Question

A wire carrying current i has the configuration as shown in figure.Two semi-infinite straight sections,both tangent to the same circle are connected by a circular arc of central angle $\theta$,along the circumference of the circle,with all sections lying in the same plane.What must $\theta$ be for B to be zero at the centre of the circle?

Answer 1

Two straight wires produces outwards magnetic field by arc of circle produces inward magnetic field.
Due to straight wires:
$B_1=2[\frac{{\mu }_o}{4\pi }\frac{i}{R}(sin{0}^o+sin{90}^o)]$

$=\frac{{\mu }_{o}i}{2\pi R}$ (outwards)

Due to circular arc
$B_2=\frac{\theta }{2\pi }\left(\frac{{\mu }_{o}i}{2R}\right)$ (inwards)

For net field to be zero.
$B_1=B_2$ or $\theta =2rad$