Why does a dielectric have a frequency dependent resistivity?

Question

This question has come about because of my discussion with Steve B in the link below.

Related: Why is glass much more transparent than water?

For conductors, I can clearly see how resistivity $\rho\,\,(=1/\sigma)$ can depend on frequency from Ohm’s law, $\mathbf{J}=\sigma\mathbf{E}$. So if the E-field is an electromagnetic wave impinging on a conductor, clearly the resistivity is frequency dependent. In a similar fashion, the frequency dependence of the electric permittivity $\epsilon=\epsilon_0n^2(\omega)$ can be derived through the frequency dependence of the electric polarization and impinging electromagnetic wave (see How Does $\epsilon$ Relate to the Dampened Harmonic Motion of Electrons?).

1. What does it mean physically for a dielectric to have a frequency dependent resistivity from (i) classical and (ii) quantum viewpoints? I am especially interested in the optical frequency range.

2. Can a simple mathematical relationship be derived similar to the frequency dependent resistivity (for conductors) and electric permittivity (for dielectrics)?

Thank you in advance for any help on this question

Answer 1

A simple model that explains the frequency dependency of the resistivity of metals reasonably well is the Drude model (http://en.wikipedia.org/wiki/Drude_model). There we have frequency dependency because the electrons in a plasma are not moving arbitrarily fast, which is consistent with Xurtio's explanation. The cutoff frequencies are usually in the optical domain. For dielectrics similar models exist, which are often a sum of Lorentzian resonances. These have their origin in resonant absorption which is a quantum physical effect.

The imaginary part of the permittivity is related to the conductivity. This can be seen as follows: Amperes law is

$\nabla \times \mathbf{H} = \mathbf{J} +i \omega \epsilon_r \epsilon_0 \mathbf E$

and insert Ohms law in differential form

$\mathbf{J} = \sigma \mathbf{E}$

then you get

$\nabla \times \mathbf{H} = i \omega (\epsilon_r \epsilon_0 -i \sigma/\omega) \mathbf E$

which is just of the same form of as original form of amperes law but without the explicit $\mathbf{J}$ term. In conclusion Ohms law can be integrated in free space Maxwells equations (without the source terms) when the relative permittivity $\epsilon_r$ is taken as a complex value ($\widetilde\epsilon_r = \epsilon_r - i \sigma/(\omega \epsilon_0)$), where an imaginary part is added related to the conductivity. This essentially models the effect of moving charges under the influence of an oscillating field (light).

So the relation between polarization ($\mathbf D = \widetilde{\epsilon}_r \epsilon_0 \mathbf E = \mathbf P + \epsilon_0 \mathbf E$) and conductivity $\sigma$ is given as

$\mathbf{P} = \epsilon_0 (\epsilon_r - i \sigma/\omega - 1) \mathbf E$.

Since the real part of the permittivity is frequency dependent, so is the conductivity. This is because of the Kramers-Kronig relations which follow from a causality relation.

Answer 2

A di-electric experiences polarization int he presence of an electric field. The magnitude of polarization will present an effective resistance (more polarization against the field = more apparent resistance).

But the polarization takes time (it's not instantaneous). So think about polarization delay vs. the change of the source electric field (i.e. the "frequency"). The faster the source field changes, the less time the dielectric has to polarize. For very slow frequencies, the polarization will be able to keep up with the changes in electric field.

Answer 3

The frequency dependent dielectric properties of a medium relate to the universal dielectric response, UDR. A topic worth reading more about. In general the percolation network nature of many systems results in frequency dependent impedence following a power law relationship. The origin of the UDR is the subject of much discussion in the scientific community, and it can be considered as an example of emergent behaviour. The UDR in diverse systems arises from multi-body interactions and can be represented as an equivalent RC network.

Answer 4

Well I hesitate to even venture into this question, because technical terms are being misused creating the problem.

Resistance, and resistivity, is something that arises from Ohm's Law.

Namely, for a certain class of materials (mostly metallic conductors) , if all other physical parameters are held constant (difficult to do), the ratio of the current flowing to the applied Voltage, is constant.

So Ohm's law simply says: R is CONSTANT .

And R does not vary with frequency either. with varying currents, Ohm's law applies to all instants of time, so with AC Voltages and Currents the two are ALWAYS in phase.

The practical problem arises, in that when you have a current flowing in a resistive medium; say a wire, there is a magnetic field set up, that surrounds the current flow, and that magnetic field is also inside the wire, and the magnitude of the field depends on the ENCLOSED current. So the center of the wire has a lower current, so it generates a smaller magnetic field.

If the current varies, then the magnetic field is restricted in its movement, or change, by the velocity of EM wave propagation (c).

As a consequence of this time lag, the current carrying conductor now exhibits, an Inductance effect, so the equivalent circuit is no longer a simple resistor with constant Ohmic resistance; it is a series circuit of a resistor in series with an inductor; approximately 3 nano-Henries per centimeter of a straight wire.

So you now have an AC impedance that is Z = R + j.2.pi.f.L

So now the current will be less, and as you raise the frequency, the inductive reactance will increase linearly with frequency, so the current will drop.

The resistance has not changed one iota; the impedance has. Eventually, you will end up with the current in the center of the wire going completely backwards, compared to the outer layers. That backwards current further diminishes the current for a given Voltage, so the wire center is now more of a nuisance than a useful conductor. So you might as well get rid of it, and use a hollow tube.

This is the essence of "Skin effect", it has nothing whatsoever to do with the resistance or the resistivity of the conductor, which remains completely frequency independent. It is the AC impedance that IS INCREASING, not THE RESISTIVITY.

If it is frequency dependent, it is NOT a RESISTOR, complying with Ohm's Law; it is a complex AC circuit involving Inductance, and also Capacitance, when you get into it.

Words, have meaning, and when scientists use the wrong words; specially ones that also have colloquial common meanings; it creates havoc for all; this question for example.