Question

A square loop of side $a$ is placed at a distance a away from a long wire carrying a current $I_1$. If the loop carries a current $I_2$ as shown in Fig. (a) then the nature of the force and its amount is :

• A. $\frac{{\mu }_0{I}_1{I}_2}{2\pi a}$, attractive
• B. $\frac{{\mu }_0{I}_1{I}_2}{4\pi }$, attractive
• C. $\frac{{\mu }_0{I}_1{I}_2}{4\pi }$, repulsive
• D. $\frac{{\mu }_0{I}_1{I}_2}{4\pi a}$, repulsive

Answer 1

The correct option is B $\frac{{\mu }_0{I}_1{I}_2}{4\pi }$, attractive

Force on any segment of the square $F=i\int dl\times B$
Let $B_1$ be the magnetic field at any point on the segment $AB,$ this is same at all points on the segment $AB$.
$\therefore \overrightarrow{F_1}=I_{2}a\frac{{\mu }_0{I}_1}{2\pi a}(-\hat{i})$ where $B_1=\frac{{\mu }_0{I}_1}{2\pi a}$
Let $B_3$ be the magnetic field at any point on the segment $CD,$ this is same at all points on the segment $CD$.
$\therefore \overrightarrow{F_3}=I_{2}a\frac{{\mu }_0{I}_1}{2\pi (a+a)}\left(\hat{i}\right)$ where $B_3=\frac{{\mu }_0{I}_1}{2\pi (a+a)}$
The force on side $BC$ is equal and opposite in direction to that on side $DA$ and they will cancel mutually( $F_2=F_4$)
Net force on the square:
$\overrightarrow{F_{net}}=\overrightarrow{F_1}+\overrightarrow{F_3}=I_{2}a\frac{{\mu }_0{I}_1}{2\pi 2a}(-\hat{i})=\frac{{\mu }_0{I}_1{I}_2}{4\pi }(-\hat{i})$
The force is attractive since it is in $-\hat{i}$ direction.