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:

[ I1a ]


[ I1b ]


[ I1c ]


[ I1d
]

where
and
are ultimately defined in terms of the Lorentz force on a charge
moving at a velocity
 viz.

[ I2
]

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]

[ I3a ]


[ I3b ]


[ I3c ]


[ I3d
]

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,
nonlocal,
and anisotropic effects:
In tensor
representation

[ I4a
]

In dyadic
representation

[ I4b
]

For the present and for most of our
discussions, we neglect
nonlocal
effects
and treat only dispersive(dissipative)
and anisotropic effects  viz.

[ I5
]

or

[ I6
]

where

[ I7
]

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

[ I8a ]


[ I8b ]


[ I8c ]


[ I8d
]

Frequency Domain Helmholtz Equations for
Vector and Scalar Potentials in a Uniform, Linear,
Isotropic Dielectric
We define the (Magnetic) Vector Potential
as

[ I9
]

which, of course, automatically satisfies
one of Maxwell's equation 
viz. [ I8d ].[3]
. We introduce the (Electric) Scalar Potential in the form

[ I10
]

which, again, automatically satisfies
another Maxwell equation  viz. [ I8a ].[4]
Therefore, for
uniform, isotropic media
Equation [ I8b ] becomes [5]

[ I11a
]

and Equation [ I8c ] becomes

[ I11b
]

Since
is as yet undefined, we usual take it in the Lorentz
gauge  viz.

[ I12]

 to simplify Equations [ I11a
] and [ I11b ]. Thus

[ I13a]

and

[ I13b]

Therefore, in the Lorentz gauge both
and
satisfy inhomogeneous (and homogeneous) Helmholz equations!
However, in the Coulomb
gaugewe take
and then Equation [ I8c ] becomes

[ I14a]

Conservation of charge requires that
so that Equation [ I11a ] becomes

[ I14b]

where

[ I15]

^{Footnotes}
[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