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.

Options

A. `(V_0 AB)/(4 sqrt 2 R) `
B. `(V_0 AB)/(sqrt2 R)`
C. `(V_0AB)/(2sqrt2 R)`
D. `(3V_0 AB)/(sqrt2R)`

Answer

A

Explanation

`F_1 = I_1lB, tau_1 = F_1 (d/2) cos 45^@ = (I_1lB) (d)/(2sqrt2)` and `F_2 = I_2 lB, tau_2 = F_2 (d/2) cos 45^@`
` = (I_2lB) (d)/(2sqrt2)`
Since `vecF_1` and `vecF_2` are in the same direction net torque exerted by the magnetic field on the loop i.e.,
` tau = tau_1 - tau_2 (I_1 - I_2) (lBd)/(2sqrt2)`
Since `I_1= (V_0)/(R ) " and " I_2 = (V_0)/(2R)`
`tau = ( (V_0)/(R ) - (V_0)/(2R) ) (lBd)/(2sqrt2) = (V_0AB)/(4 sqrt2 R)`
`(because Id = A)` are of the loop