Penetration of electromagnetic waves in insulators?

Question

The exact formula for the attenuation of E&M waves in a conducting material can easily be found in most textbooks on applied E&M.

I am wondering if such a equation (approximation or not) exists for an insulator.

Answer 1

If the insulator isn't a perfect insulator, and you know its conductivity (at the frequency of interest), you can calculate its inherent attenuation constant the same way you would with a conductor.

But practically, the attenuation in an insulator is often determined by the presence of impurities in the insulator, rather than its inherent properties. For example, iron, hydroxyl, and oxygen ion impurities are some of the major sources of attenuation in glass optical fibers. Only by extremely rigorous purification of the glass to remove these impurities before drawing the fiber are we able to manufacture optical fibers with attenuation below 1 dB/km.

So determining the attenuation of practical materials often comes down to an engineering question of determining what impurities they're contaminated with. And you might actually use an attenuation measurement (absorbance spectroscopy) to determine the impurity types and concentration, rather than the other way around.

Answer 2

As @The-Photon implies in their nice answer, the reason you won't see a separate formula for the attenuation of electromagnetic waves in insulators is because the same formula applies. The attenuation constant $\alpha$ for the electromagnetic wave $$E_y(z)=E_0e^{-\alpha z}\cos{\left(\omega t±\beta z\right)}$$ is given by $$\alpha^2 = \frac{\sigma^2\mu}{2\epsilon\left(1+\sqrt{1+\left(\frac{\sigma}{\omega\epsilon}^2\right)}\right)}$$ When the conductivity $\sigma$ is small, this reduces to $$\alpha\approx\frac{\sigma}{2}\sqrt{\frac{\mu}{\epsilon}}$$ So for a perfect classical insulator the attenuation is zero, but no real material has zero conductivity. For example, the DC conductivity of glass is is $\sim 10^{11-15}\,\Omega\,\textrm{m}$.

More importantly, again as noted by @The-photon, conductivity is frequency dependent. The relevant conductivity and permittivity are not the DC values but the frequency dependent optical conductivity and permittivity. The absorption coefficients in materials can change dramatically with frequency due to electronic bandgaps, molecular absorption bands, …. A lot of atomic, molecular, condensed matter, and plasma physics can be involved in understanding the frequency dependent values of $\sigma$ and $\epsilon$.

For example, think of an pure semiconductor at very low temperatures. It will be an insulator because no holes or electrons have enough energy to reach the conduction band. The semiconductor will be transparent to photons with energies below the semiconductor's bandgap, but photons with energies above the bandgap will be absorbed because they can kick electrons out of the valence band. Even at room temperature, the optical absorption of silicon (bandgap 1.1 eV) varies by 14 orders of magnitude going from 1500 nm (0.8 eV)to 300 nm (4.1 eV) wavelength.