Question

A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude of the magnetic field at the point P is given by $\frac{N}{12}\mu_0 aJ, $ then the value of N is

Answer 1

$B_R = B_T - B_C$
R = Remaining portion
T = Total portion and
C = Cavity
$B_R=\frac{\mu_0 I_T}{2a\pi}-\frac{\mu_0 I_C}{2(3a/2)\pi}\, \, \, \, \, \, \, \, \, \, \, \, \, \, ...(i)$
$ I_T=J(\pi a^2)$
$ I_C=J\Bigg(\frac{\pi a^2}{4}\Bigg)$
Substituting the values in Eq. (i), we have
$B_R=\frac{\mu_0}{a\pi}\Bigg[\frac{\pi a^2 J}{2}-\frac{\pi a^2 J}{12}\Bigg]=\frac{5\mu_0 aJ}{12}$
$\therefore N=5$