Where does this formula for absorption cross section come from?

[Haorong Wu]

TL;DR Summary

Where does this formula of absorption cross section come from?

My professor gave us a formula for absorption cross section, but he said he did not remember where he found it.

The formula is given by

$$\sigma_a =\frac {1} {| \mathbf E_i|^2} \int_V k \epsilon^{''}_r | \mathbf {E} |^2 dV = k \epsilon^{''}_r |\frac 3 {\epsilon_r +2}|^2 V.$$

Where does this formula come from? I want to know what those variables stand for.

He gave this formula just after the formula of Rayleigh scattering cross section. It seems these two equations are related somehow, because he also gave the counterpart-formulars in Mie scattering theory.

 

[Twigg]

My professor gave us a formula for absorption cross section, but he said he did not remember where he found it.

This made me giggle and made my soul hurt at the same time.

This looks like what you'd get if you took the imaginary part of the Clausius-Mossotti formula for electrostatic fields in a dielectric sphere (see Jackson, 3rd edition equation 4.56 and the derivation leading up to it). I'm going to go out on a limb and guess that this formula you gave is an electrostatic approximation for the absorption cross section of a dielectric sphere. The electrostatic approximation is valid when the wavelength is much larger than the radius of the sphere (Rayleigh scattering). In that scenario, the electric field due to the wave is approximately constant over the whole volume of the sphere, so you can use the Clausius-Mossotti formula (which assumes uniform fields).

As far as the symbols go, the standard conventions say:
$\epsilon_r''$ ought to be the imaginary part of the relative permeability (the real part would be $\epsilon_r'$) (phyiscally, the imaginary part corresponds to absorption of the EM wave by the material. If the permeability was entirely real, it'd just reflect or refract the light per the Fresnel equations)
V here is just the volume of the sphere
k is the wavenumber of the light
$E_i$ is just the amplitude of the incident electric field