Question

A circular loop of radius R carries a current I. Another circular loop of radius r(<<R) carries a current i and is placed at the centre of the larger loop. The planes of the two circles are at right angle to each other. Find the torque acting on the smaller loop.

Answer 1

Given:
For the outer loop,
Magnitude of current = I
Radius of the loop = R
Thus, the magnetic field at the centre due to the larger loop is given by
$B=\frac{{\mu }_{0}I}{2R}$
Let A be the area of the smaller loop and let current i pass through it.
Now,
Angle between the area vector of the smaller loop and the magnetic field due to the larger loop = 90°
Thus, the required torque is given by
$\Gamma =i(\overrightarrow{A}\times \overrightarrow{B})$
= iABsin 90°
$$\begin{gathered} =i\mathrm{\pi }{r}^2\frac{{\mathrm{\mu }}_{0}I}{2R} \\ =\frac{{\mathrm{\mu }}_0\mathrm{\pi }{r}^{2}Ii}{2R} \end{gathered}$$