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.

Explanation

force on arm AB due to magnetic field,
`F_1=I_1lB=V_0/R_1lB=(V_0)/(2R)lB`
It is acting `_|_r` to the arm AB towards observer.
Torque due to force `F_1` about axis `EF` is,
`tau_1=F_1xxd/2sin45^@=((V_0)/(2R)lB)xxd/2xx1/sqrt2=(V_0lBd)/(4sqrt2R)`
(anticlockwise)
Force on arm DC due to magnetic field,
`F_2=I_2lB=(V_0)/(R_2)lB=(V_0)/(R)lB`.
It is acting `_|_r` to arm DC towards observer,
Torque due to force `F_2` about axis `EF` is
`tau_2=F_2xxd/2sin45^@=(V_0/RlB)d/2xx1/sqrt2=(V_0lBd)/(2sqrt2R)` (clockwise)
Net torque, `tau=tau_2-tau_1` (clockwise)
`=(V_0lBd)/(2sqrt2R)[1-1/2]=(V_0lBd)/(4sqrt2R)=(V_0AB)/(4sqrt2R)`