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El documento presente te ayudará a comprender de una manera sencilla las Ecuaciones de Maxwell, útiles en gran variedad de aplicaciones a problemas de la ciencia.
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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.
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 2 of 13 E X H CONSULTING SERVICES
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
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 3 of 13 E X H CONSULTING SERVICES
"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
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 4 of 13 E X H CONSULTING SERVICES
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
.
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 5 of 13 E X H CONSULTING SERVICES
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
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 6 of 13 E X H CONSULTING SERVICES
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
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 7 of 13 E X H CONSULTING SERVICES
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
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 8 of 13 E X H CONSULTING SERVICES
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;
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 9 of 13 E X H CONSULTING SERVICES
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.
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 10 of 13 E X H CONSULTING SERVICES
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,
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 11 of 13 E X H CONSULTING SERVICES
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
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 12 of 13 E X H CONSULTING SERVICES
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
Publications, New York, 1991.
[10] Harrington, R. F., Introduction to
Electromagnetic Engineering, Dover
Publications, New York, 2003.
TECHNICAL MEMORANDUM: TUTORIAL ON MAXWELL’S EQUATIONS
– REVISION 02 JANUARY 2013 – Page 13 of 13 E X H CONSULTING SERVICES
[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