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

`b_(1)=(mu_(0)I)/(2R)xx(2pi-theta)/(2pi),B_(2)=(mu_(0)I_(2))/(2R)xx(theta)/(2pi)`
But `I_(1)(rho(Rtheta))/(A)=I_(2)(rhoR(2pi-theta))/(A)`
`impliesI_(1)theta=I_(2)(2pi-theta)jimpliesB_(1)=B_(2)`
But direction of `vec(B)_(1)` and `vec(B)_(2)` are opposite so net magnetic field at the centre is zero.