A battery is connected between two points `A and B` on the circumference of a uniform conducting ring of radius `r` and resistance `R` . One of the arcs `AB` of the ring subtends an angle `theta` at the centre . The value of the magnetic induction at the centre due to the current in the ring is

Options

A. proportional to 2 `(180^@-theta)`
B. inversely proportional to r
C. zero, only if `theta=180^@`
D. zero for all values of `theta`.

Answer

D

Explanation

Let `I_1`, `I_2` be the curernts in two parts of the ring and `R_1`, `R_2` be the resistance of two parts of ring. Then `I_1 R_1=I_2R_2`, As `Rpropl`, so `I_1l_1=I_2l_2`
Now `B_1=(mu_0)/(4pi)(I_1)/(r)theta_1=(mu_0)/(4pi)(I_1l_1)/(r^2)` `[:' theta_1=l_1/r]`
and `B_2=(mu_0)/(4pi)I_2/rtheta_2=(mu_0)/(4pi)(I_2l_2)/(r^2)` `[:' theta_2=l_2/r]`
Since `I_1l_1=I_2l_2`, so `B_1=B_2`
Here direction of `B_1` is opposite to that of `B_2`. Hence the resultant magnetic field induction at O is zero.