B.Thide - Electromagnetic Field Theory

Furthermore, assuming that = (t;x), we see that the spatial and temporal differentiation of ?G leads to

which means that @t?G transforms as ?G itself only if is time-independent, and that r ?G and r ?G transform as ?G itself only if is space-independent.

END OF EXAMPLE 1.5C

Bibliography

[1]T. W. BARRETT AND D. M. GRIMES, Advanced Electromagnetism. Foundations, Theory and Applications, World Scientific Publishing Co., Singapore, 1995, ISBN 981-02-2095-2.

[2]R. BECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc., New York, NY, 1982, ISBN 0-486-64290-9.

[3]W. GREINER, Classical Electrodynamics, Springer-Verlag, New York, Berlin, Heidelberg, 1996, ISBN 0-387-94799-X.

[4]E. HALLÉN, Electromagnetic Theory, Chapman & Hall, Ltd., London, 1962.

[5]J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X.

[6]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.

i

[7]F. E. LOW, Classical Field Theory, John Wiley & Sons, Inc., New York, NY . . . , 1997, ISBN 0-471-59551-9.

[8]J. C. MAXWELL, A dynamical theory of the electromagnetic field, Royal Society Transactions, 155 (1864).

[9]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.

[10]J. C. MAXWELL, A Treatise on Electricity and Magnetism, third ed., vol. 2, Dover Publications, Inc., New York, NY, 1954, ISBN 0-486-60637-8.

[11]D. B. MELROSE AND R. C. MCPHEDRAN, Electromagnetic Processes in Dispersive Media, Cambridge University Press, Cambridge . . . , 1991, ISBN 0-521- 41025-8.

[12]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.

[13]F. ROHRLICH, Classical Charged Particles, Perseus Books Publishing, L.L.C., Reading, MA . . . , 1990, ISBN 0-201-48300-9.

[14]J. SCHWINGER, A magnetic model of matter, Science, 165 (1969), pp. 757–761.

[15]J. SCHWINGER, L. L. DERAAD, JR., K. A. MILTON, AND W. TSAI, Classical Electrodynamics, Perseus Books, Reading, MA, 1998, ISBN 0-7382-0056-5.

[16]J. A. STRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, NY and London, 1953, ISBN 07-062150-0.

[17]J. VANDERLINDE, Classical Electromagnetic Theory, John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, and Singapore, 1993, ISBN 0-471- 57269-1.

i

2

Electromagnetic Waves

In this chapter we investigate the dynamical properties of the electromagnetic field by deriving a set of equations which are alternatives to the Maxwell equations. It turns out that these alternative equations are wave equations, indicating that electromagnetic waves are natural and common manifestations of electrodynamics.

Maxwell's microscopic equations [cf. Equations (1.45) on page 15] are

and can be viewed as an axiomatic basis for classical electrodynamics. In particular, 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 timeand 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 these second order equations which, as we shall see are wave equations, and then discuss the implications of them. We shall also show how the B wave field can be easily calculated from the solution of the E wave equation.

25

i

2.1 The Wave Equations

We restrict ourselves to derive the wave equations 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.

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

According to the operator triple product “bac-cab” rule Equation (F.64) on page 166

which is the homogeneous wave equation for E.

2.1.2 The wave equation for B

The wave equation for B is derived in much the same way as the wave equation for E. Take the curl of (2.1d) and use Ohm's law j = E to obtain

i

which, with the use of Equation (F.64) on page 166 and Equation (2.1c) on page 25 can be rewritten

Using the fact that, according to (2.1c), r B = 0 for any medium and rearranging, we can rewrite this equation as

This is the wave equation for the magnetic field. We notice that it is of exactly the same form as the wave equation for the electric field, Equation (2.7) on the facing page.

2.1.3 The time-independent wave equation for E

We now look for a solution to any of the wave equations in the form of a timeharmonic wave. As is clear from the above, it suffices to consider only the E field, since the results for the B field follow trivially. We therefore make the following Fourier component Ansatz

Introducing the relaxation time = "0= of the medium in question we can rewrite this equation as

i

which is a time-independent wave equation for E, representing weakly damped propagating waves. In the short limit we have instead

which is a time-independent diffusion equation for E.

For most metals 10 14 s, which means that the diffusion picture is good for all frequencies lower than optical frequencies. Hence, in metallic conductors, the propagation term @2E=c2@t2 is negligible even for VHF, UHF, and SHF signals. Alternatively, we may say that the displacement current "0@E=@t is negligible relative to the conduction current j = E.

If we introduce the vacuum wave number

we can write, using the fact that c = 1=p"0 0 according to Equation (1.11) on page 6,

where in the last step we introduced the characteristic impedance for vacuum

EXAMPLE 2.1 BWAVE EQUATIONS IN ELECTROMAGNETODYNAMICS

Derive the wave equation for the E field described by the electromagnetodynamic equations (Dirac's symmetrised Maxwell equations) [cf. Equations (1.50) on page 17]

under the assumption of vanishing net electric and magnetic charge densities and in the absence of electromotive and magnetomotive forces. Interpret this equation physically.

Taking the curl of (2.19b) and using (2.19d), and assuming, for symmetry reasons, that there exists a linear relation between the magnetic current density jm and the magnetic

i

field B (the analogue of Ohm's law for electric currents, je = eE)

This is the homogeneous electromagnetodynamic wave equation for E we were after.

Compared to the ordinary electrodynamic wave equation for E, Equation (2.7) on page 26, we see that we pick up extra terms. In order to understand what these extra terms mean physically, we analyse the time-independent wave equation for a single Fourier component. Then our wave equation becomes

Realising that, according to Formula (2.18) on the preceding page, 0="0 is the square of the vacuum radiation resistance R0, and rearranging a bit, we obtain the timeindependent wave equation in Dirac's symmetrised electrodynamics

From this equation we conclude that the existence of magnetic charges (magnetic monopoles), and non-vanishing electric and magnetic conductivities would lead to a shift in the effective wave number of the wave. Furthermore, even if the electric conductivity vanishes, the imaginary term does not necessarily vanish and the wave might therefore experience damping (or growth) according as m is positive (or neg-

ative) in a perfect electric isolator. Finally, we note that in the particular case that p

! = R0 m e, the wave equation becomes a (time-independent) diffusion equation

and, hence, no waves exist at all!

END OF EXAMPLE 2.1C

i

2.2 Plane Waves

Consider now the case where all fields depend only on the distance to a given plane with unit normal nˆ. Then the del operator becomes

which simplifies to the first-order ordinary differential equation for the normal component En of the electric field

This, together with (2.27a), shows that the longitudinal component of E, i.e., the component which is perpendicular to the plane surface is independent of and has a time dependence which exhibits an exponential decay, with a decrement given by the relaxation time in the medium.

Scalar multiplying (2.27b) by nˆ, we similarly find that

i

From this, and (2.27c), we conclude that the only longitudinal component of B must be constant in both time and space. In other words, the only non-static solution must consist of transverse components.

2.2.1 Telegrapher's equation

In analogy with Equation (2.7) on page 26, we can easily derive the equation

This equation, which describes the propagation of plane waves in a conducting medium, is called the telegrapher's equation. If the medium is an insulator so that = 0, then the equation takes the form of the one-dimensional wave equation

As is well known, each component of this equation has a solution which can be written

where f and g are arbitrary (non-pathological) functions of their respective arguments. This general solution represents perturbations which propagate along , where the f perturbation propagates in the positive direction and the g perturbation propagates in the negative direction.

If we assume that our electromagnetic fields E and B are time-harmonic, i.e., that they can each be represented by a Fourier component proportional to expfi!tg, the solution of Equation (2.34) becomes

i

Let us consider the lower sign in front of k in the exponent in (2.36).

This corresponds to a wave which propagates in the direction of increasing .

Inserting this solution into Equation (2.27b) on page 30, gives

or, solving for B,

Hence, to each transverse component of E, there exists an associated magnetic field given by Equation (2.40). If E and/or B has a direction in space which is constant in time, we have a plane polarised wave (or linearly polarised wave).

2.2.2 Waves in conductive media

Assuming that our medium has a finite conductivity , and making the timeharmonic wave Ansatz in Equation (2.33) on the previous page, we find that the time-independent telegrapher's equation can be written

where, in the last step, Equation (2.16) on page 28 was used to introduce the wave number k. Taking the square root of this expression, we obtain

r

K = k 1 +i = +i (2.43)

"0!

Squaring, one finds that

2"0!

i