A cylindrical cavity of diameter a exists inside a cylinder of diameter \(2 a\) as shown in the figure. Both the cylinder and the cavity are infinity long. A uniform current density \(J\) flows along the length. If the magnitude of the magnetic field at the point \(P\) is given by \(\frac{N}{12} \mu_{0} a J\), then the value of \(N\) is

Current density \(J=\frac{\text { current }}{\text { area }}=\frac{I}{A} \Rightarrow I=\mathrm{JA}\) Magnetic field \(\mathrm{B}_{\mathrm{R}}\) after removing cavity \((\mathrm{C})\) \(\mathrm{B}_{\mathrm{R}}=\mathrm{B}_{\text {total }}-\mathrm{B}_{\text {cavity }}\)
\(\frac{\mu_{0} I_{t}}{2 \pi a}-\frac{\mu_{0} I_{c}}{2 \pi\left(\frac{3}{2} a\right)} \)
\( \quad=\frac{\mu_{0}}{\pi a}\left[\frac{I_{t}}{2}-\frac{I_{c}}{3}\right](\text { here } I_{t}=J\left(\pi a^{2}\right)\) \(I_{c}=J\left(\frac{\pi a^{2}}{4}\right)) \)
\( =\frac{\mu_{0}}{\pi a}\left[\frac{\pi a^{2} J}{2}-\frac{\pi a^{2} J}{12}\right] \)
\( \text { or, } \mathrm{B}_{\mathrm{R}}=\frac{5 \mu_{0} a J}{12}\)
Comparing it with \(\frac{N}{12} \mu_{0} a J\) We get \(N=5\)