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Griffiths Problem 3.4

  1. Feb 10, 2008 #1
    [SOLVED] Griffiths Problem 3.4

    1. The problem statement, all variables and given/known data
    Prove that the field is uniquelly determined when the charge density rho is given and either V or the normal derivative [itex]\frac{\partial{V}}{\partial{n}}[/itex] is specified on each boundary surface. Do not assume the boundaries are conductors, or that V is constant over any given surface.


    2. Relevant equations



    3. The attempt at a solution
    If you know V on the surface, this is the Corollary to the First Uniqueness Theorem. I don't see how knowing the normal derivative, [itex]\frac{\partial{V}}{\partial{n}}= \nabla{V}\cdot \hat{n}[/itex], helps at all.
     
  2. jcsd
  3. Feb 10, 2008 #2
    My guess: probably one of those Stoke's theorems (curl, divergence, gradient) will simplify things.
     
  4. Feb 11, 2008 #3
    The divergence theorem and Poisson's equation tell us that [tex]\int_{S} \nabla V \cdot \hat{n} da = \int_{V}\nabla ^2V d\tau = \int_{V}\rho/\epsilon_0 d\tau [/tex]. We know rho, so we didn't even need the normal derivative to get that integral. Thus, I don't see how the curl, divergence, gradient theorems will help.
     
  5. Feb 11, 2008 #4
    anyone?
     
  6. Feb 11, 2008 #5
    Never mind. I got it.
     
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