Question

The magnetic induction of the field at the point $O$ of a loop with current $I$, whose shape is illustrated in figure above is given as $B=\frac{{\mu }_0}{4\pi }i[\frac{x\pi -\phi }{a}+\frac{\phi }{b}]$. Find $x$ The radii $a$ and $b$, as well as the angle $\phi$ are known.

Answer 1

According to Biot Savart Law, current at the center of circular arc subtending angle $\theta$ to the center=$\frac{{\mu }_{0}I}{2r}\frac{\theta }{2\pi }$

Hence magnetic field at the center of shape shown=$B=\frac{{\mu }_{0}i}{2a}\frac{2\pi -\psi }{2\pi }+\frac{{\mu }_{0}i}{2b}\frac{\psi }{2\pi }=\frac{{\mu }_{0}i}{4\pi }[\frac{2\pi -\psi }{a}+\frac{\psi }{b}]$