On Classical Electromagnetic Fields

I. Preliminaries: A Review of Some Basic Concepts and Methods (pdf copy)

The Microscopic Maxwell's Equations in the Time Domain:

 [ I-1a ] 
[ I-1b ] 
[ I-1c ] 
     [ I-1d ] 
where  and  are ultimately defined in terms of the Lorentz force on a charge  moving at a velocity  -- viz.
     [ I-2 ]

The Macroscopic Maxwell's Equations in the Time Domain:

Following common practice, we set forth a particular set of macroscopic Maxwell's equations that applies in the high frequency or optical regime.[1]
 [ I-3a ] 
[ I-3b ] 
[ I-3c ] 
     [ I-3d ] 


A Phenomenological Representation of the Linear Dielectric Response of Matter: [2]

The following is the most general phenomenological representation of the linear dielectric response of a given material which incorporates dissipative, non-local, and anisotropic effects:
In  tensor representation
     [ I-4a ]
In  dyadic representation
     [ I-4b ]
For the present and for most of our discussions, we neglect nonlocal effects and treat only dispersive(dissipative) and anisotropic effects -- viz.
     [ I-5 ]
     [ I-6 ]
     [ I-7 ]

Macroscopic Maxwell's Equations in the Frequency Domain -- Valid for Linear, Local, Anisotropic Media in the Optical Regime.

 [ I-8a ] 
[ I-8b ] 
[ I-8c ] 
     [ I-8d ] 


Frequency Domain Helmholtz Equations for Vector and Scalar Potentials in a Uniform, Linear, Isotropic Dielectric

We define the (Magnetic) Vector Potential as
     [ I-9 ]
which, of course, automatically satisfies one of Maxwell's equation -- viz. [ I-8d ].[3] .  We introduce the (Electric) Scalar Potential in the form
     [ I-10 ]
which, again, automatically satisfies another Maxwell equation -- viz. [ I-8a ].[4]   Therefore, for uniform, isotropic media Equation [ I-8b ] becomes [5]
     [ I-11a ]
and Equation [ I-8c ] becomes
     [ I-11b ]
Since  is as yet undefined, we usual take it in the Lorentz gauge -- viz.
     [ I-12]
-- to simplify Equations [ I-11a ] and [ I-11b ].  Thus
     [ I-13a]
     [ I-13b]

Therefore, in the Lorentz gauge both   and  satisfy inhomogeneous (and homogeneous) Helmholz equations!
However, in the Coulomb gaugewe take  and then Equation [ I-8c ] becomes
     [ I-14a]
Conservation of charge requires that  so that Equation [ I-11a ] becomes
     [ I-14b]
     [ I-15]


[1] All bound current effects are included in the polarization density, since magnetization density "ceases to have any physical meaning at relatively low frequencies." See Section 62 in L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press (1960).

[2] In our later treatment of nonlinear optics, we will begin by adding the following nonlinear phenomenological contributions:

[3] Since
[4] Since
[5] Using

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This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated May 12, 2000