question_answer

A current I flows in an infinitely long wire with corss section in the form of a semi-circular ring of raidus R. The magnitude of the magnetic induction along its axis is   AIEEE  Solved  Paper-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 :

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