The magnetic field at the center of an arc subtended at an angle θ is given by

Then, the magnetic field due to larger arc AB is

…(i)

which acts in inward direction according to right hand thumb rule and magnetic field due to smaller arc AB is

…(ii)

which acts in outward direction.
The resultant magnetic field

[From Eq. (i) and (ii)] …(iii)
which acts in inward direction as .
Two arcs can also be seen as the two resistances in parallel combination.
So, the potential across them will be same i.e.

…(iv)

where, R 1 and Resistance of respective segments The wire is uniform so

From Eq. (iv), we get

From Eq. (iii) and (v), we get