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Low Frequency Behavior

Near $ \omega = 0$ the qualitative behavior depends upon whether or not there is a ``resonance'' there. If there is, then $ \epsilon(\omega\approx 0)$ 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 $ \epsilon $ is nearly all real and the material is a dielectric insulator.

Suppose there are both ``free'' electrons (counted by $ f_f$ ) that are ``resonant'' at zero frequency, and ``bound'' electrons (counted by $ f_b$ ). 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 $ \epsilon_b$ is now only the contribution from all the ``bound'' dipoles.

We can understand this from ×\Vec{H} = \Vec{J} + d\Vec{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: \Vec{J} = &sigma#sigma;\Vec{E} .

If we assume a harmonic time dependence and a ``normal'' dielectric constant $ \epsilon_b$ , we get: ×\Vec{H} & = & (&sigma#sigma;- i&omega#omega;&epsi#epsilon;_b )\Vec{E}
& = & -i &omega#omega;( &epsi#epsilon;_b + i &sigma#sigma;&omega#omega; ) \Vec{E} .

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 $ \Vec{D}$ to the field $ \Vec{E}$ . In this case: ×\Vec{H} & = & - i&omega#omega;&epsi#epsilon;\Vec{E}
& = & -i &omega#omega;( &epsi#epsilon;_b + i &epsi#epsilon;_0 N e^2m f_f(&gamma#gamma;_0 - i &omega#omega;) ) \Vec{E}

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

$\displaystyle \sigma = \epsilon_0 \ \frac{n_f e^2}{m}\frac{1}{(\gamma_0 - i \omega)}.$ (11.67)

This is the Drude Model with $ n_f = f_f N$ 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 $ \omega = 0$ 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 up previous contents
Next: High Frequency Limit; Plasma Up: Dispersion Previous: Attenuation by a complex   Contents
Robert G. Brown 2017-07-11