Question

A circular loop of radius $r$ carries a current $i$. How should a long, straight wire carrying a current $3i$ be placed in the plane of the circle so that the magnetic field at the centre of the loop becomes zero?

• A. At distance $\frac{3r}{\pi }$ from the centre of the loop
• B. At distance $\frac{2r}{\pi }$ from the centre of the loop
• C. At distance $\frac{2r}{3\pi }$ from the centre of the loop
• D. Not possible

Answer 1

Answer: (A)

At distance $\frac{3r}{\pi }$ from the centre of the loop

$\overrightarrow{B}$ due to loop=$\frac{{\mu }_{0}i}{2r}$

Let the straight current carrying wire be kept at a distance R from center

Given I=3i

$\overrightarrow{B}$ due to wire = $\frac{{\mu }_{0}I}{2\pi R}=\frac{{\mu }_0\times 3i}{2\pi R}$

Now,

$\overrightarrow{B}$ due to both will balance each other

Hence,

$\frac{{\mu }_{0}i}{2r}=\frac{{\mu }_{0}3i}{2\pi R}$

$R=\frac{3r}{\pi }$

Hence the straight wire should be kept at distance $\frac{3r}{\pi }$ from the center of the loop.