Question

A wire of length $L$ is bent into an arc of a circle and found to subtend and angle of $\theta$ radians at the center. If a current of $i$ is passed through it, the magnetic induction at the center of the circle is :

• A. $\frac{{\mu }_{0}i{\theta }^2}{2\pi L}$
• B. $\frac{{\mu }_{0}i{\theta }^2}{4\pi L}$
• C. $\frac{{\mu }_{0}i\theta }{2\pi L}$
• D. $\frac{{\mu }_{0}i\theta }{4\pi L}$

Answer 1

Answer: (B)

$\frac{{\mu }_{0}i{\theta }^2}{4\pi L}$

Given,
$L=r\theta$, $r=\frac{L}{\theta }$
B due to circular area$=\frac{{\mu }_{0}i}{4\pi r}\left(\theta \right)$
$=\frac{{\mu }_{0}i}{4\pi (\frac{L}{\theta })}\left(\theta \right)$
$=\frac{{\mu }_{0}i}{4\pi L}\times {\theta }^2$