A solid, non-conducting sphere has charge $Q$ uniformly distributed over its surface and is rotated uniformly with a constant angular velocity $ω$ about its diameter (which coincides with z axis) as shown. The value of the integral $\int_{- \infty}^{+ \infty} \vec{B} \cdot \vec{d l}$ along the entire z axis (where B is due to moving charges on the sphere) is of magnitude $\frac{μ_{0} Q ω}{n π}$ . Find the value of $n$ .

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Detailed Solution

Imagine the infinite line as a closed loop with a small part of this infinite loop, having length $2 R$ , passing through the centre of the sphere. As the sphere is rotating, current threading the loop is equal to the charge on the sphere divided by time period of rotation

$i = \frac{Q}{T} = \frac{Q ω}{2 π}$

Now using Ampere's circuital law, for the closed loop,

$\oint \vec{B} \cdot \vec{d l} = μ_{0} i = \frac{μ_{0} Q ω}{2 π}$