Question

A square loop of side $2a$, and carrying current $I$, is kept in $XZ$ plane with its centre at origin. A long wire carrying the same current $I$ is placed parallel to the $z-$axis and passing through the point $(0,b,0)(b>>a)$. The magnitude of the torque on the loop about $z-$axis is given by:

• A. $\frac{{\mu }_0{I}^2{a}^2}{2\pi b}$
• B. $\frac{{\mu }_0{I}^2{a}^3}{2\pi b^2}$
• C. $\frac{2{\mu }_0{I}^2{a}^2}{\pi b}$
• D. $\frac{2{\mu }_0{I}^2{a}^3}{\pi b^2}$

Answer 1

The correct option is C $\frac{2{\mu }_0{I}^2{a}^2}{\pi b}$

Torque on the loop,
$\overrightarrow{\tau }=\overrightarrow{M}\times \overrightarrow{B}=MB\sin \theta =MB\sin 90°$
Magnetic field, $B=\frac{{\mu }_{0}I}{2\pi d}$
$\therefore \tau =I(2a{)}^2\left(\frac{{\mu }_{0}I}{2\pi d}\right)\sin 90°=\frac{2{\mu }_0{I}^2}{\pi d}\times a^2=\frac{2{\mu }_0{I}^2{a}^2}{\pi d}$