# Ampere's Law: Determining magnetic fields of a shell conductor

1. Mar 25, 2013

### Renaldo

1. The problem statement, all variables and given/known data

A current of constant density, J0, flows through a very long cylindrical conducting shell with inner radius a and outer radius b. What is the magnetic field in the regions r < a, a < r < b, and r > b? (Use any variable or symbol stated above along with the following as necessary: μ0.)

2. Relevant equations

Ampere's Law
$\oint B \bullet ds$ = μ0ienc

3. The attempt at a solution

$\oint B \bullet ds$ = μ0ienc

Solving for B:

B$\oint ds$ = μ0ienc
B2∏r = μ0ienc

B = μ0ienc/2∏r

At r < a:
B = 0 because ienc at this point = 0

At a < r < b:

B = μ0ienc/2∏r

I don't know how to get ienc. I know it has something to do with the current density.

At r > b:

ienc = itotal, but I would need an expression for the volume of the cylinder.

V = ∏b2h - ∏a2h
V = ∏h(b2-a2)
Ienc = J0V $\Rightarrow$ Ienc = J0∏h(b2-a2)

B = μ0ienc/2∏r

B = μ0J0h(b2-a2)/2r

However, by including h, I am introducing variables that the problem hasn't allowed me to use.

Last edited: Mar 25, 2013
2. Mar 25, 2013

### rude man

Current = current density times cross-sectional area. What is the cross-sectional area of your cylinder at a < r < b?
No. Volume does not enter the picture. Cross-section does.

3. Mar 25, 2013

### Renaldo

That makes sense.

So at a < r < b:

B = μ0J0(r2-a2)/2r

At r > b:

B = μ0J0(b2-a2)/2r

That worked. Thanks for your help.

Last edited: Mar 25, 2013
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