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 of the cylinder. If the magnitude of the magnetic field at the point P is given by $\frac{N}{12} μ_{0} a J$ , then the value of $N$ is……

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answer is 5.

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Detailed Solution

$B = B_{1} - B_{2}$ , where $B_{1}$ and $B_{2}$ are magnetic fields at P due to full solid cylinder carrying current and that due to the cavity-cylinder (considered carrying current) respectively.

$B_{1} = \frac{μ_{0} J π a^{2}}{2 π a} = \frac{μ_{0} J a}{2}, B_{2} = \frac{μ_{0} J π {(a / 2)}^{2}}{2 π (3 a / 2)} = \frac{μ_{0} J a}{12}$

$B = \frac{μ_{0} J a}{2} - \frac{μ_{0} J a}{12} = \frac{5 μ_{0} J a}{12}$

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