Question

The ratio of magnetic inductions at the centre of a circular coil of radius ${}^{\prime }{a}^{\prime }$ and on its axis at a distance equal to its radius, will be

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{\sqrt{2}}{1}$
• C. $\frac{1}{2\sqrt{2}}$
• D. $\frac{2\sqrt{2}}{1}$

Answer 1

The correct option is D $\frac{2\sqrt{2}}{1}$

Let us consider, $a=$ radius of circular coil
$i=$ current flow in the circular coil
The magnetic field at the centre of a circular coil is given by
$B_C=\frac{{\mu }_{0}i}{2a}$
The magnetic field on its axis at a distance equal to the radius of the circular coil is given by
$B_{\text{axis}}=\frac{{\mu }_{0}i{a}^2}{2(a^2+a^2{)}^{3/2}}=\frac{{\mu }_{0}i{a}^2}{2(2a^2{)}^{3/2}}$
$B_{\text{axis}}=\frac{{\mu }_{0}i}{2a{.2}^{3/2}}=\frac{{\mu }_{0}i}{4\sqrt{2}a}$
Then, ratio $=\frac{B_C}{B_{\text{axis}}}=\frac{{\mu }_{0}i}{2a}\times \frac{4\sqrt{2}a}{{\mu }_{0}i}$
$=2\sqrt{2}$