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

The magnetic field for an infinitely long cylinder is given by, $B_{\text{surface}}=\frac{{\mu }_{o}JR}{2}$
$B_{\text{out}}=\frac{{\mu }_{o}J{R}^2}{2r}$​
$r=$ distance from the axis of the cylinder.
$R=$ Radius of the cylinder.


Assuming the bigger cylinder to carry a positive current density and the smaller cylinder carry a negative current density of magnitude $J$ each.


Net Magnetic field
$B=B_1(2a\text{diameter)}+B_2(a\text{diameter)}$
Therefore, magnitude of magnetic field at point P
$B=\frac{{\mu }_0(J\times a)}{2}-\frac{{\mu }_0(J\times \frac{a^2}{4})}{2\times \frac{3a}{2}}=\frac{5}{12}{\mu }_{0}Ja$
Thus, on comparing with given relation
$N=5$