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:
$${\bf{j}}({\bf{r}}) = \begin{cases}<br /> j_0\left(\frac{y}{t}\right)^2{\bf{\hat{x}}}, & -\frac{t}{2} < y < \frac{t}{2} \\<br /> 0, & \textrm{elsewhere}<br /> \end{cases}$$
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 ${\bf{A}}({\bf{r}}) = \frac{\mu_0}{4\pi}\int\frac{{\bf{j}}({\bf{r'}})}{|{\bf{r}}-{\bf{r'}}|} \,d\tau '$, 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?