Magnetic vector potential from an infinite current sheet with thickness t

[superiority]

Homework Statement

An infinite sheet of copper conductor, thickness t, lies in the xz-plane. The sides of the sheet intersect the y-axis at y=\pm\frac{t}{2}. The current density in the sheet is given by:
{\bf{j}}({\bf{r}}) = \begin{cases}<br /> j_0\left(\frac{y}{t}\right)^2{\bf{\hat{x}}}, &amp; -\frac{t}{2} &lt; y &lt; \frac{t}{2} \\<br /> 0, &amp; \textrm{elsewhere}<br /> \end{cases}<br />
Determine the magnetic vector potential A inside and outside the sheet and hence find B.

Homework Equations

\nabla^2A_x = \mu_0 j_x = j_0\left(\frac{y}{t}\right)^2
\nabla^2A_y = \mu_0 j_y = 0
\nabla^2A_z = \mu_0 j_z = 0
\nabla \cdot {\bf{A}} = 0
\nabla\times {\bf{A}} = {\bf{B}}

The Attempt at a Solution

I know the standard equation for finding magnetic vector potential is {\bf{A}}({\bf{r}}) = \frac{\mu_0}{4\pi}\int\frac{{\bf{j}}({\bf{r&#039;}})}{|{\bf{r}}-{\bf{r&#039;}}|} \,d\tau &#039;, but that only applies when J goes to 0 at infinity. When the current is constant over an infinite region, it diverges.

I know that in calculating magnetic vector potential for an infinite wire, you can substitute in some finite limit c, calculate the integral, and then (I believe) take the limit as c\rightarrow\infty and possibly add some constant term (allowed because of the freedom in A), but I can't figure out what the appropriate integral would be in this situation, or even if that same strategy would work at all. I'm not even sure which direction A would point (though B is fairly straightforward, I think: straight up at the edge on the positive y-side, diminishing as you move to infinity or to the origin, then changing direction and doing the same thing backwards).

 

[Kirjava]

Well you've written down the differential equations for A, so have you tried solving them?

 

[superiority]

Okay, I need boundary conditions to find the solutions, right? I know that A is continuous everywhere. (Incidentally, I realized that I forgot to write \nabla^2{\bf{A}}_x = 0 outside the sheet). A is 0 at y = \infty. So that gives me

\frac{\partial^2 A^{in}_x}{\partial x^2} + \frac{\partial^2 A^{in}_x}{\partial y^2} + \frac{\partial^2 A^{in}_x}{\partial z^2} = j_0\left(\frac{y}{t}\right)^2
\frac{\partial^2 A^{in}_y}{\partial x^2} + \frac{\partial^2 A^{in}_y}{\partial y^2} + \frac{\partial^2 A^{in}_y}{\partial z^2} = 0
\frac{\partial^2 A^{in}_z}{\partial x^2} + \frac{\partial^2 A^{in}_z}{\partial y^2} + \frac{\partial^2 A^{in}_z}{\partial z^2} = 0
\frac{\partial^2 A^{out}_x}{\partial x^2} + \frac{\partial^2 A^{out}_x}{\partial y^2} + \frac{\partial^2 A^{out}_x}{\partial z^2} = 0
\frac{\partial^2 A^{out}_y}{\partial x^2} + \frac{\partial^2 A^{out}_y}{\partial y^2} + \frac{\partial^2 A^{out}_y}{\partial z^2} = 0
\frac{\partial^2 A^{out}_z}{\partial x^2} + \frac{\partial^2 A^{out}_z}{\partial y^2} + \frac{\partial^2 A^{out}_z}{\partial z^2} = 0
\frac{\partial A^{in}_x}{\partial x} + \frac{\partial A^{in}_y}{\partial y} + \frac{\partial A^{in}_z}{\partial z} = 0
\frac{\partial A^{out}_x}{\partial x} + \frac{\partial A^{out}_y}{\partial y} + \frac{\partial A^{out}_z}{\partial z} = 0
{\bf{A}}^{in}\left(y = \tfrac{t}{2}\right) = {\bf{A}}^{out}\left(y = \tfrac{t}{2}\right)
\lim_{y \to \infty}{\bf{A}}^{out}(y) = 0

But the problem is that I don't actually know how to solve for the components of A from this point.

 

[Kirjava]

In problems like these it pays enormously to think about the symmetries of your setup. You have a sheet which is infinite in the x and z directions. Would it make sense for the answer to depend on these coordinates?