question_answer

An infinitely long straight conductor is bent into the shape as shown below. It carries a current of I ampere and the radius of the circular loop is R metre. Then, the magnitude of magnetic induction at the centre of the circular loop is

A)  \[\frac{{{\mu }_{0}}I}{2\pi R}\]            

B)  \[\frac{{{\mu }_{0}}nI}{2R}\]

C)  \[\frac{{{\mu }_{0}}nI}{2\pi R}\left( \pi +1 \right)\]      

D)  \[\frac{{{\mu }_{0}}nI}{2\pi R}\left( \pi -1 \right)\]  

Correct Answer: C

Solution :

Magnetic field due to long wire at 0 point \[{{B}_{1}}=\frac{{{\mu }_{0}}}{2\pi }\left( \frac{I}{R} \right)\]  (upward direction) Magnetic field due to loop at O point \[{{B}_{2}}=\frac{{{\mu }_{0}}}{4\pi }.\frac{I.2\pi R}{{{R}^{2}}}\] \[{{B}_{2}}=\frac{{{\mu }_{0}}}{2}.\frac{I}{R}\](in upward direction) Hence, resultant magnetic field at centre O \[B={{B}_{1}}+{{B}_{2}}\] \[B=\frac{{{\mu }_{0}}I}{2\pi .R}(\pi +1)T\]