Field from:

The Field near an Infinite Cylinder

Remember when we were looking at electric fields inside and outside charged spherical shells? We used Gauss' Law to show that the field inside the shell was zero, and outside the shell the electric field was the same as the field from a point charge with a charge equal to the charge on the shell and placed at the center of the shell.

The analagous situation for Ampere's Law is a long cylindrical shell carrying a uniformly-distributed current I. Inside the hollow part of the cylinder the magnetic field is zero (an amperian loop encloses no current) and outside the cylinder the magnetic field is the same as that from a long straight wire placed on the axis of the cylinder:
B =
μo I

The most interesting case is the field inside the solid part of the shell. Let's say the shell has an inner radius a and an outer radius b. What is the field in the region a < r < b?

First we need the current density, J, the current per unit area. The area of the shell is:

A = πb2 - πa2
Therefore J =
πb2 - πa2

Apply Ampere's Law to an amperian loop of radius r in the solid part of the cylindrical shell.

Bdl = B dl = B dl = μo Ienc

The left-hand side of the equation is easy to calculate.

B dl = B 2πr

The right-hand side is a little trickier. How much current is enclosed by the loop?

The area enclosed is:

Aenc = πr2 - πa2

The enclosed current is then Ienc = J Aenc
Ienc = J (πr2 - πa2) =
I (r2 - a2)
b2 - a2

Putting all the pieces of Ampere's Law together gives:
B 2πr =
μo I (r2 - a2)
b2 - a2
B =
μo I (r2 - a2)
2πr (b2 - a2)
in the region a < r < b