Vector Potential A: Discontinuity at the surface current

1. Mar 26, 2015

Sleepycoaster

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. Mar 28, 2015

BvU

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. Mar 28, 2015

Sleepycoaster

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. Mar 29, 2015

BvU

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 .