A long cylindrical conductor of radius R carries a current I

A long, cylindrical conductor of radius R carries a current I as shown in the figure below. The current density J, however, is not uniform over the cross-section of the conductor but is a function of the radius according to J = 2br, where b is a constant. Find an expression for the magnetic field magnitude B at the following distances, measured from the axis. (Use the following variables as necessary: mu_0, r_1, r_2, b, R.) r_1 R

Solution

(a) for r1 < R

lets first find the current value for this cross section,

dI = J.A = 2 b r (2 pi r ) dr

integrating and r is from 0 to r1.

I = 4 pi b r1^3 / 3

now Using Ampere\'s Law,

B. L = u0 I

B(2 pi r1) = u0 ( 4 pi b r1^3 / 3)

B = 2 u0 b r1^2 / 3

(b) for r2 > R

for this current will be

(limit will be from 0 to R )

I = 4 pi b R^3 / 3

applying Ampere\'s law,

B( 2 pi r2) = u0 ( 4 pi b R^3 / 3)

B = 2 u0 b R^3 / 3 r2