question_answer

In the given figure, then magnetic field induction at point O will be

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

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

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

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

Correct Answer: B

Solution :

Magnetic field due to a straight wire of semi-infinite length is \[\frac{{{\mu }_{0}}i}{4\pi r}\] Magnetic field due to semicircular area is given by \[\frac{{{\mu }_{0}}i\,\pi }{4\pi r}\] \[\therefore \,\,\,\,\,\,\,\,\,\,B={{B}_{a}}+{{B}_{b}}+{{B}_{c}}\] \[=\,\,\,\,\,\,\,\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}+\frac{{{\mu }_{0}}i}{4\pi r}+\frac{{{\mu }_{0}}}{4\pi }\frac{i}{r}=\left( \frac{{{\mu }_{0}}i}{4\pi r} \right)(\pi +2)\]