Question

A current $I$ flows in a circular arc of wire which subtends an angle $\frac{3\pi }{2}$ radians 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

Answer: (D)

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