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Homework Help: Continuity across boundaries in Electromag

  1. Jan 3, 2010 #1
    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 [tex]\omega[/tex]. Then, by Maxwell's equations, the electric field satisfies
    [tex]-\nabla^2 \mathbf{E} = \frac{\omega^2}{c^2} \epsilon(\omega) \mathbf{E} [/tex] for some appropriate function [tex] \epsilon(\omega)[/tex], where c is the speed of light.

    Consider the boundary between two such regions (which may contain surface charge). Prove that [tex] \epsilon \mathbf{E}^{\bot}[/tex] 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.
    [tex] E^{\bot}_{\text{above}} - E^{\bot}_{\text{below}} = \frac{\sigma}{\epsilon_0} [/tex]

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

    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?
  2. jcsd
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