A uniform dielectric hollow cylinder of mass M, radius R, length $l$ ( $l > > R$ ) carrying uniform charge of surface charge density $σ$ can rotate without friction about a fixed horizontal axle that coincides with the axis of the cylinder. Several turns of a light thin insulating cord are wrapped on the cylinder and a block of mass m is suspended from the free end of the cord. Initially the block is held at rest as shown in the figure and it is released. Neglect charge transferred to the cord and fringing of magnetic field at the ends of the cylinder and assuming magnetic field B is uniform inside the cylinder. Acceleration due to gravity is g and permeability of the medium inside the cylinder is $μ_{0}$
After the block has fallen through a height ‘h’, the kinetic energy of the cylinder + block system is
Case 1: $K_{1}$ , if the cylinder does not have any charge
Case 2: $K_{2}$ , if the cylinder has charge density $σ$

Question

Infinity Learn · Accepted Answer

$K_{1} > K_{2}$

,

If magnetic field inside cylinder in Case 2 at the given instant is B, then $K_{1} - K_{2}$ = $\frac{B^{2}}{2 μ_{0}} π R^{2} l$

,

In Case 2, acceleration of the block after it is released is $\frac{m g}{m + M + π μ_{0} σ^{2} R^{2} l}$

,

In Case 2, Electric field produced by time varying magnetic field inside cylinder cannot exert torque on cylinder itself

Detailed Solution