Question

A square loop $ABCD$ having side $l$ carrying a current $i$, is placed at a distance $l/2$; coplanar and parallel with a long straight conductor $XY$ carrying a current $I$, the net force on the loop will be

• A. $\frac{2{\mu }_{0}IiL}{3\pi }$
• B. $\frac{{\mu }_{0}IiL}{2\pi }$
• C. $\frac{2{\mu }_{0}Ii}{3\pi }$
• D. $\frac{{\mu }_{0}Ii}{2\pi }$

Answer 1

The correct option is C $\frac{2{\mu }_{0}Ii}{3\pi }$

Force on arm $AB$ due to current in conductor $XY$ is
$F_1=\frac{{\mu }_0}{4\pi }\frac{2IiL}{\frac{L}{2}}=\frac{{\mu }_{0}Ii}{\pi }$
acting towards $XY$ in the plane of loop.
Force on arm $CD$ due to current in conductor $XY$ is

$F_2=\frac{{\mu }_0}{4\pi }\frac{2IiL}{3\left(\frac{L}{2}\right)}=\frac{{\mu }_{0}Ii}{3\pi }$
acting away from $XY$ in the plane of loop,
$\therefore$ Net force on the loop= $F_1-F_2$
$=\frac{{\mu }_{0}Ii}{\pi }\left(1-\frac{1}{3}\right]=\frac{2}{3}\frac{{\mu }_{0}Ii}{\pi }$