Question

Two long parallel wires separated by $0.1\, m$ carry currents of $1\, A$ and $2\, A$ respectively in opposite directions. A third current-carrying wire parallel to both of them is placed in the same plane such that it feels no net magnetic force. It is placed at a distance of

• A. 0.5 m from the $1^{st}$ wire, towards the $2^{nd}$ wire.
• B. 0.2 m from the $1^{st}$ wire, towards the $2^{nd}$ wire.
• C. 0.1 m from the $1^{st}$ wire, towards the $2^{nd}$ wire.
• D. 0.2 m from the $1^{st}$ wire, away from the $2^{nd}$ wire.

Answer 1

Answer: (C)

0.1 m from the $1^{st}$ wire, towards the $2^{nd}$ wire.

Answer 2

According to question, the third wire should not feel magnetic force due to wire (1) and (2). It can be only possible in the condition that fields due to wire (1) and (2) must act opposite to each other and must be equal.
The magnetic field due to long straight wire,
$B =\frac{\mu_{0} \cdot I}{2 \pi x}$
$\therefore B_{1} =B_{2} \frac{\mu_{d}}{2 \pi x} =\frac{\mu_{d_{2}}}{2 \pi(0 \cdot 1+x)}$
Here $i_{1} =1 A$
$i_{2} =2 A$
$\frac{\mu_{0} \times 1}{2 \pi x} =\frac{\mu_{0} \times 2}{2 \pi(0 \cdot 1+x)}$
$2 x =(0 \cdot 1+x)$
$x =0 \cdot 1 \,m$