question_answer

A circular current carrying coil has a radius R. The distance from the centre of the coil on the axis of the coil, where the magnetic induction is\[\frac{1}{8}\] of its value at the centre of coil is :

A) \[\sqrt{3}R\]                                     

B) \[\frac{R}{\sqrt{3}}\]

C) \[\left( \frac{2}{\sqrt{3}} \right)R\]                         

D) \[\frac{R}{2\sqrt{3}}\]

Correct Answer: A

Solution :

                The magnetic field at the axis of a current carrying coil of radius R, and at a distance\[x\] from the centre of the coil is \[B=\frac{{{\mu }_{0}}}{4\pi }.\frac{2\pi i{{R}^{2}}}{{{({{R}^{2}}+{{x}^{2}})}^{3/2}}}\] Given,                   \[B={{B}_{centre}}\times \frac{1}{8}\] \[\Rightarrow \]               \[\frac{{{\mu }_{0}}}{4\pi }\times \frac{2\pi i{{R}^{2}}}{{{({{R}^{2}}+{{x}^{2}})}^{3/2}}}=8{{R}^{3}}\]  \[\Rightarrow \]               \[{{({{R}^{2}}+{{x}^{2}})}^{3/2}}=8{{R}^{3}}\] \[\Rightarrow \]               \[x=\sqrt{3}R\]