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Electrodynamics question

  1. Sep 27, 2012 #1
    1. The problem statement, all variables and given/known data

    I'd type this out but there's a bit too much formulae.
    http://www.maths.tcd.ie/~tristan/MA3431/Homework_1.pdf

    It's problem 2. I'm just wondering if my solution is correct.

    Thanks in advance!

    2. Relevant equations



    3. The attempt at a solution

    In this problem I'm taking Ro = |xo - x'|

    The volume in question is charge-free so the charge density, ρ(x'), is zero so the first term on the right hand side of the potential vanishes.

    Also, ∂/∂n(1/Ro) = -1/R2o

    Substituting this into the potential function gives the required result.


    For the second part of 2 we use the divergence theorem (which I wont state here due to my lack of latex skills) as told.

    We know that ∂[itex]\Phi[/itex]/∂n = ∇[itex]\Phi[/itex].n and from the definition of the electric field E we end up with -E.n.

    When this is used in the divergence theorem we end up with the volume integral of ∇.E which is equal to ρ/ε which vanishes in a charge free volume.

    We now have the required expression.


    (Haven't gotten to part 3 yet, will be posted soon.)

    I know I haven't explained everything in a great way, but it's a lot easier on paper than it is online to write out loads of partials and surface integrals.
     
  2. jcsd
  3. Sep 28, 2012 #2

    vela

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    Perhaps it was just a typo, but the lefthand side should be written
    $$\left.\frac{\partial}{\partial n}\left(\frac{1}{R}\right)\right|_{R=R_0}.$$ The way you wrote it, you're differentiating the constant 1/R0, so the lefthand side would be equal to 0.

    Other than that, what you've done sounds fine.
     
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