Question

A circular current carrying coil has a radius R The distance from the centre of the coil off the axis of the coil, where the magnetic induction is I/8th of its value at the centre of the coil is

• A. $\sqrt{3}R$
• B. $R/\sqrt{3}$
• C. $\left(\frac{2}{\sqrt{3}}\right)R$
• D. $\frac{R}{2\sqrt{3}}$

Answer 1

Answer: (A)

For a circular coil of radius a carrying a current $i$ , the magnetic field at point $P$ , distance $x$ from coil is given by
$B=\frac{{\mu }_{0}i{a}^2}{2{(a^2+x^2)}^{3/2}}N{A}^{-1}{m}^{-1}$ .. (i)
At the centre of coil $x=0$
$\therefore$ $B=\frac{{\mu }_{0}i}{2a}N{A}^{-1}{m}^{-1}$ .. (ii)
Given, $B=\frac{1}{8}B$
$\therefore$ $\frac{{\mu }_{0}i{a}^2}{2{(a^2+x^2)}^{3/2}}=\frac{1}{8}\left(\frac{{\mu }_{0}i}{2a}\right)$
$\Rightarrow$ $\frac{a^2}{{(a^2+x^2)}^{3/2}}=\frac{1}{8a}$
$\Rightarrow$ $8a^3={(a^2+x^2)}^{3/2}$
$\Rightarrow$ $a^2+x^2=4a^2$
$\Rightarrow$ $x=\sqrt{3\cdot a}$
Given, $a=R$ $x=\sqrt{3}R$