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

Answer: (C)

$R\sqrt{3}$

From above formula magnetic field at center z=0

$B=\frac{{\mu }_{\circ }Ni}{2R}$....(i)

Given magnetic field $B^{\prime }$ at distance Z from center is $\frac{B}{8}$

$B^{\prime }=\frac{1}{8}\frac{{\mu }_{\circ }Ni}{2R}$....(ii)

From above concept $B^{\prime }=\frac{{\mu }_{\circ }Ni{R}^2}{2\left(\right(R^2+Z^2{)}^{\frac{3}{2}}}$ ....(iv)

From equation (ii) and (iv)

$\frac{{\mu }_{\circ }Ni{R}^2}{2\left(\right(R^2+Z^2{)}^{\frac{3}{2}}}=$ $\frac{1}{8}\frac{{\mu }_{\circ }Ni}{2R}$

$8R^3=(R^2+Z^2{)}^{\frac{3}{2}}$

taking cube root on both sides

$2R=\sqrt{R^2+Z^2}$

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