Physics - Magnetism Question with Solution | TestHub

Magnetic field on the axis of a circular coil, at a location $x$ away from the center of the coil is given by, $B=\frac{{\mu }_{0}I{R}^2}{2{\left(R^2+x^2\right)}^{\frac{3}{2}}}$ For, $x=0$ the induction at the centre of the coil is $B_0=\left(\frac{{\mu }_{0}I}{2R}\right)$ $\begin{array}{rl}& \therefore \frac{1}{8}\times \left(\frac{{\mu }_{0}I}{2R}\right)=\frac{{\mu }_{0}I{R}^2}{2{\left(R^2+x^2\right)}^{\frac{3}{2}}}\\ & \Rightarrow (2R{)}^3={\left(R^2+x^2\right)}^{\frac{3}{2}}\\ & \Rightarrow (2R)={\left(R^2+x^2\right)}^{\frac{1}{2}}\\ & \Rightarrow (2R{)}^2=\left(R^2+x^2\right)\\ & \Rightarrow x=\sqrt{3}R\end{array}$