A non-conducting disc of radius $r$ is uniformly charged with surface charge density $σ$ on one side only and it is pivoted at its centre in a vertical plane and it can rotate freely about a horizontal axis passing through the centre, in a time-varying uniform magnetic field $B = 2 t$ as shown in the figure. A block of mass $m$ is suspended from one end of a massless, inextensible and insulating thread wrapped over the periphery of disc. If the block remains stationary, the mass of the block is equal to $\frac{π σ r^{3}}{n g}$ . Find the value of $n$ .

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Detailed Solution

$E \cdot 2 π r = 2 \cdot π r^{2} \Rightarrow E = r$

Torque applied on an elemental ring,

$d τ = d q \cdot E \cdot r$

Total torque due to $\vec{E}$

$τ_{E} = \int σ \cdot 2 π r d r \cdot r \cdot r = \frac{2 π σ r^{4}}{4}$

Torque due to weight $= m g r$

$\Rightarrow \frac{π σ r^{4}}{2} = m g r$

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