Question

The magnetic field at a distance $x$ on the axis of a circular coil of radius $R$ is $\frac{1}{8}$ th of that at the centre. The value of $x$ is

• A. $\frac{R}{\sqrt{3}}$
• B. $\frac{2R}{\sqrt{3}}$
  
• C. $R\sqrt{3}$
  
• D. $R\sqrt{2}$

Answer 1

The correct option is C $R\sqrt{3}$


Magnetic field on the axis of current carrying circular loop,

$B_1=\frac{{\mu }_0}{4\pi }.\frac{2M}{(r^2+x^2{)}^{\frac{3}{2}}}\dots \left(i\right)$
Magnetic field at the centre of current carrying circular loop,

$B_2=\frac{{\mu }_0}{4\pi }.\frac{2M}{r^3}\dots \left(ii\right)$

From Eqs. (i) and (ii),

$\frac{B_2}{B_1}=\frac{(r^2+x^2{)}^{\frac{3}{2}}}{r^3}$

$\frac{B_2}{B_1}={\left(\frac{x^2+R^2}{R^2}\right)}^{\frac{3}{2}}$

$\Rightarrow {\left(\frac{x^2+R^2}{R^2}\right)}^{\frac{3}{2}}=8$

$\Rightarrow \frac{x^2+R^2}{R^2}=4$

$\Rightarrow x=\sqrt{3}R$