Two long straight parallel wires A and B are 6 cm apart. They carry currents of 6 A and 3 A in the same direction. A third long straight wire carrying a current is placed parallel to them, in the same plane. The net magnetic force on the third wire is zero when its distance from A is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4 cm
The force per unit length between two long parallel wires is \( f=\mu_0I_aI_b/2\pi d \). Wires A and B carry currents in the same direction, so they both attract the third wire or they both repel it. Between A and B these two forces act in opposite directions and can cancel. Outside the pair they act in the same direction and cannot cancel.
Let the third wire carry a current \( I \) and lie at a distance \( x \) from A. Its distance from B is then \( (6-x) \) cm.
Given:
\( I_A=6\ \mathrm{A} \)
\( I_B=3\ \mathrm{A} \)
\( d=6\ \mathrm{cm} \)
For zero net force, the two forces per unit length must be equal:
\( \begin{aligned} \frac{\mu_0I_AI}{2\pi x} &= \frac{\mu_0I_BI}{2\pi(6-x)} \\ \frac{6}{x} &= \frac{3}{6-x} \\ 36-6x &= 3x \\ x &= 4\ \mathrm{cm} \end{aligned} \)
From the above, the third wire is 4 cm from A and 2 cm from B. It lies nearer to B, the wire with the smaller current. The answer does not depend on the size or the direction of the current in the third wire.
NCERT (2026-27):
Force Between Two Parallel Currents, the Ampere – “Let fba represent the magnitude of the force Fba per unit length. …”
\( \displaystyle f_{ba}=\frac{\mu_0 I_a I_b}{2\pi d} \)\( \qquad (4.19) \)