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

Options

A. `2mgR//Br`
B. `mgR//Br`
C. `mgR//2Br`
D. `3mgR//Br`

Answer

b

Explanation

PC and CQ are in parallel. So, 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`
Hence, `F_1=F_2=Bir`
Consider the FBD of ring and the blocks
FBD of ring: Consider torque about 'C',
`T_1r+F_1r//2+F_2r//2=T_2r....(i)`
FBD of blocks : `T_1=mg and T_2=mg`
So, `mgr+Bir^2//2+Bir^2//2=2mgr`
or `i=(mg)/(Br) or V/R=(mg)/(Br) or V=(mgR)/(Br)`
Net torque of tension
`=(T_2-T_1)r =mgr=(BVr^2)/R`