Question

A current $i$ ampere flows in a circular arc of wire which subtends an angle $\left(\frac{3\pi }{2}\right)$ radius at its centre, whose radius is $R$. The magnetic induction $B$ at the centre is

• A. $\frac{{\mu }_{0}i}{R}$
• B. $\frac{{\mu }_{0}i}{2R}$
• C. $\frac{2{\mu }_{0}i}{R}$
• D. $\frac{3{\mu }_{0}i}{8R}$

Answer 1

The correct option is D $\frac{3{\mu }_{0}i}{8R}$

For a loop, magnetic induction at centre

$B=\frac{{\mu }_0}{4\pi }\times \frac{2\pi i}{r}$

When loop subtends angle $\theta$ at centre, then

$B=\frac{{\mu }_0}{4\pi }\times \frac{\theta i}{r}$ (Here angle is $\theta$ instead of 2$\pi$)

In the given problem, $\theta =3\pi /2$ and $r=R$

$\therefore B=\frac{{\mu }_0}{4\pi }\times \frac{3\pi }{2}\times \frac{i}{R}=\frac{3{\mu }_{0}i}{8R}$