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)/(4sqrt(2)R)`
B. `(V_(0)AB)/(sqrt(2)R)`
C. `(V_(0)AB)/(2sqrt(2)R)`
D. `(3V_(0)AB)/(sqrt(2)R)`

Answer

1

Explanation

`F_(1)=I_(1)lB,tau_(1)=F_(1)((d)/(2))cos45^(@)=(I_(1)lB)(d)/(2sqrt(2))`
and `D_(2)=I_(2)lB,tau_(2)=F_(2)((d)/(2))cos45^(@)=(I_(2)lB)(d)/(2sqrt(2))`
Since `vec(F)_(1)` and `vec(F)_(2)` are in the same direction net torque exerted by the magnetic field on the lop, i.e.
`tau=tau_(1)-tau_(2)(I_(1)-I_(2))(lBd)/(2sqrt(2))`
Since `I_(1)=(V_(0))/(R)` and `I_(2)=(V_(0))/(2R)`
`tau=((V_(0))/(R)-(V_(0))/(2R))(lBd)/(2sqrt(2))=(V_(0)AB)/(4sqrt(2)R)`
`( :' Id=A,)` area of the loop.