Question
A cylindrical cavity of diameter 'a' exists inside a long solid cylinder of diameter '2a' as shown in figure. Both the cylinder and the cavity are taken to be infinitely long. The axis of the cavity is parallel to the axis of the cylinder and is at distance a/2 from it. A uniform current of current density J (A m−2) flows through the cylinder along its length and not through the cavity. The magnitude of the magnetic field at a point P on the surface of the cylinder lying farther from the axis of the cavity, is :
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Solution
[Key Concept: A cavity carrying no current is equivalent to the full cylinder with current density J plus a cavity-sized cylinder with current density −J.]
We can treat cavity as superposition of a full solid cylinder carrying current density as +J and a smaller cylinder (the cavity) carrying current density -J.
$$\begin{matrix} & \left( {\overrightarrow{B}}_{\text{net~}} \right)_{\text{at point~}P} = {\overrightarrow{B}}_{\text{Big cylinder~}} + {\overrightarrow{B}}_{\text{small cylinder~}} \\ & \ = \frac{\mu_{0}I_{1}}{2\pi R} + \frac{\mu_{0}I_{2}}{2\pi r^{'}} \\ & \ = \frac{\mu_{0}\text{ }J\pi a^{2}}{2\pi a} - \frac{\mu_{0}\text{ }J\left( \frac{\pi a^{2}}{4} \right)}{2\pi\left( \frac{3a}{2} \right)} \\ & \ = \frac{\mu_{0}Ja}{2} - \frac{\mu_{0}Ja}{12} \\ & \ = \frac{5\mu_{0}Ja}{12} \end{matrix}$$
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