1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
    Cheers.
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted