A current carrying circular coil of radius $R$ has a point $P$ situated on its axis at a distance $x$ from…
A current carrying circular coil of radius $R$ has a point $P$ situated on its axis at a distance $x$ from its centre $\mathrm{O}$ of the coil. The magnetic induction at point $P$ is $\left(\frac{1}{8}\right)^{\text {th }}$ of magnetic field at its centre O. The value of $x$ is
$\frac{R}{2 \sqrt{3}}$
$\sqrt{3} R$
$\frac{R}{\sqrt{3}}$
$\frac{2}{\sqrt{3}} R$
Solution
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 $B_0=\left(\frac{\mu_0 I}{2 R}\right)$
$\begin{aligned}
& \therefore \frac{1}{8} \times\left(\frac{\mu_0 I}{2 R}\right)=\frac{\mu_0 I R^2}{2\left(R^2+x^2\right)^{\frac{3}{2}}} \\
& \Rightarrow(2 R)^3=\left(R^2+x^2\right)^{\frac{3}{2}} \\
& \Rightarrow(2 R)=\left(R^2+x^2\right)^{\frac{1}{2}} \\
& \Rightarrow(2 R)^2=\left(R^2+x^2\right) \\
& \Rightarrow x=\sqrt{3} R
\end{aligned}$