Near
the qualitative behavior depends upon whether or not
there is a ``resonance'' there. If there is, then
can begin with a complex component that
attenuates the propagation of EM energy in a (nearly static) applied
electric field. This (as we shall see) accurately describes *conduction* and *resistance*. If there isn't, then
is
nearly all real and the material is a dielectric insulator.

Suppose there are both ``free'' electrons (counted by
) that are
``resonant'' at zero frequency, and ``bound'' electrons (counted by
). Then if we start out with:
&epsi#epsilon;(&omega#omega;) & = & &epsi#epsilon;_0(1 + N e^2m
&sum#sum;_i f_i(&omega#omega;_i^2 - &omega#omega;^2 - i &omega#omega;&gamma#gamma;_i))

& = & &epsi#epsilon;_0(1 + N e^2m
&sum#sum;_b f_b(&omega#omega;_b^2 - &omega#omega;^2 - i &omega#omega;&gamma#gamma;_b))

& & + N e^2m
&sum#sum;_f f_f(- &omega#omega;^2 - i &omega#omega;&gamma#gamma;_f)

& = & &epsi#epsilon;_b + i &epsi#epsilon;_0 N e^2 f_fm &omega#omega;
(&gamma#gamma;_0 - i &omega#omega;)
where
is now *only* the contribution from all the
``bound'' dipoles.

We can understand this from × = + d dt (Maxwell/Ampere's Law). Let's first of all think of this in terms of a plain old static current, sustained according to Ohm's Law: = &sigma#sigma; .

If we assume a harmonic time dependence and a ``normal'' dielectric
constant
, we get:
×
& = & (&sigma#sigma;- i&omega#omega;&epsi#epsilon;_b )

& = & -i &omega#omega;( &epsi#epsilon;_b + i &sigma#sigma;&omega#omega; )
.

On the other hand, we can instead set the static current to *zero*
and consider all ``currents'' present to be the *result* of the
polarization response
to the field
. In this case:
×
& = & - i&omega#omega;&epsi#epsilon;

& = & -i &omega#omega;( &epsi#epsilon;_b +
i &epsi#epsilon;_0 N e^2m f_f(&gamma#gamma;_0 - i &omega#omega;) )

Equating the two latter terms in the brackets and simplifying, we obtain the following relation for the conductivity:

(11.67) |

This is the

We conclude that the distinction between dielectrics and conductors is a matter of perspective away from the purely static case. Away from the static case, ``conductivity'' is simply a feature of resonant amplitudes. It is a matter of taste whether a description is better made in terms of dielectric constants and conductivity or complex dielectric.