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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 [itex]y=\pm\frac{t}{2}[/itex]. The current density in the sheet is given by:
[tex]{\bf{j}}({\bf{r}}) = \begin{cases}
j_0\left(\frac{y}{t}\right)^2{\bf{\hat{x}}}, & -\frac{t}{2} < y < \frac{t}{2} \\
0, & \textrm{elsewhere}
\end{cases}
[/tex]
Determine the magnetic vector potential A inside and outside the sheet and hence find B.
Homework Equations
[tex]\nabla^2A_x = \mu_0 j_x = j_0\left(\frac{y}{t}\right)^2[/tex]
[tex]\nabla^2A_y = \mu_0 j_y = 0[/tex]
[tex]\nabla^2A_z = \mu_0 j_z = 0[/tex]
[tex]\nabla \cdot {\bf{A}} = 0[/tex]
[tex]\nabla\times {\bf{A}} = {\bf{B}}[/tex]
The Attempt at a Solution
I know the standard equation for finding magnetic vector potential is [itex]{\bf{A}}({\bf{r}}) = \frac{\mu_0}{4\pi}\int\frac{{\bf{j}}({\bf{r'}})}{|{\bf{r}}-{\bf{r'}}|} \,d\tau '[/itex], 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 [itex]c\rightarrow\infty[/itex] 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).