The Field Inside a Current-Carrying Wire

Let's use Ampere's Law to find the field inside a long straight wire of radius R carrying a current I. Assume the wire has a uniform current per unit area:

J = I/πR2

To find the magnetic field at a radius r inside the wire, draw a circular loop of radius r. The magnetic field should still go in circular loops, just as it does outside the wire.

Apply Ampere's Law:

Bds = 2πrB = μo Ienc

The current passing through our loop is the current per unit area multiplied by the area of the loop:

Ienc = Js πr2 = Ir2/R2

Therefore 2πrB = μo Ir2/R2

B = μoIr/2πR2

So, inside the wire the magnetic field is proportional to r, while outside it's proportional to 1/r.