An infinitely long hollow conducting cylinder with inner radius `(r )/(2)` and outer radius `R` carries a uniform current ra density along its length . The magnitude of the magnetic field , ` | vec(B)|` as a function of the radial distance `r` from the axis is best represented by

Options

A.
B.
C.
D.

Explanation

Let I be the current through the long hollow conducting cylinder, as shown in figure.
When point P is inside the hollow cylinder (i.e., `r lt R//2`), then
`oint vecB.dvecl=mu_0 xx`current threading the closed path
`B2pir=mu_0xx0=0` or `B=0`
when point P is inbetween inner and outer part of conducting cylinders. (i.e., `R//2 lerltR`)
`B2pir=mu_0(Ipi[r^2-(R//2)^2])/(pi[R^2-(R//2)^2]`

`B=(mu_0)/(2pir)(I[r^2-(R//2)^2])/(3R^2//4)=(2mu_0I)/(3piR^2)[r-(R^2)/(4r)]`
`=(2mu_0)/(3piR^2)Jpi[(R^2-(R//2)^2][r-(R^2)/(4r)]`
`[:' I=Jpi(R^2-R^2//4)]`
`=(mu_0J)/(2)[r-(R^2)/(4r)]`, i.e., `Bprop(r-R^2//4r)`
When point P is outside the cylinder, i.e., `rgtR`, then
`B2pir=mu_0I` or `B=(mu_0I)/(2pir)`, i.e., `Bprop1/r`
Therefore, option (d) is true.