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Maxwell's Equations, Yet Again

Suppose we are given a system of classical charges that oscillate harmonically with time. Note that, as before, this can be viewed as the special case of the Fourier transform at a particular frequency of a general time dependent distribution; however, this is a very involved issue that we will examine in detail later in the semester.

The form of the charge distribution we will study for the next few weeks is:

$\displaystyle \rho(\mbox{\boldmath$x$},t)$ $\textstyle =$ $\displaystyle \rho(\mbox{\boldmath$x$}) e^{-i \omega t}$ (11.1)
$\displaystyle \mbox{\boldmath$J$}(\mbox{\boldmath$x$},t)$ $\textstyle =$ $\displaystyle \mbox{\boldmath$J$}(\mbox{\boldmath$x$}) e^{-i \omega t} .$ (11.2)

The spatial distribution is essentially ``arbitrary''. Actually, we want it to have compact support which just means that it doesn't extend to infinity in any direction. Later we will also want it to be small with respect to a wavelength.


Robert G. Brown 2007-12-28