Magnetic field of a semi infinite sheet of current

[lys04]

Homework Statement

Finding the magnetic field due to a semi infinite sheet of current, infinitely long in the z direction and have finite width in the x direction, let the width be L and have current flowing in the positive z direction.

Relevant Equations

$$\vec{B}=\nabla \times \vec{A}$$

Here's what I'm thinking:
Since the width is L and the current is flowing in positive z direction, there is a surface current density of $$\vec{K}=\frac{I}{L} \vec{z}$$
Find the vector potential due to one infinitely long wire in the z direction
Add a lot of them together to form a finite width sheet
Then find the magnetic field from the vector potential.
But I'm not sure how to do the first step. Is the formula $$ \vec{A}=\frac{\mu_0}{4\pi}\int \frac{\vec{I}}{r} da'$$ relevant? I'm not sure since my current at infinity doesn't go to 0.

 

[kuruman]

Why bother with the magnetic vector potential? Just consider the superposition of fields from very long wires. You did not specify the point relative to the sheet where the magnetic field is to be found.

 

[renormalize]

But I'm not sure how to do the first step.

For a current-carrying cylindrical wire that's infinite in the $z$-direction, just use Ampere's Circuital Law to get the magnetic field $\vec B$ outside the wire. Here's a simple illustration from https://www.sciencefacts.net/amperes-law.html:

Here $\vec B$ points in the $\hat{\phi}$-direction of cylindrical-coordinates centered on the wire. That was easy!
The tricky part will be to form a finite-width current sheet by carefully summing the various fields that arise from $N$ wires placed parallel to each other and spaced a distance $d$ apart to get $\vec B_{\text{total}}(N,d)$, and then taking the limit as $N\rightarrow\infty,d\rightarrow 0$.

 

[lys04]

The tricky part will be to form a finite-width current sheet by carefully summing the various fields that arise from N wires placed parallel to each other and spaced a distance d apart to get B→total(N,d), and then taking the limit as N→∞,d→0.

Yeah, I'm not sure how to do that since in the case where the width was infinite by symmetry the fields below and above would only have horizontal components and the vertical ones would all cancel out.