A long cylidrical conductor of radius `R` carries a current i as shown in figure. The current desity `J` is a function of radius according to `J=br`, where `b` is a constant. Find an expression for the magnetic field `B`

a. at a distasnce `r_1ltR` and
b.at a distance `r_2gtR,` measured from the axis.

Answer

B, C

Explanation

`i=int_0^r(2pir dr)j=int_0^r(2pir dr)(br)=(2pibr^3)/3`
`a. For r_1ltR`
`B=mu_0/(2pi) i_m/r_1`
`=mu_0/(2pi) ((2pibr_1^3/3)/r_1)`
`=(mu_0br_1^2)/3`
b. For `r_2ltR`
`B=mu0/(2pi) i_m/r_2`
`=(mu_0)/(2pi) ((2pibR^3/b)/r_2)=(mu_0bR^30/(3r_2)`