Magnetic Field of a Thin Conducting Plate

[dww52]

A very, large, thin conducting plate lies in the x-y plane. The plate carries a current in the y direction. The current is uniformly distributed over the plate with σ amperes flowing across each meter of length perpendicular to the current. Use Ampere's Law to find the magnetic field at some distance from the plate. (Hint: The magnetic field lines are parallel to the plate.)

Homework Equations

Ampere's Law:
∫B_|| ds = μ₀*I

3. The Attempt at a Solution :
I'm completely lost on where to even start this question. I guess that you may have to use a Gausian cylinder or rectangular prism.

 

[Redbelly98]

Welcome to PF!

A very, large, thin conducting plate lies in the x-y plane. The plate carries a current in the y direction. The current is uniformly distributed over the plate with σ amperes flowing across each meter of length perpendicular to the current. Use Ampere's Law to find the magnetic field at some distance from the plate. (Hint: The magnetic field lines are parallel to the plate.)

Homework Equations

Ampere's Law:
∫B_|| ds = μ₀*I

3. The Attempt at a Solution :
I'm completely lost on where to even start this question. I guess that you may have to use a Gausian cylinder or rectangular prism.

For Ampere's Law, instead of a surface you are looking for a closed loop to draw somewhere. In general, the geometry of the problem might suggest some simple shape -- often a circle or rectangle -- that will make things work out.

 

[dww52]

So if I use a rectangular wire around the plate, do I only include the top and bottom portions of the length, since these are the only sections where the B is perpendicular to the path?

Also how do I calculate the current, is it just equal to σ?

 

[Redbelly98]

So if I use a rectangular wire around the plate, do I only include the top and bottom portions of the length, since these are the only sections where the B is perpendicular to the path?

Yes, use a rectangular path. But have a look again at Ampere's Law, you are looking for sections where B is parallel to the path:

Ampere's Law:
∫B|| ds = μ0*I

"B||" means the component of B that is parallel to the the length element ds.

Also how do I calculate the current, is it just equal to σ?

σ is the current in 1 meter of plate. So if the rectangle encloses 1 meter of the plate, yes. If the rectangle encloses some other length, then no.

Have you drawn the rectangular loop yet? The thin plate should appear as a line in the figure, with the current directed either out of or into the page.