Vector Potential A: Discontinuity at the surface current

  • #1

Homework Statement


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.]

Homework 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

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?
 

Answers and Replies

  • #2
BvU
Science Advisor
Homework Helper
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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
 
  • #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.
 
  • #4
BvU
Science Advisor
Homework Helper
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3,752
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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