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Vector Potential A: Discontinuity at the surface current

  1. Mar 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove Eqn. 1 (below) using Eqns. 2-4. [Suggestion: I'd set up Cartesian coordinates at the surface, with z perpendicular to the surface and x parallel to the current.]

    2. Relevant equations

    I used ϑ for partial derivatives.

    Eqn. 1: ϑAabove/ϑn - ϑAbelow/ϑn = -μ0K
    Eqn. 2:A = 0
    Eqn. 3: Babove - Bbelow = μ0(K × n-hat)
    Eqn. 4: Aabove = Abelow

    3. The attempt at a solution

    Conceptually, I'm mostly stuck at the partial derivatives with respect to n. n is just a normal vector to a plane surface. It will flip completely as soon as you go from looking at points below the surface to points above the surface.

    I've taken Eqn. 3 and plugged in B = × A to get:
    × Aabove - × Abelow = μ0(K × n-hat)

    It looks pretty close, but by Eqn. 4, the two terms on the left should be equal and thus everything is zero. That's hardly going to help.

    The usefulness of Eqn. 2 seems dubious to me, but it would be useful if I need find A using Poisson's equation, which is only possible by Eqn. 2.

    2A = -μ0J

    But then again, the surface is 2D so J doesn't really fit.

    I need a nudge in the right direction. Help?
     
  2. jcsd
  3. Mar 28, 2015 #2

    BvU

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    Hello coaster,

    I notice you didn't get much response. Speaking for myself, I didn't react because you made it difficult to understand what this is about. Perhaps you can provide a description and some context.

    Also I haven't seen many ##\partial\over \partial\hat n## in my career (there must be a reason for that! think about what it's supposed to mean...), so I don't know what you mean and where you get that equation.

    All the best,
    BvU
     
  4. Mar 28, 2015 #3
    Thanks for replying. This is a problem from a book on Electricity and Magnetism that my university is using. I don't really understand the partial derivative over n-hat myself, and the book doesn't mention it in detail.

    I'll drop this topic and ask my professor if he knows.
     
  5. Mar 29, 2015 #4

    BvU

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    Either that, or you check out a few "magnetic vector potential examples", e.g. here : last eqn in 5.6 shows that the partial derivative isn't ##
    \partial\over \partial\hat n## but ##\partial\vec A \over \partial n##, by which they mean its normal derivative - so in your case ##{\partial A_x \over \partial z},{\partial A_y \over \partial z},{\partial A_z \over \partial z}## (two of which are 0).

    Also Griffiths eqn 5.76 .
     
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