# Image charge inside dielectric with complex permittivity?

A common textbook problem in electromagnetism is to place a point charge $$q$$ outside a dielectric material with permittivity $$\epsilon$$ .

In the case of a semi-infinite dielectric material, the induced image charge $$q'$$ is $$q' = -q \frac{\epsilon-1}{\epsilon+1}$$

My question is, what is the right way to deal with the image charge when the permittivity $$\epsilon$$ is complex (i.e. $$\epsilon = \epsilon_1 + i \epsilon_2$$)?

Is the following expression the right solution? If so, what would be the interpretation of the imaginary part?

$$q' = -q \,\,\mathrm{Re}[\frac{\epsilon-1}{\epsilon+1}]$$

By taking the permittivity to be complex, I am presumably making the implicit assumption that the external charge is "oscillating" at a fixed frequency $$\omega$$ such that $$\epsilon(\omega) = \epsilon_1 + i \epsilon_2$$. At some point the magnetic field caused by the oscillating charge may be important, but I am interested mainly in the regime where the magnetic field can be neglected.

As you say, with a complex permittivity, you are assuming that the charge is oscillating. This, of course, means that if you start with a charge $$q$$, it will eventually change to a charge $$-q$$. That is, $$-2q$$ charge had to flow in from somewhere. So an image charge is not well defined for this case. Instead, the problem that is usually studied is image charges for an oscillating dipole either electric or magnetic. You can then put these together to get any current conserving charge density.