Figure shows a current loop having two circular arcs joined by two radial lines. Find the magnetic field $B$ at the centre $O$.

MAgnetic field at point $O$ due to wires $CB$ and $AD$ will be zero. Magnetic field due to wire $BA$ will be,

$B_1=\left(\frac{\theta }{2\pi }\right)\left(\frac{{\mu }_{0}i}{2a}\right)$

Direction of field $B_1$ is coming out of the plane of the figure. Similarly, field at $O$ due to arc $DC$ will be, $B_2=\left(\frac{\theta }{2\pi }\right)\left(\frac{{\mu }_{0}i}{2b}\right)$

Direction of field $B_2$ is going into the plane of the figure. The resultant field at $O$ is $B=B_1-B_2=\frac{{\mu }_{0}i\theta \left(b-a\right)}{4\mathrm{\pi a}b}$