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```88 MMET'98 Proceedings
Department of Electrical Engineering
Purdue University at Indianapolis
723 West Michigan Street, Sk 16QB
Indianapolis, Indiana 46202-5 132, USA
lyshev@ engr. iupui.edu
The electromagnetic field model is governed by four Maxwell's equations, which are given in
the point form for tiiie-varying fields as
Jff(x,y,z,t)
V X E ( x , y , z , t ) =-p--
d t '
p ( x , y , z , t ) V - E ( x , y , z , t ) = , V . H ( x , y , z , t ) = Q ,
E
where E is the electric field intensity; pi is the magnetic field intensity; 9 is the current density; p is
the charge density.
The development of analytic methods to solve Maxwell's equations in the coordinate systems
used is our particular interest. The common coordinate systems applied (rectangular, circularly
cylindrical, and spherical) are studied, and complete analytic solutions to Maxwell's equations are
given. The rectangular and cylindrical Coordinate systems are C Q ~ ~ O I I ~ Y used. This paper
demonstraltes that an analytic solution to Maxwell's equations, if it exists, can be fmd in the chosen
coordinate system by the superposition of TM and TE fields, and the reciprocity theorem can be used.
In the circularly cylindlrical system, the TM and TE potentials satisfy the Helmholtz equation, and the
first-, second-, and third-kind Bessel functions are applied. Only the spherical coordinate system has
Kharkov, Ukraine, VIIth International Conference on Mathematical Methods in Electromagnetic Theory
http://iupui.edu
89
MMET'98 Proceedings
complete coordinate surfaces of fmite size, and therefore, in this system the bsundw value ~ ~ o b ~ e ~
cm be solved. Hence, the spheroidal configuration, which has @ -symmetry, is of a great importance.
'The wave or ~~~~o~~ equation for the field quantities (E or H ) in kms of the ~ ~ e - v ~ ~ ~ ~ sources
cm be expressed and solved in the spherical coordmate system.
Kharkov, Ukraine, VIIth International Conference on Mathematical Methods in Electromagnetic Theory
MMET'98 Proceedings
E , = - j q . P n w a i + V p
By making use of E , the following equations result
v x v x PT,wai - -
H , = B x PTMai, E , = 9
jctx
V x H , = jwEE2, V x E , = -jw@, - M ,
From quation
E , = v x P+, q
V2 pTk;al = y 2 PTEa, + M , .
The thorough analysis per€omed indicates, that for TM and TEi fields &e equations
H , = V x PTMa,
agld E , = V X Prk;a,
should be solved by using the following procedure. Bn the Cartesian coordinate systems, as one
~ b t ~ i h ~ BTM md PTE, H , a d E , results.
one finds PTE by S O I V ~ ~ equation
e, E , can be found by solving
V x H , = j m E , +I,, E , --ja
while E , isobtained sing .PTk and H , is found fi.om
coordhae system, ch is given h terms p, \$, z ~ the TM and \$E field
equations age
N = V x PTMai
and E = V x PTEa,,
and the scalar potential satisfies Helmholtz equations are given by
V 2 P T M = Y 2 P T M , y 2 = -W2W .
The following partial differential equation for PTM results
and the solution of t h i s equation is found using hyperbolic functions. In particular,
where CD,(p) @,(e) and al,(z) are the Bessel functions.
R Z M = Q P ( P P 4 (@PZ (z> ,
If the spherical coordinate system is used, the field components are found for TM fields as
and for TE fields we have
Kharkov, Ukraine, VIIth International Conference on Mathematical Methods in Electromagnetic Theory
MMET’98 Proceedings 91
+ J , v x E , =2? -qesrfi,H,
E,=-N,, H , = E L , J 1 = - M 2 , & , = & , & = E Z 9
or
er
E I - - H 2 7 H 1 - --E 2 3 J ! = M 2 ? E ~ = & , p, = E ~ ,
E , = H, H , = E , , J , = - M 2 , F , = -be,, pl = - E , e
In psrticular, by applying these transformations, solution of (4) is identical to (3).
4. Cowclusions
This paper addresses the problem of solution of well’s equatioiis in the rectmplar,
cal coordinate systems. Sinusoidal steady-state and
searched. By uskg the results, it is shown that Maxwell’
solved, mQ Lqportant problems have been focused and solved. In particular, the complexity of the
S O ~ ~ ~ O I I depiends on the coordinate system used, and spherical ~ o o r d ~ ~ ~ system has advantages to
find analytic solutions to Maxwell’s equations.
Kharkov, Ukraine, VIIth International Conference on Mathematical Methods in Electromagnetic Theory