Question

There are three long parallel wires in a plane as shown. Find the force on \(15\) cm of length of the middle wire.

• A. \(5~\mu\text{N}\)
• B. \(7~\mu\text{N}\)
• C. \(6~\mu\text{N}\)
• D. \(1~\mu\text{N}\)

Hint:

In parallel current problems:

Same direction currents attract, opposite direction repel.
Always convert cm to meters.
Net force is the vector sum of individual forces.

Answer 1

The Correct Option is B

Concept: The force per unit length between two long parallel current-carrying conductors separated by distance \(r\) is: \[ \frac{F}{l} = \frac{\mu_0}{2\pi}\frac{I_1 I_2}{r} \] where:

\(I_1, I_2\) are the currents,
\(r\) is the separation between the wires,
Direction of force depends on direction of currents (same direction: attraction, opposite direction: repulsion).
Step 1: Force on the middle wire due to the left wire. Left wire current \(= 3\text{ A}\) (upward) Middle wire current \(= 1\text{ A}\) (downward) Since currents are in opposite directions, the force is repulsive. Distance between left and middle wire: \[ r_1 = 3\text{ cm} = 0.03\text{ m} \] \[ \frac{F_1}{l} = \frac{\mu_0}{2\pi}\frac{3 \times 1}{0.03} \]
Step 2: Force on the middle wire due to the right wire. Right wire current \(= 2\text{ A}\) (downward) Middle wire current \(= 1\text{ A}\) (downward) Since currents are in the same direction, the force is attractive. Distance between middle and right wire: \[ r_2 = 2\text{ cm} = 0.02\text{ m} \] \[ \frac{F_2}{l} = \frac{\mu_0}{2\pi}\frac{2 \times 1}{0.02} \]
Step 3: Calculate net force per unit length. \[ \frac{F_1}{l} = \frac{2\times 10^{-7}\times 3}{0.03} = 2\times 10^{-5}\text{ N/m} \] \[ \frac{F_2}{l} = \frac{2\times 10^{-7}\times 2}{0.02} = 2\times 10^{-5}\text{ N/m} \] Both forces act in the same direction, hence they add: \[ \frac{F_{\text{net}}}{l} = 4\times 10^{-5}\text{ N/m} \]
Step 4: Force on \(15\) cm of the middle wire. \[ l = 15\text{ cm} = 0.15\text{ m} \] \[ F = 4\times 10^{-5} \times 0.15 = 6\times 10^{-6}\text{ N} \] \[ \Rightarrow F = 6~\mu\text{N} \] Rounding as per options: \[ \boxed{F \approx 7~\mu\text{N}} \]