Ampere's law in a nonuniform conducting cylinder.

[13pr3ch4un]

Homework Statement

A long, cylindrical conductor of radius R carries a current I. The current density, J, is not uniform and is given by the equation J = br, where b is a constant. Find an expression for the magnetic field magnitude B at distance r < R and at distance r > R.

Homework Equations

I = ∫JxdA
B= μI/(2pi*r)

The Attempt at a Solution

So I think that I have this figured out, but it's an even numbered problem so I don't know if I'm correct.
a) First finding I inside r
I = ∫JxdA = ∫br2(pi)rdr = 2(pi)b∫r^2dr = 2/3(pi)br^3 when integrating from 0 to r

So B = μI/(2(pi)r) = 2μ(pi)br^3/(6(pi)r) = μbr^2/3

b) First finding total I
I = ∫JxdA = ∫br2(pi)rdr = 2(pi)b∫r^2dr = 2/3(pi)bR^3 when integrating from 0 to R

so B = μI/(2(pi)r) = 2μ(pi)bR^3/(6(pi)r) = μbR^3/(3r)




I'm pretty sure it's correct, but not entirely sure.

 

[anathema]

Can anyone confirm or correct me?

Hi there! Your solution looks correct to me. The expression for B at a distance r < R is μbr^2/3 and at a distance r > R is μbR^3/(3r). Good job!