A cell is connected between two points of a uniformly thick circular conductor and `i_1 and i_2` are the currents flowing in two parts of the circular conductor of radius a The magnetic field at the centre of the loop will be

Options

A. zero
B. `(mu_0)/(4pi)(I_1-I_2)`
C. `(mu_0)/(2a)(I_1+I_2)`
D. `(mu_0)/a(I_1+I_2)`

Answer

A

Explanation

Let `l_1,l_2` be the lengths of the two parts PRQ and PSQ of the conductor and `rho` be the resistance per unit length of the conductor . The resistance of the portion PRQ will be `R_1 =l_1rho`

The resistance of the portion PSQ will be `R_2 = l_2 rho` pot diff across P and ` Q = I_1 , R _1 = I_2 R_2`
`I_1 l_1 rho = I_2 rho ` or `I_1 l_1 =I_2 l_2 " "....(1)`
Magnetic field induction at the centre O due to currents through circular conductors PRQ and PSQ will be
`=B_1 -B_2 =(mu_0)/2 (I_1l_1sin 90^@)/(r^2)-(mu_0)/(4pi)(I_2l_2sin 90^@)/r^2=0`