Question

Two circular coils of equal radius carrying currents $i_1,i_2$ in opposite directions are at a large distance $d$. The distance from the first coil where the resultant magnetic induction is zero is:

• A. $\frac{d}{1+\frac{i_2}{i_1}}$
• B. $\frac{d}{1+\sqrt{\frac{i_2}{i_1}}}$
• C. $\frac{d}{1+{\left(\frac{i_2}{i_1}\right)}^{\frac{1}{3}}}$
• D. $\frac{d}{1+{\left(\frac{i_2}{i_1}\right)}^2}$

Answer 1

The correct option is C $\frac{d}{1+{\left(\frac{i_2}{i_1}\right)}^{\frac{1}{3}}}$

$B=\frac{{\mu }_{0}i}{2\pi }\left(\frac{2\pi r^2}{r^3}\right)$

On the axis of the coil

So, $B_1=\frac{{\mu }_0{i}_1}{2\pi }\left(\frac{2\pi r^2}{(x^3}\right)$

$B_2=\frac{{\mu }_0{i}_2}{2\pi }\left(\frac{2\pi r^2}{(d-x{)}^3}\right)$

So, equation $B_1$ and $B_2$

$\frac{i_1}{(x{)}^3}=\frac{i_2}{(d-x{)}^3}$

$\Rightarrow (\frac{i_2}{i_1}{)}^{1/3}=\frac{d-x}{x}$

$\Rightarrow x=\frac{d}{1+(\frac{i_2}{i_1}{)}^{1/3}}$