate in magnetostatics are given in Table 1 There are numerous practical applications of static fields. Both st electric and magnetic fields are used in the design of many devices. example, we can use a static electric field to accelerate a particle  a static magnetic field to deflect it. This scheme can be employe Read Free Foron 30this Days Sign up to vote title  the design of an oscilloscope and/or an ink-jetNot printer. usefulWe have dev  Useful  Cancel anytime. Special offer for students: Only $4.99/month. Chapter 6 to address some of the applications of static fields. On student has mastered the fundamentals of static fields, he/she should


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it carries a current i (t ). More often than not, the relationship is without proof as follows: v=L

di dt

Someone with a discerning mind may have wondered about t gin of this equation. It is a consequence of a lifetime of work by Faraday (1791–1867) toward an understanding of a very comple nomenon called magnetic induction. We will begin our discussion of time-varying fields by Faraday’s law of induction and then explain how it led to t velopment of generators (sources of three-phase energy), moto workhorses of the industrialized world), relays (magnetic contr mechanisms), and transformers (devices that transfer electric e from one coil to another entirely by induction). One of the four known Maxwell equations is, in fact, a statement of Faraday’s induction. At this time it will suffice to say that Faraday’s law r the induced electromotive force (emf) e (t ) in a coil to the time-v You're Reading a Preview magnetic flux  (t ) linking that coil as Unlock full access with a free trial.

d  e = − Download With Free Trial dt

The significance of the negative sign ( Lenz’s law) and the deri of (1.7) from (1.8) will be discussed in detail in this text. We will also explain why Maxwell felt it necessary to Ampère’s law for time-varying fields. The inclusion of displac current (current through a capacitor) enabled Maxwell to predi fields should propagate in free space with the velocity of ligh modification of Ampère’s law is considered to be one of the mo nificant contributions by James Clerk Maxwell (1831–1879) in th of electromagnetic field theory.  and t Faraday’s law of induction, theupFree modified Amp` ere law, Read Foron 30 Days Sign to vote this title  Gauss laws (one for the time-varying electric field and the oth Not useful  Useful  Cancel anytime. the time-varying magnetic field) form a set of four equations; are now called Maxwell’s equations. These equations are


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1.7 Applications of time-varying fields

field (B), it experiences a force ( F ) such that 



F = q (E + u × B) 







We will refer to this equation as the Lorentz force equation. With the help of the four Maxwell equations, the equation of co nuity, and the Lorentz force equation we can now explain all the eff of electromagnetism.

1.7 Applications of time-varying fields .................................

Among the numerous applications of electromagnetic field theory will consider those pertaining to the transmission, reception, and pr agation of energy. This selection of topics is due to the fact that solution of Maxwell’s equations always leads to waves. The natur the wave depends upon the medium, the type of excitation (source), You're Reading a Preview the boundary conditions. The propagation of awith wave may Unlock full access a free trial.either be in an unbounded reg (fields exist in an infinite cross section, such as free space) or bounded region (fieldsWith existFree in a Trial finite cross section, such as a wave Download or a coaxial transmission line). Although most of the fields transmitted are in the form of sph cal waves, they may be considered as plane waves in a region far aw from the transmitter (radiating element, such as an antenna). How “far away” is depends upon the wavelength (distance traveled to c plete one cycle) of the fields. Using plane waves as an approximat we will derive wave equations from Maxwell’s equations in term electric and magnetic fields. The solution of these wave equations describe the behavior of a plane wave in an unbounded medium. W simplify the analysis by imposing restrictions such that (a) the wav  a uniform plane wave, (b) thereRead are no sources of currents and cha Free Foron 30 Days Sign up to vote this title  in the medium, and (c) the fields vary sinusoidally in time. We will t Useful Not useful  Cancel anytime. Special offer for students: Only $4.99/month. determine (i) the expressions for the fields, (ii) the velocity with w they travel in a region and (iii) the energy associated with them W


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We devote Chapter 9 to the discussion of transmission of ener one end to the other via a transmission line. We will show that one end of the transmission line is excited by a time-varying sour transmission of energy is in the form of a wave. The wave equ in this case will be in terms of the voltage and the current at any along the transmission line. The solution of these wave equation tell us that a finite time is needed for the wave to reach the other for practical transmission lines, the wave attenuates exponentiall the distance. The attenuation is due to the resistance and conduc of the transmission line. This results in a loss in energy along the length of the transmission line. However, at power frequencies 60 Hz) there is a negligible loss in energy due to radiation becau spacing between the conductors is extremely small in compariso the wavelength. As the frequency increases so does the loss of signal along the of the transmission line. At high frequencies, the energy is trans from one point to another via waveguides. Although any hollow tor can be used as a waveguide, the most commonly used wave You're Reading a Preview have rectangular or circular cross sections. We will examine th essary conditions thatwith must betrial. satisfied for the fields to exist, Unlock full access a free field expressions, and compute the energy at any point inside the guide. The analysisWith involves the solution of the wave equation Download Free Trial the waveguide subjected to external boundary conditions. The an is complex; thus, we will confine our discussion to a rectangular guide. Although the resulting equations appear to be quite involv difficult to remember, we must not forget that they are obtained b ply applying the boundary conditions to a general solution of the equation. A transmission line can be used to transfer energy from ver frequencies (even dc) to reasonably high frequencies. The wave on the other hand, has a lower limit on the frequency called the frequency. The cutoff frequency depends upon the dimensions waveguide. Signals below the cutoff cannotpropaga Read Free Foron 30this Days Sign up to frequency vote title  side the waveguide. Another between a tra Useful difference Not useful  major  Cancel anytime. sion line and a waveguide is that the transmission line can su the transverse electromagnetic (TEM) mode. In practice,


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Electromagnetic Field Theory Fundamentals

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