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.

 

[asteriatic]

But I'm not sure how to incorporate the current into the equation. Can anyone provide some guidance or clarification?

I would first approach this problem by reviewing the relevant equations and concepts related to Ampere's Law and magnetic fields. Ampere's Law states that the integral of the magnetic field around a closed path is equal to the permeability of free space (μ0) multiplied by the current passing through the surface enclosed by the path. In this case, we have a thin conducting plate with a uniform current distribution, so we can use a rectangular path with one side parallel to the plate and the other side perpendicular to the current.

Next, we need to consider the direction of the magnetic field lines. Since the current is flowing in the y direction, the magnetic field lines will be parallel to the plate. This means that the magnetic field will only have a component in the z direction, and we can simplify our path to just the two parallel sides.

Now, we can use Ampere's Law to find the magnetic field at a distance from the plate. We know that the magnetic field lines are parallel to the plate, so the integral of the magnetic field along the path will simply be B|| times the length of the path. We also know that the current is uniformly distributed, so we can use the current density (σ) to find the total current passing through the path.

Putting all of this together, we get the following equation:
B|| * L = μ0 * σ * L
where B|| is the magnetic field, L is the length of the path, μ0 is the permeability of free space, and σ is the current density.

We can rearrange this equation to solve for the magnetic field:
B|| = μ0 * σ
This tells us that the magnetic field is directly proportional to the current density and the permeability of free space. As the current density or the permeability increases, so does the magnetic field.

In conclusion, using Ampere's Law, we can determine the magnetic field at a distance from a thin conducting plate carrying a current in the y direction. By considering the direction of the magnetic field lines and the uniform current distribution, we can simplify the problem and use a rectangular path to find the magnetic field. The resulting equation shows that the magnetic field is directly proportional to the current density and the permeability of free space.