question_answer

In the figure shown, the magnetic field induction at the point 0 will be

A) \[\frac{{{\mu }_{0}}i}{2\pi r}\]                                   

B) \[\left( \frac{{{\mu }_{0}}}{4\pi } \right)\left( \frac{i}{r} \right)(\pi +2)\]

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

D)        \[\frac{{{\mu }_{0}}i}{4\pi r}(\pi +1)\]

Correct Answer: B

Solution :

Field due to a straight wire of infinite length is \[\frac{{{\mu }_{0}}i}{4\pi r}\]if the point is on a line perpendicular to its length while at the centre of a semicircular oil is\[\frac{{{\mu }_{0}}\pi i}{4\pi r}\]. \[\therefore \]  \[B={{B}_{a}}+{{B}_{b}}+{{B}_{c}}\] \[=\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}+\frac{{{\mu }_{0}}}{4\pi }\frac{\pi i}{r}+\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}\] \[=\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}(\pi +2)\]out of page