A conducting ring of mass m and radius r has a weightless conducting rod PQ of length 2r and resistance 2R attached to it along its diametre. It is pivoted at its centre C with its plane vertical, and two blocks of mass m and 2m are suspended by means of a light inextensible string passing over it as shown in Fig. The ring is free to rotate about C and the system is placed in magnetic field B (into the plane of the ring). A circuit is now complete by connecting the ring at A and C to a battery of emf V. It is found that for certain value of V , the system remains static.

The value of V can be related with m, B and r as

Explanation

Resistance of `PQ=2R`
Since equivalent resistance of the circuit is `R//2`, in static condition current through the battery is `I=2V//R`
The current through `CP` and `CQ` is `i=I//2=V//R`
Consider the FBD of ring and the blocks
Consider torque about `'C'`
`tau_(N)=0`
`tau_(mg)=0`, `tau_(F2)=(Bir^(2))/(2)(hatk)`
`tau_(T1)=T_(1)rhatk`
`tau_(T_(2))=T_(2)r(-hatk)`
`tau_(f_(1))=(Bir^(2))/(2)hatk` ltbr gt `:' tau_(N)+tau_(mg)+tau_(F_(1))+tau_(F_(1))=0`
`rArr T_(1)r-T_(2)r+(Bir^(2))/(2)+(Bir^(2))/(2)=0` .....(`i`)
As system is in static equilibrium
`T_(1)=mg`.....(`i`)
and `T_(2)=2mg`.....(`ii`)
Putting the values of `T_(1)` and `T_(2)` in equation (`i`)
`mgr-2mgr+Bir^(2)=0`
`rArri=(mg)/(Br)rArr(V)/(R )=(mg)/(Br)rArrV=(mg)/(Br)`