# Continuity across boundaries in Electromag

1. Jan 3, 2010

### fantispug

1. The problem statement, all variables and given/known data
This question is adapted from an implicit assumption in Ashcroft and Mermin question 1.5.

Consider a medium with no net charge (but possibly a net current) in which Ohm's law holds. Let an electromagnetic wave travel through the medium with angular frequency $$\omega$$. Then, by Maxwell's equations, the electric field satisfies
$$-\nabla^2 \mathbf{E} = \frac{\omega^2}{c^2} \epsilon(\omega) \mathbf{E}$$ for some appropriate function $$\epsilon(\omega)$$, where c is the speed of light.

Consider the boundary between two such regions (which may contain surface charge). Prove that $$\epsilon \mathbf{E}^{\bot}$$ is continuous across the boundary.

2. Relevant equations
The surface charge between two regions is given by the discontinuity of the perpendicular component of the electric field.
$$E^{\bot}_{\text{above}} - E^{\bot}_{\text{below}} = \frac{\sigma}{\epsilon_0}$$

3. The attempt at a solution
I don't think it's true. Given the perpendicular component of $$\epsilon \mathbf{E}$$ is continuous across the boundary, using the definition of $$\epsilon$$ we see
$$-\nabla^2 (E^{\bot}_{\text{above}} - E^{\bot}_{\text{below}}) = 0$$
that is
$$\nabla^2 \sigma = 0$$.

I don't see any a priori reason this should be true (given this I could run the argument backwards to prove the assertion). I think this would be the crux of the argument; and indeed I can not see another way to approach the problem.

Any help on what I'm missing?
Cheers.