Magnetic Field and infinite cylindrical shell

[Varnson]

Homework Statement

An infinite cylindrical shell has inner radius 'a' and outer radius 'b'. The shell carries a charge density J(s) = Cs^2 + Ds. Where 'C' and 'D' are constants. Find B everywhere.

Homework Equations

The Attempt at a Solution

My thought was to find the field do to the inner radius, the field do to outer radius, and then subtract inner from outer. Is this the correct thought process?

 

[OlderDan]

Homework Statement

An infinite cylindrical shell has inner radius 'a' and outer radius 'b'. The shell carries a charge density J(s) = Cs^2 + Ds. Where 'C' and 'D' are constants. Find B everywhere.

Homework Equations

The Attempt at a Solution

My thought was to find the field do to the inner radius, the field do to outer radius, and then subtract inner from outer. Is this the correct thought process?

Not really. There is a current density at all radii between the inner and outer surfaces of the cylinder. By symmetry, you can predict the direction of the field in space. Which equation or law is relevant to finding the magnetic field from a current distribution? You will want to look at the three regions separately: ousie the cylinder, inside the cylinder, and between the walls.

 

[Varnson]

After I made the post, I looked at the problem again and realized that I would need to look at those 3 cases. The equation I was planning on using was I = } J * da. Where '}' is the integral and '*' is the dot product. I am also thinking that I enclosed for the inside the cylinder = 0 because it is a shell. Am I correct on assuming that? Sorry for the lack of clarity with the equations I write, these are my first few posts, and I am not sure how to incorporate easy to read equations like the other posts have. Thanks for the help!

 

[FrogPad]

After I made the post, I looked at the problem again and realized that I would need to look at those 3 cases. The equation I was planning on using was I = } J * da. Where '}' is the integral and '*' is the dot product. I am also thinking that I enclosed for the inside the cylinder = 0 because it is a shell. Am I correct on assuming that? Sorry for the lack of clarity with the equations I write, these are my first few posts, and I am not sure how to incorporate easy to read equations like the other posts have. Thanks for the help!

The equation your are trying to represent with 'ASCII art' is:
I = \int_S \vec J \cdot d \vec s

Just click on the equation above to see how to write it. There are a few guides on the forum to help you out. The language is called LaTeX.

Since you have a lot of symmetry what law can you use to help you out in finding the B field?

 

[Varnson]

Is the law you are talking about me being able to use Ampere's Law?

 

[OlderDan]

Is the law you are talking about me being able to use Ampere's Law?

That would be the one. The integral is simple if you use the symmetry. The more difficult part is the integral of Jda in the one case, but I think you can handle that.

 

[Varnson]

What symmetry could I take advantage of?

 

[estel]

It's cylindrical, so take the current enclosed in a loop of radius "s" around the cylinder, which lies so that the area of the loop is perpendicular to the current. The magnetic field will then be parallel to the Amperian loop.

 

[OlderDan]

What symmetry could I take advantage of?

estel is correct, but to be more explicit suppose you are at some point a distance s from the axis of the cylinder. If you move to another point at the same distance from the axis, how will the magnetic field change? OR suppose you stay in the same place and rotate the wire about its axis. What happens to the field at your location? What do you know about the direction of the field based on your experience with simpler current densities, such as the current in a wire?

 

[Varnson]

I figured out the problem, thanks for the help! I am sure I will be back on the forum in the near future.