Question

A circular current carrying coil has a radius $R$. The distance from the centre of the coil on the axis where the magnetic induction will be ${\left(\frac{1}{8}\right)}^{th}$ of its value at the centre of the coil is

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

Answer 1

Answer: (B)

$R\sqrt{3}$

Magnetic field on the axis of a circular coil
$B_1=\frac{{\mu }_{0}i{R}^2}{2(R^2+z^2{)}^{3/2}}$
Magnetic field at the center of the circular coil:
$B_2=\frac{{\mu }_{0}i}{2R}$
Given, $B_1=\frac{1}{8}{B}_2$
$\Rightarrow \frac{{\mu }_{0}i{R}^2}{2(R^2+z^2{)}^{3/2}}=\frac{1}{8}\frac{{\mu }_{0}i}{2R}$
$\Rightarrow 8R^3=(R^2+z^2{)}^{3/2}$
$\Rightarrow 64R^6=(R^2+z^2{)}^3$
$(4R^2{)}^3=(R^2+z^2{)}^3$
$\Rightarrow 4R^2=R^2+z^2$
$\Rightarrow z=\sqrt{3}R$