There exists a long conductor along z-axis carrying a current `I_0` along positive z-direction. A loop having total resistance R is placed symmetrical about x-axis and y-axes in x-y plane as shown in Fig. potential difference `V_(BA)=V` is applied. Radii of arcs are a and b, respectively, as shown in the figure.

The total torque acting on the loop is nearly

Options

A. `(mu_(0)VI_(0))/(piR)(b-a)hati`
B. `-2(mu_(0)VI_(0))/(piR)(b-a)hati`
C. `(mu_(0)VI_(0))/(2piR)(b-a)hati`
D. None of these

Answer

B

Explanation

Current from B to A (through NM or QP):
`I=V/(R//2)=(2V)/R`
Force due to `I_(0)` on N'N and `Q'Q` wilbe out of the paper and on M'M and P'P will be in the paper . This will produce the torque on loop along -ve x-axis.
Torque on M'M (along -ve x-axis)
`tau=int((mu_(0)I_(0))/(2pir)Idr)r sin ((pi)/6)=(mu_(0)I_(0))/(4pi)=(b-a), -(mu_(0)I_(0)2v)/(4Rp)(b-a)`
Total torque `=4tau=2(mu_(0)I_(0)V)/(piR)(b-a)` (along negative x-axis)