Tutorial on Maxwell's Equations- Part 1

TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS

– REVISION 02 JANUARY 2013 – of 11 E X H CONSULTING SERVICES

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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

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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 MAGNETIC DIPOLE 38

C RADIATION FIELDS FROM A HALF-WAVELENGTH DIPOLE 40



"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



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.

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

OF THE 19TH

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

.



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 SYMBOL DIMENSIONS NOTE

Electric Field Intensity E

Volt/meter

Electric Flux Density D

Coulomb/meter2 ED

Magnetic Field Intensity H

Ampere/meter

Magnetic Flux Density B

Tesla (Weber/meter2) HB

Conduction Current Density cJ

Ampere/meter2 EJc

Displacement Current Density dJ

Ampere/meter2

t

DJd

Magnetic Vector Potential A

Volt-Second/meter AB

Conductivity Siemens/meter OhmSiemen 1

Voltage V Volt CoulombJouleVolt

Current I Ampere SecondCoulombAmpere

Power W Watt VoltAmpSecond

JouleWatt

Capacitance F Farad VoltCoulombFarad

Inductance L Henry CoulombSecondVoltHenry

2

Resistance Ω Ohm AmpereVoltOhm

Permittivity (free space) Farad/meter 36

101085.8

912

Permeability (free space) Henry/meter 7104

Speed of Light c meter/second 8100.3

1

oo

c

Free Space Impedance Ohm

120

o

oo

Poynting Vector P

Watt/meter2 HEP



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.

Figure 3.1-a: Volume Bounded by a Closed Surface

Figure 3.1-b: Open Surface Bounded by a Closed Contour

1 Schey, H. M., div grad curl and all that, 3rd ed., W. W. Norton &

Co., New York, 1997.

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:

The 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 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:

zyx uz

Tu

y

Tu

x

TT

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:

z

A

y

A

x

AA zyx

The respective components of the vector

zzyyxx uAuAuAA

are differentiated with

respect to the associated variable. The divergence

2 Flux is Latin for flow.



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 3.2: 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:

v

sdAA s

v

lim0

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

.

Figure 3.3: Closed Surface (sphere) Enclosing 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.

sv

sdAdvA

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:

zxy

yzx

x

yz

uy

A

x

A

ux

A

z

A

uz

A

y

AA

In matrix notation:



zyx

zyx

AAA

zyx

uuu

A

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 z-

components 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 3.4: 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:

n

c aldA

Ass

lim 0

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 right-

hand 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 which depicts a

point, P, lying in the plane of a vector field A

.

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:

NO ROTATION 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:

cs

ldAsdA

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:



xfb

a

b

a

dxxfdx

d

As a proof of Stokes’ theorem, consider Figure 3.6

where an open surface is bounded by a closed

contour.

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:

nn sA

In the limit as the incremental area tends to zero, one

may write:

s

n

nn ss dAAns

0lim

Similarly taking the closed line integral around each

incremental area, one may write:

c

n

nl

ldAlA

0lim

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

s

cldA

As

lim 0

3 Recall that in the limit as S→0, S→dS.

cS

c

sldAd

ldAdA s

ss

s

lim

0

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:

AAA

A

V

2

0

0

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:

ldVdVcs

s

Intuitively, one may write:

0 ldVc

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 (nA )

and multiplied by a similarly indexed incremental

line (nL

) which connects the points and the product

subsequently summed, an equation may be written

for the path length:

LdALALimPathn

nnLn

0



If the start and end points of the path are the same,

the equation may be written as a closed-path integral:

D. E. Q. 0 LdAPath

A voltage analogy asserts that the total voltage around a closed loop is zero.

A corollary to the vector identity 0 V states

that if a vector field is curl-free, then that vector field

may be expressed as the gradient of a scalar field.

VEE

then , 0 If

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:

sv

sdAdvA

Invoking Stokes theorem on the right-hand term

yields:

ldAdAdvAcsv

s

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.

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 right-

hand rule; and since it is a common path in opposite

directions, the net result is zero.

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]

Figure 3.7: Volume Separated at a Common Path

A corollary to the vector identity 0 A

states that if a vector field is divergence-free, then

that vector field may be expressed as the curl of

another vector field.

AB then ,B If

0

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:

AAA

2

AAA

2

Expanding the right-hand term results in the

definition of the vector Laplacian:

zzyyxx AaAaAaA 2222

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.

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



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.

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.

[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, Addison-

Wesley, 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


[10] Harrington, R. F., Introduction to

Electromagnetic Engineering, Dover


[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

http://www.archive.org/details/electricandmagne01maxwrich

http://www.archive.org/details/electricandmag02maxwrich

Tutorial on Maxwell's Equations- Part 1

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