question_answer

A current I flows in an infinitely long wire with cross section in the form of a semi-circular ring of radius R. The magnitude of the magnetic induction along its axis is              [AIEEE 2011]

A)             \[\frac{{{\mu }_{0}}I}{4\pi R}\]

B) \[\frac{{{\mu }_{0}}I}{{{\pi }^{2}}R}\]

C)             \[\frac{{{\mu }_{0}}I}{2{{\pi }^{2}}R}\]

D) \[\frac{{{\mu }_{0}}I}{2\pi R}\]

Correct Answer: B

Solution :

[b]

\[B=\int{dB\sin \theta }\]

\[B=\int{\frac{{{\mu }_{0}}di}{2\pi R}\sin \theta }\]

\[=\frac{{{\mu }_{0}}}{2\pi R}\left( \frac{i}{\pi R} \right)R\int\limits_{0}^{\pi }{\sin \theta d\theta }\]

\[=\frac{{{\mu }_{0}}i}{{{\pi }^{2}}R}\]