Tutorial de Ecuaciones de Maxwell

TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS

– REVISION 02 JANUARY 2013 – Page 1 of 13 E X H CONSULTING SERVICES

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.



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 16

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 34

B RADIATION FIELDS FROM A MAGNETIC DIPOLE 37

C RADIATION FIELDS FROM A HALF-WAVELENGTH DIPOLE 39



"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



PART 2

4.0 MAXWELL’S EQUATIONS FOR STATIC FIELDS

The exploration of Maxwell’s Equations begins with the

study of static electricity, i.e. the study of electrically

charged particles at rest. The simplest example is that of a

single charge of value +q at the center of an imaginary sphere of radius r as shown in Figure 4.1.

Figure 4.1: Charged Particle at the Center of Sphere

The electric field intensity vector E

, at the surface of an

imaginary sphere of radius r may be written using

Coulomb’s law:

Volt/meter 4 2

rr

qE

o

In this case, the electric field is in a radial direction from

the charge – in accordance with the unit vector ( r

) – and is directly proportional to the value of charge and

inversely proportional to the square of the distance

between the charge and the observation point on the

surface of the sphere. Although discrete lines are depicted

to indicate the electric field intensity direction, one

should recognize that the electric field intensity is

continuous over the surface of the imaginary sphere. The

electric flux density vector is simply:

2terCoulomb/me

4 2r

r

qED o

Using the Divergence theorem:

vs

dvDdD s

vvs

ss

dvDdvqdD

rddrd

rddrrr

qdD

vs

s

s

sin where

sin4

2

2

2

vD

This significant result is Maxwell’s first equation! The

equation states that the divergence of the electric flux

density over a closed surface that bounds a volume is

equal to the enclosed volume charge density, v.

Noting that the electric field intensity vector has a single

radial direction, one may conclude that the field has no rotational component and therefore:

E 0

(for the static case)

The result is Maxwell’s third equation and applies to the

static case.

Now consider a small magnet enclosed within an

imaginary sphere of radius, r, as shown in Figure 4.2.

Also illustrated are the magnetic flux density lines

emanating from one pole of the magnet and terminating at the opposite pole. Observing from an additional

perspective, the magnetic flux density lines form closed

loops around the poles of the magnet.

Figure 4.2: Magnet Enclosed within Imaginary Sphere

Because the net magnetic flux density over the surface of

the sphere is zero, using the divergence theorem one may

write:

0 vsdvBdB s



0 B

The significant result in this case is the second of Maxwell’s Equations and is valid in all cases. If one

considers the divergence of the electric flux density

vector, where the source of the electric flux density

vector was found to be the enclosed electric charge, this

is an intuitively satisfying result since magnetic charges

have not been found to exist. Once again, although the

discrete magnetic flux density lines are illustrated, one

should recognize that the magnetic flux density is

continuous over the surface of the imaginary sphere.

AMPERE’S LAW

Ampere’s law states that the line integral of the tangential

component of the magnetic intensity vector around a

closed path is equal to the net current enclosed by the

path. Figure 4.3 illustrates the geometry using a circular

path; however, the law also applies to an arbitrary path.

Figure 4.3: Illustration of Ampere’s Law

Mathematically, one may write Ampere’s Law:

encc

IldH

Applying Stokes’ theorem:

encc

IdHldHs

s

The enclosed current may be written as a density:

s

enc SdJI

Now equating the integrands of the surface integrals:

JH

This significant result is the fourth of Maxwell’s

Equations for the static case. The equation also provides

three observations of intuitive significance with respect to

the vector curl operation.

1. The plane of the circumferential magnetic

intensity vector is normal to the current density

vector. This is an expectation derived directly

from the curl operation.

2. The curl of the magnetic intensity vector is the

current density vector which is the source of the magnetic intensity vector. In a manner of

speaking, the curl of the magnetic intensity

vector finds its source.

3. Because the magnetic field intensity is

circumferential, i.e. rotational, one may

conclude that it has a non-zero curl.

BIOT-SAVART LAW

The Biot-Savart law provides a mathematical statement

of the Oersted observation that compass needles are

deflected in the presence of current carrying wires. The

Biot-Savart law may be written and interpreted with the

aid of Figure 4.4.

Figure 4.4: Geometry of Biot-Savart Law

The Biot-Savart law asserts that the incremental magnetic

intensity Hd

, produced at a point, P, from an

incremental current element lId

, is directly proportional

to the current and inversely proportional to the square of the distance from the current element to the observation

point. In addition, the direction of the magnetic intensity

is that of the cross product of the incremental length and

the unit vector along the line to the observation point. As

one would expect, the direction of the incremental field is

normal to the plane formed by the incremental current

element and the unit vector from the current element to

the observation point.

24

aldIHd

This is the magnetic equivalent of Coulomb’s law and

may be utilized to gain additional insight with respect to



the vector curl operation in the following manner.

Postulate a circular loop of radius , with current I and located at the origin of the coordinate system in the Z=0

plane; the problem is to calculate the magnetic intensity

vector at the center of the loop. With respect to Figure

4.5, the solution requires closed contour integration of the

lId

product around the circumference of the loop.

Figure 4.5: Magnetic Field Intensity at the Origin of a Current

Loop in the Plane, Z=0.

The following equations may be written:

loop theof origin at the

and

2

44

2

0

2

0 2

z

zz

z

aI

H

dI

adI

aH

aaaaadald

In summary, Maxwell’s Equations for the static case are

summarized in Table II.

Table II: Maxwell’s Equations for Static Electromagnetic Fields

Equation

Number

Differential

Form Integral Form Comment

1. vD

.encvs

QdvdD vs

Gauss’s Law; divergence finds the

source of the electric field vector, Qenc

2. 0 B

0s sdB

Gauss’s law for magnetic flux density

3. 0 E

0 ldEc

No rotation of the static electric field

intensity

4. cJH

.enccc

IdJldHs

s

Ampere’s law; curl finds the source of

the magnetic intensity vector, Ienc

5. sv

sdAdvA

Divergence theorem relates a volume integral to a surface integral

6. ldAdAcs

s

Stokes’ theorem relates a surface integral to a line integral

A short commentary describes a significant attribute pertinent to each equation.

1. The divergence of the electric flux density vector

over a surface is equal to the volume charge

density as the surface tends to zero; the integral of

the electric flux density vector over a closed

surface is equal to the charge enclosed by the

surface.

2. The divergence of the magnetic flux density vector

is zero; field lines form closed loops; magnetic

charges do not exist.

3. A static electric field has no rotation.

4. The flux of the curl of the magnetic intensity

vector over an open surface is equal to the current

density vector through the surface that is bounded

by the closed contour; the line integral of the

magnetic intensity vector around closed path that

bounds an open surface is equal to the current

through the surface.

5. The divergence of a vector field from a closed

volume is equal to the integral of the vector field

over the closed surface that bounds the volume;



relates a volume integral to a surface integral;

relationship between the differential and integral

forms of Maxwell’s divergence equations.

6. The integral of the curl of a vector field over an

open surface is equal to the line integral of the

vector field along the closed path that bounds the

open surface; relates a surface integral to a line

integral; relationship between the differential and

integral forms of Maxwell’s curl equations.

What has been demonstrated to this point in the investigation is that static charges produce electrostatic

fields and that constant velocity charges or steady currents

produce magneto-static fields. In the next section, what

will be demonstrated is that time varying currents produce

electromagnetic fields and waves.

5.0 MAXWELL’S EQUATIONS FOR DYNAMIC FIELDS

Dynamic electromagnetic fields are those fields that vary with time. As will be demonstrated, the electric and

magnetic field intensity vectors must exist simultaneously

under dynamic conditions. Maxwell was the first to

discover this phenomenon and mathematically pursued

the electric and magnetic field coupling to the

electromagnetic wave propagation conclusion. Maxwell

was an accomplished mathematician and used his talent to

integrate the experimental results of Ampere and Faraday

into a concise mathematical formulation. Some passages

from Maxwell [9.] chastise Ampere and Faraday for their

lack of mathematical diligence beyond the experiments.

MAXWELL WRITES ON FARADAY:

“The method which Faraday employed in his researches consisted in a constant appeal to experiment as a means of

testing the truth of his ideas and a constant cultivation of ideas under the direct influence of experiment. In his published researches we find these ideas expressed in a language which is all the better fitted for a nascent science because it is somewhat alien from the style of physicists who have been accustomed to establish mathematical forms of thought.”

MAXWELL WRITES ON AMPERE:

“The method of Ampere, however, though cast into an

inductive form, does not allow us to trace the formation of the ideas which guided it. We can scarcely believe that Ampere really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shewn us, and that when he had afterwards built up a perfect demonstration he removed all traces of scaffolding by which he had raised it.”

MAXWELL WRITES ON HIS WORK:

“It is mainly with the hope of making these ideas the basis of a mathematical method that I have undertaken this treatise.

FARADAY’S LAW OF INDUCTION

In 1831, Michael Faraday discovered that a time varying

magnetic field produces an electromotive force (emf) in a closed path that is coupled or otherwise linked to the time

varying magnetic field. Stated mathematically:

volts dt

demf

A time varying magnetic field may result from the

following factors:

1. Time changing field linking a stationary closed

path

2. Relative motion between a steady field and a

closed path

3. A combination of the above

The emf may also be defined in terms of the integration along a closed path:

volts ldEemf

Replacing the time varying flux density with the magnetic

flux density vector, one may write:

s

sdBdt

dldEemf

Recall Stokes’ theorem:

s

sdEldEc

Equating the integrands of the surface integrals:

t

BE

This is Maxwell’s first curl equation in differential form

for the time varying condition. It should be noted that the

source of electric field intensity vector is the time

changing magnetic flux density1 vector and that the curl

of the electric field intensity vector is the same direction as the changing magnetic flux density vector.

1 Recall that the curl of a vector field finds its source.



PRINCIPLE OF THE CONSERVATION OF CHARGE

The principle of charge conservation states that the time

rate of decrease of charge within a volume must be equal

to the net rate of current flow through the closed surface

that bounds the volume; mathematically:

IdJdt

dQ

ssc

Applying the divergence theorem to the above equation

yields:

vs

dvJdJ sc

Writing the equation for the charge leaving a volume:

v

dvdt

d

dt

dQv

By simple substitution, one may write:

vv

dvdt

ddvJ

dt

dv

tJ v

This is the equation of continuity of current which

basically asserts that there can be no accumulation of

current at a point and is the basis of Kirchhoff’s current

law. In the physical sense, the divergence of the current

density is equal to the time rate of decrease in volume charge density. Recall that for the static case, Maxwell’s

equation for the curl of the magnetic intensity vector –

Ampere’s law – found the source to be the conduction

current density vector:

JH

Executing the divergence of the above equation:

0 JH

This is a clear contradiction because it was just

demonstrated that the divergence of the current density

was equal to the time rate of change of volume charge

density.

Maxwell recognized the contradiction under time varying

conditions and mathematically reconciled the

inconsistency in the following manner:

Dtt

JJ

JJH

vd

d

0

t

DJ d

The displacement current term is the result of a time

varying electric field a typical example of which is the

current within a capacitor when an alternating voltage is

impressed across the plates.

Maxwell’s second curl equation may now be written:

t

DJH

The current within a capacitor provides a unique example

of the displacement current concept and also serves to

equate the conduction and displacement currents in a

typical circuit. Figure 5.1 illustrates an impressed AC

voltage across the plates of a capacitor.

Figure 5.1-1:

Figure 5.1-2: Displacement Current Example

Two surfaces are utilized with a common closed loop

integration path. Because the path of integration is common, the current must be equal regardless of the

surface.

Table III summarizes the general form of Maxwell’s

equations for time varying electromagnetic fields. Notice

that the general form of the equations is also applicable to

the static case upon removal of the time dependence. The

integral forms of Maxwell’s equations illustrate well the

underlying physical significance while the differential, or

point form, are utilized more frequently in problem

solving. I have included a fifth equation – the continuity

equation – that is not normally represented as one of

Maxwell’s equations, however, the basic principle,



significance and relevance to the final form cannot be

overemphasized.

Table III: Maxwell’s Equations for Dynamic Electromagnetic Fields

Equation

Number

Differential

Form

Integral

Form Comment

1. vD

.encvs

QdvdD vs

Gauss’s Law; divergence finds the

source of the electric field vector

2. 0 B

0s sdB

Gauss’s law for magnetic flux

density

3.

t

BE

s

sdBdt

dldE

c

Faraday’ law

4. t

DJH c

sdt

DJldH

s cc

Ampere’s law with Maxwell’s

correction for displacement current

5. t

J v

vvsdv

dt

ddvJdJ vc s

Conservation of charge or continuity

equation



TIME HARMONIC FIELDS

A time harmonic field is one that varies periodically

or in a sinusoidal manner with time. As in the case of

general AC circuit analysis, the phasor representation

of a sinusoidal signal provides a convenient and

efficient method of signal representation. Consider a

vector field that is a function of position and time,

e.g.:

tzyxA ,,,

The vector field tzyxA ,,,

, may be conveniently

written as:

tjeAtzyxA ~Re,,,

Using the phasor notation, the differential and

integral forms of Maxwell’s equations for the time

harmonic case may be written as illustrated in Table

IV.

Table IV Maxwell’s Equations for Time Harmonic Fields

Differential Form for Time Harmonic Fields Integral Form for Time Harmonic Fields

EjJH

HjE

B

D

c

v

~~~

~~0

~

~

s

s

s

s

s

s

s

s

dEjJldH

dHjldE

dB

QdD

cc

c

~~~

~~

0~

~

END OF PART 2

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

Documents

Tutorial de Ecuaciones de Maxwell