Question

Consider the situation shown in figure. The straight wire is fixed but the loop can move under magnetic force. The loop will
(a) remain stationary
(b) move towards the wire
(c) move away from the wire
(d) rotate about the wire.

Answer 1

(b) move towards the wire


Force acting on the wire per unit length carrying current i₂ due to the wire carrying current i₁ placed at a distance d is given by
$F=\frac{{\mathrm{\mu }}_{\mathrm{o}}{i}_1{i}_2}{2\mathrm{\pi }d}$
So, forces per unit length acting on sides AB and CD are as follows:
$$\begin{gathered} F_{\mathrm{AB}}=\frac{{\mathrm{\mu }}_{\mathrm{o}}{i}_1{i}_2}{2\mathrm{\pi }d}(\mathrm{Towards}\mathrm{the}\mathrm{wire}) \\ {F}_{\mathrm{CD}}=\frac{{\mathrm{\mu }}_{\mathrm{o}}{i}_1{i}_2}{2\mathrm{\pi }(d+a)}(\mathrm{Away}\mathrm{from}\mathrm{the}\mathrm{wire}) \end{gathered}$$
Here, F_AB > F_CD because force is inversly proportional to the distance from the wire and wire AB is closer to the wire carrying current i_1.
The forces per unit length acting on sides BC and DA will be equal and opposite, as they are equally away from the wire carrying current i₁, with current i₂ flowing in the opposite direction.
∴ F_BC = $-$F_DA
Now,
Net force:
$$\begin{gathered} F=F_{\mathrm{AB}}\mathit{+}{F}_{\mathrm{BC}}\mathit{+}{F}_{\mathrm{CD}}\mathit{+}{F}_{\mathrm{DA}} \\ \Rightarrow F=\frac{{\mathrm{\mu }}_{\mathrm{o}}{i}_1{i}_2}{2\mathrm{\pi }d}+F_{\mathrm{BC}}-\frac{{\mathrm{\mu }}_{\mathrm{o}}{i}_1{i}_2}{2\mathrm{\pi }(d+a)}-F_{\mathrm{BC}} \\ \Rightarrow F=\frac{{\mathrm{\mu }}_{\mathrm{o}}{i}_1{i}_2}{2\mathrm{\pi }}\left(\frac{1}{d}-\frac{1}{d+a}\right) \\ \Rightarrow F=\frac{{\mathrm{\mu }}_{\mathrm{o}}{i}_1{i}_{2}a}{2\mathrm{\pi }d(d+a)} \end{gathered}$$

(Towards the wire)
Therefore, the loop will move towards the wire.