Discussion Forum

1)

A cylindrical cavity of diameter c 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:

Option A

Explanation:

$B_{R}=B_{T}-B_{C}$

 R= Remaining portion

T= Total portion and

C= cavity

 $B_{R}= \frac{\mu_{0}l_{T}}{2a \pi}- \frac{\mu_{0}l_{C}}{2(3a/2)\pi}$.......(i)

 $l_{T}=J(\pi a^{2})$

$l_{C}=J\left(\frac{\pi a^{2}}{4}\right)$

Substituting the values in Eq. (i), we
have

 $B_{R}=\frac{\mu_{0}}{a \pi}\left[\frac{l_{T}}{2}-\frac{l_{C}}{3}\right]$

 

$=\frac{\mu_{0}}{a \pi}\left[\frac{\pi a^{2}J}{2}-\frac{\pi a^{2} J}{12}\right]$

$=\frac{5\mu_{0}aJ}{12}$

 $\therefore$  N=5