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.

Explanation

(a) `r_(1)lt R`
Consider a circle of radius `r` and thickness `dr`
`dA=2pir dr`
`di_(1)=jdA=2 cr. 2pirdr`
Current inside circle of radius `r_(1)(r_(1)lt R)`
`i_(1)=4picint_(0)^(r_(1))r^(2)dr=(4picr_(1)^(3))/3`
`B.2pir_(1)=mu_(0)i_(1)=(4picr_(1)^(3)mu_(0))/3`
`B=(2cr_(1)^(2)mu_(0))/3`
(b) `r_(2)gtR`
Current inside circle of raduis `r_(2)`, current exists only in circle of radius `R`
`i_(2)=4picint_(0)^(R) r^(2)dr=(4picR^(3))/3`
`B.2pir_(2)=mu_(0)i_(2)=(mu_(0).4picR^(3))/3`
`B=(2cR^(3)mu_(0))/(3r_(2))`