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(1.68)
= E(cos - isin) + icB(cos - isin) = e-i(E + icB) = e-iF
from which it is easy to see that
2
F F" = F = e-iF eiF" = |F|2 (1.69)
while
F F = e2iF F (1.70)
Furthermore, assuming that = (t,x), we see that the spatial and temporal differenti-
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20 CLASSICAL ELECTRODYNAMICS
ation of F leads to
" F " "F
"t F a" = -i e-iF + e-i (1.71a)
"t "t "t
" F a" " F = -ie-i" F + e-i" F (1.71b)
" F a" " F = -ie-i" F + e-i" F (1.71c)
which means that "t F transforms as F itself if is time-independent, and that " F
and " F transform as F itself if is space-independent.
END OF EXAMPLE 1.4
Bibliography
[1] R. BECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc.,
New York, NY, 1982, ISBN 0-486-64290-9.
[2] W. GREINER, Classical Electrodynamics, Springer-Verlag, New York, Berlin,
Heidelberg, 1996, ISBN 0-387-94799-X.
[3] E. HALLN, Electromagnetic Theory, Chapman & Hall, Ltd., London, 1962.
[4] J. D. JACKSON, Classical Electrodynamics, third ed., Wiley & Sons, Inc.,
New York, NY . . . , 1999, ISBN 0-471-30932-X.
[5] L. D. LANDAU, AND E. M. LIFSHITZ, The Classical Theory of Fields,
fourth revised English ed., vol. 2 of Course of Theoretical Physics, Pergamon
Press, Ltd., Oxford . . . , 1975, ISBN 0-08-025072-6.
[6] J. C. MAXWELL, A Treatise on Electricity and Magnetism, third ed., vol. 1,
Dover Publications, Inc., New York, NY, 1954, ISBN 0-486-60636-8.
[7] D. B. MELROSE, AND R. C. MCPHEDRAN, Electromagnetic Processes in Dis-
persive Media, Cambridge University Press, Cambridge . . . , 1991, ISBN 0-521-
41025-8.
[8] W. K. H. PANOFSKY, AND M. PHILLIPS, Classical Electricity and Magnetism,
second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962,
ISBN 0-201-05702-6.
[9] J. SCHWINGER, A magnetic model of matter, Science 165 (1969).
[10] J. SCHWINGER, L. L. DERAAD, JR., K. A. MILTON, AND W. TSAIYANG ,
Classical Electrodynamics, Perseus Books, Reading, MA, 1965, ISBN 0-7382-
0056-5.
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1.4 BIBLIOGRAPHY 21
[11] J. A. STRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc.,
New York, NY and London, 1953, ISBN 07-062150-0.
[12] J. VANDERLINDE, Classical Electromagnetic Theory, John Wiley & Sons, Inc.,
New York, Chichester, Brisbane, Toronto, and Singapore, 1993, ISBN 0-471-
57269-1.
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2
Electromagnetic
Waves
In this chapter we shall investigate the dynamical properties of the electromag-
netic field by deriving an set of equations which are alternatives to the Maxwell
equations. It turns out that these alternative equations are wave equations, in-
dicating that electromagnetic waves are natural and common manifestations of
electrodynamics.
Maxwell s microscopic equations (1.43) on page 14, which are usually
written in the following form
(t,x)
" E = (2.1a)
0
"B
" E = - (2.1b)
"t
" B = 0 (2.1c)
"E
" B = 0j(t,x) + 00 (2.1d)
"t
can be viewed as an axiomatic basis for classical electrodynamics. In partic-
ular, these equations are well suited for calculating the electric and magnetic
fields E and B from given, prescribed charge distributions (t,x) and current
distributions j(t,x) of arbitrary time- and space-dependent form.
However, as is well known from the theory of differential equations, these
four first order, coupled partial differential vector equations can be rewritten as
two un-coupled, second order partial equations, one for E and one for B. We
shall derive the second order equation for E, which, as we shall see is a homo-
geneous wave equation, and then discuss the implications of this equation. We
shall also show how the B field can be easily calculated from the solution of
the E equation.
23
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24 ELECTROMAGNETIC WAVES
2.1 The wave equation
Let us derive the wave equation for the electric field vector E and the magnetic
field vector B in a volume with no net charge, = 0, and no electromotive force
EEMF = 0 and compare the results.
2.1.1 The wave equation for E
In order to derive the wave equation for E we take the curl of (2.1b) and using
(2.1d), to obtain
" " "
" (" E) = - (" B) = -0 j + 0 E (2.2)
"t "t "t
According to the operator triple product bac-cab rule Equation (F.67) on
page 155
" (" E) = "(" E) - "2E (2.3)
Furthermore, since = 0, Equation (2.1a) on the previous page yields
" E = 0 (2.4)
and since EEMF = 0, Ohm s law, Equation (1.26) on page 10, yields
j = E (2.5)
we find that Equation (2.2) can be rewritten
" "
"2E - 0 E + 0 E = 0 (2.6)
"t "t
or, also using Equation (1.9) on page 5 and rearranging,
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