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

Answer: 5

$B_R=B_T-B_C$
R = Remaining portion
T = Total portion and
C = Cavity
$B_R=\frac{{\mu }_0{I}_T}{2a\pi }-\frac{{\mu }_0{I}_C}{2(3a/2)\pi }...(i)$
$I_T=J(\pi a^2)$
$I_C=J\left(\frac{\pi a^2}{4}\right)$
Substituting the values in Eq. (i), we have
$B_R=\frac{{\mu }_0}{a\pi }[\frac{\pi a^{2}J}{2}-\frac{\pi a^{2}J}{12}]=\frac{5{\mu }_{0}aJ}{12}$
$\therefore N=5$