An infinitely long hollow conducting cylinder with inner radius `(R)/(2)` and outer radius `R` carries a uniform current 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.

Answer

D

Explanation

`r=` distance of a point from centre
For `r le R//2` using Ampere's circuital law, `ointB.dl` or `B1=mu_(0)(I_("in"))`
or `B(2pir)=mu_(0)(I_("in"))` or `B=(mu_(0))/(2pi)(I_("in"))/r`…….i
Since `I_("in")=0implies:.B=0` for `R/2lerleR I_("in")=[pir^(2)-pi(R/2)^(2)]sigmas`
Here `sigma=` current per unit area. Substituting in eq. (i) we have

`B=(m_(0))/(2pi)([pir^(2)-pi(R^(2))/4]sigma)/r=(mu_(0)sigma)/(2r)(r^(2)-(R^(2))/4)` at `r=R/2, B=0` at `r=R,B=(3mu_(0)sigmaR)/8` For `rgeR`
`I_("in")=I_("Total")=I` (say) Therefore, substituting in eq. (i) we have `B=(mu_(0))/(2pi) . I/r` or `B prop 1/r`