Question

The parts of two concentric circular arcs joined by two radial lines and carries current $i$. The arcs subtend an angle $\theta$ at the center of the circle. The magnetic field at the centre O, is

• A. $\frac{{\mu }_{0}i(b-a)\theta }{4\pi ab}$
• B. $\frac{{\mu }_{0}i(b-a)}{\pi -\theta }$
• C. $\frac{{\mu }_{0}i(b-a)\theta }{\pi ab}$
• D. $\frac{{\mu }_{0}i(a-b)}{2\pi ab}$

Answer 1

The correct option is A $\frac{{\mu }_{0}i(b-a)\theta }{4\pi ab}$

The magnetic field at the center of a circular arc of radius $r$ is given by:

$B_{arc}\left(center\right)=\frac{{\mu }_{0}i}{2r}\left(\frac{\theta }{2\pi }\right)$.

Note that we are just considering the magnitude of the magnetic field here and based on the direction of the current (clockwise or counter-clockwise) the magnetic field will also change its direction. Since our current loop is complete, definitely the magnetic field will be the difference between the two field generated by individual arcs.

Hence, $\Delta B_{arc}=B_{arc}\left(a\right)-B_{arc}\left(b\right)=\frac{{\mu }_{0}i\theta }{4\pi }(\frac{1}{a}-\frac{1}{b})=\frac{{\mu }_{0}i(b-a)\theta }{4\pi ab}$.