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 `l_1` and `l_2` be the lengths of the two circular arcs ACB and ADB respectively. `rho` be the resistance per unit length of the circular conductor.
Resistance of circular arc `ACB, R_1=l_1rho`
Resistance of circular arc `ADB, R_2=l_2rho`
Let `I_1`, `I_2` be the current in arms ACB and ADB respectively. As these two arcs are connected in parallel, so the potential difference across them must be equal i.e.,
`I_1R_1=I_2R_2`
or `I_1l_1rho=I_2l_2rho`
or `I_1l_1=I_2l_2` ...(i)
Magnetic field induction at O due to currrent in arc ADB is,
`B_2=(mu_0)/(4pi)I_2/rtheta_2=(mu_0)/(4pi)(I_2l_2)/(r^2)`
It is acting downwards, perpendicular to the plane of circular conductor.
As `I_1l_1=I_2l_2`,
so `B_1=B_2` and `vecB_1=-vecB_2`
`:.` Resultant magnetic field at `O=vecB_1+vecB_2`
`=(-vecB_1)+vecB_2=0`