tutorial on maxwell's equations- part 2
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Tutorial on Maxwell's EquationsTRANSCRIPT
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 1 of 12 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
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THE INFORMATION WITHIN THIS DOCUMENT IS DISCLOSED WITHOUT EXCEPTION TO THE GENERAL PUBLIC.
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TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
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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 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
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"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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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
.
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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
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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:
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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.
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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 field 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.
The following equation 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
Figure 4.5: Magnetic Field Intensity at the Origin of a
Current Loop in the Plane, Z=0
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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:
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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.
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
1 Recall that the curl of a vector field finds its source.
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.
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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.
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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
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