As shown in figure a cell is connected across two points A and B of a uniform circular conductor of radius r. Prove that the magnetic field induction at its centre O will be zero.

Explanation

Let `I_(1)` and `I_(2)` be the lengths of the segments of the uniform conductor and `R_(1)`,` R_(2)` their respective resistance.
and `i_(1)/i_(2)=R_(2)/R_(1)=l_(2)/l_(1)`
So, `B_(1)/B_(2)=i_(1)l_(1)/i_(2)l_(2)`
`(l_(2)/l_(1)) (l_(1)/l_(2))=1`
i.e..,`B_(1)=B_(2)`
But the currents in the two segments are in opposite directions with respect to O. Therefore the magnetic induction `vec B_(1)` is out of the paper and B, is into the paper. So they are opposite to each other and they cancel each other. Vectorially `vec B_(1) =-B_(2)`.Therefore, B= net magnetic induction `B_(1) +B_(2) =0`.