Question

A rectangular conducting loop consists of two wires on two opposite sides of length l joined together by rods of length d. The wires are each of the same material but with cross-sections differing by a factor of 2. The thicker wire has a resistance R and the rods are of low resistance, which in turn are connected to a constant voltage source `V_0`. The loop is placed in a uniform magnetic field B at `45^@` to its plane. Find `tau`, the torque exerted by the magnetic field on the loop about an axis through the centres of rods.

Answer 1

Thicker wire has a resistance R, then the other wire has a resistance 2R as the wires are of the same material but with cross-sections differing by a factor 2.
Now, the force and hence, torque on first wire is given by
`F_(1)=i_(1)iB=(V_(0))/(2R)lB,tau_(1)=(d)/(2sqrt(2))F_(1)=(V_(0)ldB)/(2sqrt(2)R)`
Similarly, the force hence torque on otehr wire is given by
`F_(2)=i_(2)lB=(V_(0))/(2R)lB,tau_(2)=(d)/(2sqrt(2))F_(2)=(V_(0)ldB)/(4sqrt(2)R)`
So, net torque `tau=tau_(1)-tau_(2)`
`tau=(1)/(4sqrt(2))(V_(0)ldB)/(R)`