Consider two long concentric current-carrying cylinders. The inner cylinder is solid with a radius R and has a current I uniformly distributed over the cross-sectional area of the cylinder. The outer cylinder is a thin cylindrical shell of radius 2R and current 2I in a direction opposite to the current in the inner cylinder.
In a cross-sectional view the current is out of the page in the inner cylinder and into the page in the outer cylinder. What is the magnetic field everywhere? In the cross-sectional view define a positive field as a field with counter-clockwise field lines.
We need to consider three regions:
For r < R we only need to deal with the inner cylinder. Using the result for inside a long straight wire we get:
|For r < R,||B(r)||=||
This is positive because the current out of the page produces a counter-clockwise field.
For R < r < 2R the field is the field outside the inner cylinder. The outer cylinder contributes nothing to the field.
Outside a cylinder with a uniform current density the field looks like the field from a long straight wire:
|For R < r < 2R,||B(r)||=||
For r > 2R both cylinders contribute to the field. One way to get the answer is to determine the fields from each cylinder separately and add.
|The field from the inner cylinder is||
|The field from the outer cylinder is||
|For r > 2R,||B(r)||=||
The negative sign means the field lines are clockwise.
At what point does the field have the largest magnitude?
The field is zero at r = 0, increases until r = R, decreases from there until r = 2R, at which point (only because of the numbers we have) it just flips sign and keeps decreasing in magnitude as r increases. The peak field is at r = R.
The graph of the field is shown above. The sign comes simply from the direction we chose to be positive in this problem, that positive means the field lines are counter-clockwise on the cross-sectional view.