A ring of radius R having uniformily distibuted charge Q is mounted on a rod suspended by two identical strings. the tension in the strings in equilibrium is To . now , a vertical magnetic field is switched on and ring is rotated at constant angular velocity w. find the maximum w with which the ring can be rotated if the strings can withstand a maximum tension of 3To/2 .

when the magnetic field is turned on, will there be different tensions in the two joining strings ?? why ??

In equilibrium: 2To​=mg
or To​=2mg​ ...(i)

Magnetic moment,M=iA=(2πω​Q)(πR2)
τ=MBsin90o=2ωBQR2​

LetT1​andT2​be the tensions in the two strings when magnetic field is switched on(T1​>T2​).
For translational equilibrium of ring is vertical direction,
T1​+T2​=mg ...(ii)

For rotational equilibrium,
(T1​−T2​)2D​=τ=2ωBQR2​
or T1​−T2​=2ωBQR2​ ...(iii)

Solving equations (ii) and (iii) we have
T1​=2mg​+2DωBQR2​
AsT1​>T2​and maximum values ofT1​can be23To​​, we have
23To​​=To​+2Dωmax​BQR2​
∴ωmax​=BQR2DTo​​