TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
RADAR SENSOR SYSTEMS FREQUENCY SYNTHESIS FREQUENCY CONVERSION
TECHNICAL MEMORANDUM: A SHORT TUTORIAL ON MAXWELL’S EQUATIONS AND
RELATED TOPICS Release Date: 2013
PREPARED BY:
KENNETH V. PUGLIA – PRINCIPAL 146 WESTVIEW DRIVE WESTFORD, MA 01886-3037 USA
STATEMENT OF DISCLOSURE THE INFORMATION WITHIN THIS DOCUMENT IS DISCLOSED WITHOUT EXCEPTION TO THE GENERAL PUBLIC. E X H CONSULTING SERVICES BELIEVES THE CONTENT TO BE ACCURATE; HOWEVER, E X H CONSULTING SERVICES ASSUMES NO RESPONSIBILITY WITH RESPECT TO ACCURACY OR USE OF THIS INFORMATION BY RECIPIENT. RECIPIENT IS ENCOURAGED TO REPORT ERRORS OR OTHER EDITORIAL CRITIQUE OF CONTENT.
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
TABLE OF CONTENTS PARAGRAPH
PAGE
PART 1 1.0 INTRODUCTION
4
2.0 CONTENT AND OVERVIEW
4
3.0 SOME VECTOR CALCULUS
6
PART 2 4.0 MAXWELL’S EQUATIONS FOR STATIC FIELDS
10
5.0 MAXWELL’S EQUATIONS FOR DYNAMIC FIELDS
14
6.0 ELECTROMAGNETIC WAVE PROPAGATION
17
PART 3 7.0 SCALAR AND VECTOR POTENTIALS
21
8.0 TIME VARYING POTENTIALS AND RADIATION
27
APPENDICES APPENDIX
Page
A
RADIATION FIELDS FROM A HERTZIAN DIPOLE
35
B
RADIATION FIELDS FROM A M AGNETIC DIPOLE
38
C
RADIATION FIELDS FROM A H ALF-WAVELENGTH DIPOLE
40
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
"WE HAVE STRONG REASON TO CONCLUDE THAT LIGHT ITSELF – INCLUDING RADIANT HEAT AND OTHER RADIATION, IF ANY – IS AN ELECTROMAGNETIC DISTURBANCE IN THE FORM OF WAVES PROPAGATED THROUGH THE ELECTRO-MAGNETIC FIELD ACCORDING TO ELECTRO-MAGNETIC LAWS." James Clerk Maxwell, 1864, before the Royal Society of London in 'A Dynamic Theory of the Electro-Magnetic Field'
"… THE SPECIAL THEORY OF RELATIVITY OWES ITS ORIGINS TO MAXWELL'S EQUATIONS OF THE ELECTROMAGNETIC FIELD …" "… SINCE MAXWELL'S TIME, PHYSICAL REALITY HAS BEEN THOUGHT OF AS REPRESENTED BY CONTINUOUS FIELDS, AND NOT CAPABLE OF ANY MECHANICAL INTERPRETATION. THIS CHANGE IN THE CONCEPTION OF REALITY IS THE MOST PROFOUND AND THE MOST FRUITFUL THAT PHYSICS HAS EXPERIENCED SINCE THE TIME OF NEWTON …" ALBERT EINSTEIN
"…MAXWELL'S IMPORTANCE IN THE HISTORY OF SCIENTIFIC THOUGHT IS COMPARABLE TO EINSTEIN'S (WHOM HE INSPIRED) AND TO NEWTON'S (WHOSE INFLUENCE HE CURTAILED)…" MAX PLANCK
"… FROM A LONG VIEW OF THE HISTORY OF MANKIND - SEEN FROM, SAY TEN THOUSAND YEARS FROM NOW – THERE CAN BE LITTLE DOUBT THAT THE MOST SIGNIFICANT EVENT OF THE 19TH CENTURY WILL BE JUDGED AS MAXWELL'S DISCOVERY OF THE LAWS OF ELECTRODYNAMICS …" RICHARD P. FEYNMAN
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
1.0 INTRODUCTION Given the accolades of such prestigious scientists, it is prudent to periodically revisit the works of genius; particularly when that work has made such a profound scientific and humanitarian contribution. Over the years, I have been intensely fascinated by the totality of Maxwell’s Equations. Part of the attraction is the extent of features and aspects of their physical interpretation. It is still somewhat surprising to me that four ostensibly innocuous equations could so completely encompass and describe – with the exception of relativistic effects – all electromagnetic phenomenon. Herein was the motivation for this investigation: a more intuitive understanding of Maxwell’s Equations and their physical significance. One of the significant findings of the investigation is the extraordinary application uniqueness of vector calculus to the field of electromagnetics. In addition, I was reminded that our modern approach to circuit theory is, in reality, a special case – or subset – of electromagnetics, e.g., the voltage and current laws of Kirchhoff and Ohm, as well as the principles of the conservation of charge, which were established prior to Maxwell’s extensive and unifying theory and documentation in “A Treatise on Electricity and Magnetism” in 1873. Although not immediately recognized for its scientific significance, Maxwell’s revelations and mathematical elegance was subsequently recognized, and in retrospect, is appreciated – one might say revered – to a greater extent today with the benefit of historical perspective. James Clerk Maxwell (1831-1879), a Scottish physicist and mathematician, produced a mathematically and scientifically definitive work which unified the subjects of electricity and magnetism and established the foundation for the study of electromagnetics. Maxwell used his extraordinary insight and mathematic proficiency to leverage the significant experimental work conducted by several noted scientists, among them:
Charles A. de Coulomb (1736-1806): Measured electric and magnetic forces. André M. Ampere (1775-1836): Produced a magnetic field using current – solenoid. Karl Friedrich Gauss (1777-1855): Discovered the Divergence theorem – Gauss’ theorem – and the basic laws of electrostatics. Alessandro Volta (1745-1827): Invented the Voltaic cell. Hans C. Oersted (1777-1851): Discovered that electricity could produce magnetism. Michael Faraday (1791-1867): Discovered that a time changing magnetic field produced an electric field, thus demonstrating that the fields were not independent.
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Completing the sequence of significant events in the history of electromagnetic science:
James Clerk Maxwell (1831-1879): Founded modern electromagnetic theory and predicted electromagnetic wave propagation. Heinrich Rudolph Hertz (1857-1894): Confirmed Maxwell’s postulate of electromagnetic wave propagation via experimental generation and detection and is considered the founder of radio.
I hope you enjoy and benefit from this brief encounter with Maxwell’s work and that you subsequently acknowledge and appreciate the profound contribution of Maxwell to the body of scientific knowledge. 2.0 CONTENT AND OVERVIEW The exploration begins with a review of the elements of vector calculus, which need not cause mass desertion at this point of the exercise. The topic is presented in a more geometric and physically interpretive manner. The concepts of a volume bounded by a closed surface and an open surface bounded by a closed contour are utilized to physically interpret the vector operations of divergence and curl. Gauss’ law and Stokes theorem are approached from a mathematical and physical interpretation and used to relate the differential and integral forms of Maxwell’s equations. The myth of Maxwell’s ‘fudge factor’ is dispelled by the resolution of the contradiction of Ampere’s Law and the principle of conservation of charge. Various forms of Maxwell’s equations are explored for differing regions and conditions related to the time dependent vector fields. Maxwell’s observation with respect to the significance of the E-field and H-field symmetry and coupling are mathematically expanded to demonstrate how Maxwell was able to postulate electromagnetic wave propagation at a specific velocity – ONE OF THE MOST PROFOUND SCIENTIFIC DISCLOSURES TH OF THE 19 CENTURY . The investigation concludes with
the development of scalar and vector potentials and the significance of these potential functions in the solution of some common problems encountered in the study of electromagnetic phenomenon. The presentation will consider only simple media. Simple media are homogeneous and isotropic. Homogeneous media are specified such that r and r do not vary with position. Isotropic media are characterized such that r and r do not vary with magnitude or direction of E or H. Therefore: r and r are constants. Vectors are conventionally represented with arrows at the top of the letter representing the vector quantity, e.g. A .
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS The International System of Units, abbreviated SI, is used. A summary of the various scalar and vector field quantities and constants and their dimensional units are presented in Table I. Recognition of the dimensional character of the various quantities is quite useful in the study of electromagnetics.
The study of electromagnetics begins with the concept of static charged particles and continues with constant motion charged particles, i.e., steady currents, and discloses more significant consequential results with the study of time variable currents. Faraday was the first to observe the results of time varying currents when he discovered the phenomenon of magnetic induction.
Table I. Field Quantities, Constants and Units PARAMETER
Magnetic Flux Density
SYMBOL E D H B
Tesla (Weber/meter2)
B H
Conduction Current Density
Jc
Ampere/meter2
J c E
Displacement Current Density
Jd
Ampere/meter2
D Jd t
Magnetic Vector Potential
A
Volt-Second/meter
B A
Conductivity
Siemens/meter
Voltage
V
Volt
Current
I
Ampere
Power
W
Watt
Capacitance
F
Farad
Inductance
L
Henry
Resistance
Ω
Ohm
Permittivity (free space)
Farad/meter
8.85 1012
Permeability (free space)
Henry/meter
4 107
Speed of Light
c
meter/second
Free Space Impedance
Ohm
Poynting Vector
P
Watt/meter2
Electric Field Intensity Electric Flux Density Magnetic Field Intensity
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DIMENSIONS
NOTE
Volt/meter
D E
Coulomb/meter2 Ampere/meter
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Siemen 1 Volt Joule
Ohm
Coulomb
Ampere Coulomb Watt Joule
Second
Amp Volt
Second
Farad Coulomb Henry Volt Second
Ohm Volt
c
1
o o
o
Volt
2
Coulomb
Ampere 109 36
3.0 108
o 120 o
P EH
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS 3.0 SOME VECTOR CALCULUS “Much of vector calculus was invented for use in electromagnetic theory and is ideally suited to it”.1 Vector calculus uniquely describes electromagnetic phenomenon in a concise and almost elegant manner. All engineering students have had an introduction to vector analysis to the extent of addition, dot (·) and cross () products of vectors. These are operations that are included within the study of vector algebra. However, with the more advanced differential vector operations of gradient ( A ), divergence ( A ) and curl ( A ) – and their complementary operations of integration – one must expand and embrace the three dimensional quality of vector calculus. The nature of electromagnetic fields embodies both spatial position and time. Unfortunately, this can be overwhelming to the student upon an initial encounter with the study of electromagnetic fields. In addition to the formal mathematical definitions of the vector differential operators, an intuitive explanation is offered in the following material. In the following discussion, there are references to volumes bounded by closed surfaces and open surfaces bounded by closed contours. Those references are defined graphically in Figure 3.1.
In some cases, the geometric references are imaginary and only serve to define other concepts and provide visual clarification. In other cases, the geometric references provide definition to conductors, dielectrics and further define specific spatial relationships. Verbal definitions of the vector differential operators – gradient, divergence and curl – are as follows: gradient of a scalar field (T) is a directional derivative vector that represents the magnitude and direction of the maximum space rate of change of the scalar field, T. Room temperature and landscape elevation are examples of three dimensional scalar quantities for which calculation of a gradient may be required.
The
The
Divergence of a vector field is a spatial derivative, scalar value that represents the outward flux2 of the vector field at a point. The divergence of a vector field is a measure of the spreading of a vector field at a point.
The
curl of a vector field is a spatial derivative vector with magnitude equal to the strength of field rotation and direction normal to the surface that maximizes the rotation at a point. The curl operation is a measure of the field rotation at a point or at a surface.
The formal mathematical definitions of the vector differential operators follow. GRADIENT:
T Figure 3.1-a: Volume Bounded by a Closed Surface
T T T ux uy uz x y z
The scalar function T(x,y,z) is differentiated with respect to the constituent associated variables. The partial differential in each case is multiplied by the corresponding unit vector. DIVERGENCE:
A A A A x y z x y z
Figure 3.1-b: Open Surface Bounded by a Closed Contour
The respective components of the vector A Axux Ayu y Azuz are differentiated with respect to the associated variable. The divergence
1
Schey, H. M., div grad curl and all that, 3rd ed., W. W. Norton & Co., New York, 1997.
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2
Flux is Latin for flow.
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS operator indicates differentiation with respect to x of the x-component of the field, differentiation with respect to y of the y-component of the field and differentiation with respect to z of the z-component of the field; therefore, to have a non-zero divergence, the field must vary in magnitude along a line having the same direction as the field. This concept is graphically illustrated in Figure 3.2 where a vector field is shown and exhibits a non-zero divergence.
Figure 3.3: Closed Surface (sphere) Enclosing the Source of the Vector Field A .
Figure 3.2: Vector Field Exhibiting Non-Zero Divergence
In the interest of completeness and to satisfy the mathematical traditionalists amongst the readers, the more formal definition of the vector divergence operation is tendered:
A lim
v 0
A ds s
v
The formal definition of divergence of the vector field A at a point is the net outward flux (flow) of A per unit volume as the volume approaches zero. The circle about the surface integral sign indicates that the integral is to be executed over the entire closed surface S that bounds the volume v. What should be noted here is that as the volume approaches zero, the formula applies at a point. Further, the dot-product of the vector field with the differential surface represents the flux of that vector field over the incremental surface. To attain a more physical interpretation, consider Figure 3.3 where a closed surface – a sphere in this case – encloses a charge, q+, which is the source of the vector field A .
The divergence theorem, attributed to Gauss and also known as Gauss’ theorem, is an important and useful identity in vector calculus and may be obtained with a little manipulation of the divergence definition by integrating the differential volume.
A dv A ds v
s
Simply stated, the volume integral of the divergence of a vector field is equal to the net outward flux of the vector field over the closed surface that bounds the volume. Another significant tool provided by the divergence theorem is that a volume integral of the divergence of a vector field may be converted to a surface integral of the vector and vice versa. As subsequently shown, Gauss’ theorem is also utilized to relate the differential and integral forms of Maxwell’s divergence equations. CURL: The curl operation is more commonly and concisely written:
Az Ay u x A y z A A x z u y x z Ay Ax u z x y In matrix notation:
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
ux A x Ax
uy y Ay
uz z Az
The curl operator indicates differentiation with respect to x of the y- and z-components of the vector, differentiation with respect to y of the x- and zcomponents of the vector and differentiation with respect to z of the x- and y-components of the vector. Therefore, to have a non-zero curl, a vector must vary in magnitude along a line normal to the direction of the field. This concept is graphically illustrated in Figure 3.4 where a vector field exhibits a non-zero curl and zero divergence.
Figure 3.5: Illustration of the Vector Differential Curl Operation
The curl of the vector field A is a vector with magnitude equal to the strength of rotation at the point and with direction normal to the plane of the surface, as the surface tends to zero. Because the surface tends to zero, the curl is defined as a vector point function.
Note: if the vector field has no rotation at a point, the curl of the vector field at that point is zero; in other words: Figure 3.4: Field with Non-Zero Curl and Zero Divergence.
Once again, to present the formal mathematical definition of the differential vector curl operation, the following formula is offered:
A lim
s 0
A dl a c
s
NO CURL
As was the case with the divergence and the theorem attributed to Gauss, another perspective may be gained from the fundamental theorem associated with the vector curl operation, more widely known as Stokes’ theorem and written mathematically:
A d s A dl
n
s
The verbal definition states that the curl of a vector field A , denoted by A , is a vector that results from the closed integral of the dot product of the vector with the closed contour that bounds the open surface of the plane of the vector as the surface approaches zero, i.e. at a point. The magnitude is equal to the maximum net circulation of A per unit area as the area tends to zero and with direction normal to the surface. Because the normal vector to a surface may point in one of two directions, the righthand rule is utilized to indicate positive curl. Unfortunately, this definition provides little intuitive insight; therefore, a more physical definition is attempted with the aid of Figure 3.5 whichdepicts a point, P, lying in the plane of a vector field A .
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NO ROTATION
c
The verbal definition of Stokes’ theorem may be stated as follows: The flux of the curl of a vector field over a surface is equal to the total rotation of the vector around the closed contour that bounds the surface. Stokes’ theorem also provides a relationship between a line integral around a closed contour and the flux of the curl through the surface bounded by the contour. As subsequently demonstrated, Stokes’ theorem is also utilized to relate the differential and integral forms of Maxwell’s curl equations. Another interesting observation from Stokes’ theorem is that the left-hand term defines an integral of a differential ( A ) over a region – the surface – and is equal to the integration of the function along the boundary – the contour that defines the surface. Stated in elementary (first year) calculus texts: the integral of a differential is the value of the function at the boundary; remember:
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
A ds lim S s 0
b
d b dx f x dx f x a a
s
As a proof of Stokes’ theorem, consider Figure 3.6 where an open surface is bounded by a closed contour.
A dl ds c
s
A dl c
Q.E.D. (admittedly with some degree of dispensation) Before proceeding, there are three vector identities that facilitate the manipulation, simplification and solution of otherwise intractable problems. Specifically, the following vector identities are invaluable and are used extensively in electromagnetic theory and problem solving:
V 0 A 0 A A 2 A
Figure 3.6: Proof of Stokes’ Theorem
The surface is subdivided into a number of incremental areas, Sn, for which the flux of the vector A , may be written at each point:
A n sn
In the limit as the incremental area tends to zero, one may write:
lim
sn 0
A n sn A ds
n
s
Similarly taking the closed line integral around each incremental area, one may write:
lim
l 0
A l
n
n
A dl c
From the mathematical definition of the differential vector curl operator, the right sides of each equation are equal. A more intuitive – heuristic – approach involves direct integration of the definition of the vector curl operation:3
A lim
s 0
A dl c
s
The first identity, expressed as the curl of the gradient of a scalar field is equal to zero. This may be immediately proven simply by execution of the indicated vector operators in Cartesian coordinates. However, that would provide no intuitive or physical understanding. Another approach employs Stokes’ theorem which stipulates that the surface integral over an open surface is equal to the line integral of the contour that bounds the surface; written mathematically:
s V ds c V dl Intuitively, one may write:
c V dl
0
Recalling from the gradient – defined as the maximum space rate of change of a scalar field – that if we integrate or sum all the vector changes around a closed path, the net change is zero. For example, suppose that you are hiking a mountainous region and that you continuously sample the maximum directional change in altitude as you traverse a path that brings you back to the starting point. Intuitively, and in fact, the net change in altitude is zero. Recall that the gradient represents the magnitude and direction of the rate of change of the scalar field at a point. If each of the gradient points is indexed ( An ) and multiplied by a similarly indexed incremental line ( Ln ) which connects the points and the product subsequently summed, an equation may be written for the path length:
Path LimLn 0 An Ln A dL n
3
Recall that in the limit as S→0, S→dS.
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS If the start and end points of the path are the same, the equation may be written as a closed-path integral:
Path A dL 0 Q. E. D.
A voltage analogy asserts that the total voltage around a closed loop is zero. A corollary to the vector identity V 0 states that if a vector field is curl-free, then that vector field may be expressed as the gradient of a scalar field. Figure 3.7: Volume Separated at a Common Path
If E 0 , then E V
The choice of E and V is not without significance as will be demonstrated.4 The second identity, which asserts that the divergence of the curl of a vector field is equal to zero, may also be proven via execution of the indicated vector operators in Cartesian coordinates, and once again no physical or intuitive experience is gained; however, using the divergence theorem, one may write:
A dv A ds v
s
Invoking Stokes theorem on the right-hand term yields:
A dv A ds A dl v
s
c
A corollary to the vector identity A 0 states that if a vector field is divergence-free, then that vector field may be expressed as the curl of another vector field.
If B 0 , then B A The choice of B and A is not without significance as will be demonstrated.
The proof of the third identity may also be demonstrated but is more often utilized for the definition of the vector Laplacian. Rewriting the identity:
A A 2 A 2 A A A
Note that we have related the divergence from a volume of the curl of a vector field to the flux of the curl of a vector field over a surface and to the closed path line integral of the same vector field – all terms of which are equal to zero.
Expanding the right-hand term results in the definition of the vector Laplacian:
Consider the graphic of Figure 3.75 where the closed volume has been separated into two open surfaces bounded by the closed integration paths. Because the flux of the curl out of the surface – or the divergence of the curl from the volume – is normal to the surfaces and pointing outward, the paths of integration must be opposite according to the righthand rule; and since it is a common path in opposite directions, the net result is zero.
Clearly, the vector Laplacian represents the second derivative of the respective constituent components of the vector field and mathematically resembles a combination of a divergence and gradient operation. As will be demonstrated in a later section, the vector Laplacian serves a primary function in the development of the vector wave equations.
2 A ax2 Ax a y2 Ay az2 Az
Although not comprehensive in development or presentation, this brief review of vector calculus should be sufficient for the exploration and understanding of Maxwell’s Equations and with this background material completed, the investigation may begin. END OF PART 1
4
The relationship between the gradient of the scalar potential, V, and the electric field intensity vector, E, applies to the static case only. 5
The example is from Cheng [1]
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS ACKNOWLEDGEMENT The author gratefully acknowledges Dr. Tekamul Büber for his diligent review and helpful suggestions in the preparation of this tutorial, and Dr. Robert Egri for suggesting several classic references on electromagnetic theory and historical data pertaining to the development of potential functions. The tutorial content has been adapted from material available from several excellent references (see list) and other sources, the authors of which are gratefully acknowledged. All errors of text or interpretation are strictly my responsibility. AUTHOR’S NOTE This investigation began some years ago in an informal way due to a perceived deficiency acquired during my undergraduate study. At the conclusion of a two semester course in electromagnetic fields and waves, my comprehension of the material was vague and not well integrated with other parts of the electrical engineering curriculum. In retrospect, I was unable to envision and correlate the relationship of the EM course material with other standard course work, e.g. circuit theory, synthesis, control and communication systems. It was not until sometime later that I realized the value of EM theory as the basis for most electrical principles and phenomenon. In addition to my mistaken belief of EM theory as an abstraction, the profound contribution of Maxwell – and others of his period and later – to the body of scientific knowledge could hardly be acknowledged and appreciated. Experimentation – as demonstrated by Ampere and Faraday – advances the art; while Maxwell’s intellect and proficiency in applied mathematics and imagination, has yielded a unified theory and initiated the scientific revolution of the 20th century.
[5] Sadiku, M. N. O., Elements of Electromagnetics, 3rd ed., Oxford University Press, New York, 2001. [6] Paul, C. R., Whites, K. W., and Nasar, S. A., Introduction to Electromagnetic Fields, 3rd ed., McGraw-Hill, New York, 1998. [7] Feynman, R. P., Leighton, R. O., and Sands, M., Lectures on Physics, vol. 2, AddisonWesley, Reading, MA, 1964. [8] Maxwell, J. C., A Treatise on Electricity and Magnetism, Vol. 1, unabridged 3rd ed., Dover Publications, New York, 1991. [9] Maxwell, J. C., A Treatise on Electricity and Magnetism, Vol. 2, unabridged 3rd ed., Dover Publications, New York, 1991. [10] Harrington, R. F., Introduction to Electromagnetic Engineering, Dover Publications, New York, 2003. [11] Schey, H. M., div grad curl and all that, 3rd ed., W. W. Norton & Co., New York, 1997. Maxwell’s original “Treatise on Electricity and Magnetism” is available on-line: http://www.archive.org/details/electricandmagne01maxwrich http://www.archive.org/details/electricandmag02maxwrich
REFERENCES [1] Cheng, D. K., Fundamentals of Engineering Electromagnetics, Prentice Hall, Upper Saddle River, New Jersey, 1993. [2] Griffiths, D. J., Introduction to Electrodynamics‡, 3rd ed., Prentice Hall, Upper Saddle River, New Jersey, 1999. [3] Ulaby, F. T., Fundamentals of Applied Electromagnetics, 1999 ed., Prentice Hall, Prentice Hall, Upper Saddle River, New Jersey, 1999. [4] Kraus, J. D., Electromagnetics, 4th ed., McGraw-Hill, New York, 1992.
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