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

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