Vector Potential A: Discontinuity at the surface current

[Sleepycoaster]

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: ϑA_above/ϑn - ϑA_below/ϑn = -μ₀K
Eqn. 2: ∇ ⋅ A = 0
Eqn. 3: B_above - B_below = μ₀(K × n-hat)
Eqn. 4: A_above = A_below

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:
∇ × A_above - ∇ × A_below = 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?

 

[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

 

[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.

 

[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 http://maktabkhooneh.org/files/library/eng/electrical/7.pdf eqn 5.76 .