Question

Two parallel wires separated by a distance of 10 cm carry currents of 10 A and 40 A along the same direction. Where should a third current by placed so that it experiences no magnetic force?

Answer 1

Let the third wire W₃ having current i in upward direction be placed x cm from the 10 A current wire.


The magnetic force per unit length on a wire due to a parallel current-carrying wire is given by
$\frac{F}{l}=\frac{{\mathrm{\mu }}_0{i}_1{i}_2}{2\mathrm{\pi }d}$
According to the question, wire W₃ experiences no magnetic force.
∴ $\frac{F}{l}\mathrm{due}\mathrm{to}\mathrm{wire}{\mathrm{W}}_1=\frac{F}{l}\mathrm{due}\mathrm{to}\mathrm{wire}{\mathrm{W}}_2$
$$\begin{gathered} \Rightarrow \frac{{\mathrm{\mu }}_{0}10i}{2\mathrm{\pi }x}=\frac{{\mathrm{\mu }}_{0}40i}{2\mathrm{\pi }(10-x)} \\ \Rightarrow 10-x=4x \\ \Rightarrow x=2\mathrm{cm} \end{gathered}$$
Thus, wire W₃ is placed 2 cm from the 10 A current wire.