An infinite cylindrical wire of radius R and having current density varying with its radius r as, `J = J_(0)[1-(r//R)]`. Then answer the following questions.

Graph between the magnetic field and radius is

Options

A. R
B. `3R//4`
C. `r//2`
D. `R//4`

Answer

B

Explanation

Current with in radius r:` I= int_()^(r) J dA`
`implies I= int_(0)^(r) J_(0)[1-(R)/(R)] 2pi r dr = 2pi j_(0)[(r^2)/(2)-(r^3)/(3R)]`
Apply ampere's law `B2pi r = mu_(0)I`
`B 2 pi r = mu_(0)2pi J_(0)((r^2)/(2)-(r^3)/(3R)) implies B=mu_(0)J_(0)[(r)/(2)-(r^(2))/(3R)]`
to find max B: `(dB)/(dr) = 0 implies r=3R//4`
`B_(max)=mu_(0)J_90[(3R//4)/(2)-((3R//4)^(2))/(3r)]=(3R)/(16) mu_(0)J_(0)`.