question_answer

Current \[I\] is flowing in conductor shaped as shown in the figure. The radius of the curved part is r and the length of straight portion is very large. The value of the magnetic field at the centre O will be

A)  \[\frac{{{\mu }_{0}}I}{4\pi r}\left( \frac{3\pi }{2}+1 \right)\]       

B) \[\frac{{{\mu }_{0}}I}{4\pi r}\left( \frac{3\pi }{2}-1 \right)\]

C)  \[\frac{{{\mu }_{0}}I}{4\pi r}\left( \frac{\pi }{2}+1 \right)\]                          

D)  \[\frac{{{\mu }_{0}}I}{4\pi r}\left( \frac{\pi }{2}-1 \right)\]

Correct Answer: A

Solution :

\[{{B}_{A}}=0\] \[{{B}_{B}}=\frac{{{\mu }_{0}}}{4\pi }\frac{(2\pi -\pi /2)I}{\tau }\otimes \]                 \[=\frac{{{\mu }_{0}}}{4\pi }\frac{3\pi I}{2\,r}\] \[{{B}_{C}}=\frac{{{\mu }_{0}}I}{4\pi r}\otimes \] So, net magnetic field at the centre                 \[={{B}_{A}}+{{B}_{B}}+{{B}_{C}}\]                 \[=0+\frac{{{\mu }_{0}}}{4\pi }\frac{3\pi I}{2\,r}+\frac{{{\mu }_{0}}I}{4\pi r}=\frac{{{\mu }_{0}}}{4\pi }\frac{I}{r}\left( \frac{3\pi }{2}+1 \right)\]