Question

Consider a long wire where the current density is not uniform but instead increases as you approach the center of the wire, so that at a distance r from the center the current density is J(r)= I/ 2PiRr. Find the magnetic field strength both inside and outside of this wire.
I know I need to use biot-savart law, but im not sure how to do this with a non uniform density

Answer 1

Integrate the current from r = 0 to r = r, using the current density.

I(r) = Integral (r=0 to r) of J*2*pi*r dr

= I r/R where R is the radius of the wire


Then use Ampere's law that says the integral of H (=B/mu) around the loop equals the current flowing through the loop. The value of B will be uniform around that circular loop because of symmetry.

B(r) = mu*2*pi*(r/R)*(I/r)
= mu*2*pi*(I/R)
which is independent of r.

Outside the wire, use Ampere's law again:

B(r) = mu*2*pi(*I/r)