question_answer

An infinitely long straight conductor is bent into the shape as shown in the figure. It carries a current of i ampere and the radius of the circular loop is r metre. Then the magnetic induction at its centre will be                              [MP PMT 1999]

A)            \[\frac{{{\mu }_{0}}}{4\pi }\frac{2i}{r}(\pi +1)\]

B)             \[\frac{{{\mu }_{0}}}{4\pi }\frac{2i}{r}(\pi -1)\]

C)            Zero

D)            Infinite

Correct Answer: B

Solution :

                   The given shape is equivalent to the following diagram The field at \[O\] due to straight part of conductor is\[{{B}_{1}}=\frac{{{\mu }_{o}}}{4\pi }.\frac{2i}{r}\]¤. The field at \[O\]due to circular coil is\[{{B}_{2}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{2\pi i}{r}\otimes \]. Both fields will act in the opposite direction, hence the total field at O. i.e. \[B={{B}_{2}}-{{B}_{1}}=\left( \frac{{{\mu }_{o}}}{4\pi } \right)\times (\pi -1)\frac{2i}{r}=\frac{{{\mu }_{o}}}{4\pi }.\frac{2i}{r}(\pi -1)\]