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 infinitity long. A uniform current density `j` flows along the length . If the magnitude of the magnetic field at the point `P` is given by `(N)/(12) mu_(0)aJ`, then the value of `N` is

Explanation

Current dwensity `J = ( current )/( area) = (I)/( pi( 2a^(2))) = (I')/( (pia^(2))`
rArr `I = (I') = (I)/(4)`
Let us consider the cavity to have current `I')` flowing in both the directions.
the magnetic field at `P` due to the current flowing through the cylinder
` B_(1) = (mu_(0))/( 4 pi) ( 2I)/(a)`
The magnetic field at `P` due to the current `( I')` flowing in opposite direction is
` B_(2) = (mu_(0))/( 4 pi) (3I')/( 3a//2) = ( mu_(0))/( 4 pi) (2(I//4))/(3a//2) = (mu_(0))/( 4 pi) (I)/( 3a)`
:. The net magnetic field is
`B = B_(1) - B_(2) = (mu_(0))/( 4 pi) (I)/(a)[ 2 - (1)/(3)] = (mu_(0))/(4 pi) (I)/(a)xx(5)/(3)
:. `(B = mu_(0))/( 4 pi) (J pi a^(2))/(a)xx(5)/(3) = mu_(0) ( 5 Ja)/(12)