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## Low Frequency Behavior

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 Drude Model with the number of free'' electrons per unit volume. It is primarily useful for the insight that it gives us concerning the conductivity'' being closely related to the zero-frequency complex part of the permittivity. Note that at it is purely real, as it should be, recovering the usual Ohm's Law.

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.    Next: High Frequency Limit; Plasma Up: Dispersion Previous: Attenuation by a complex   Contents
Robert G. Brown 2017-07-11