Question

Consider two long, straight, parallel conductors A and B separated by a distance \(d\). They carry steady current \(I\) in the same direction. The force acting on unit length of conductor B due to conductor A is:

• A. \(\dfrac{\mu_0}{2\pi}\dfrac{I^2}{d}\) and is attractive
• B. \(\dfrac{\mu_0}{2\pi}\dfrac{I^2}{d}\) and is repulsive
• C. \(\dfrac{\mu_0}{4\pi}\dfrac{I^2}{d}\) and is attractive
• D. \(\dfrac{\mu_0}{4\pi}\dfrac{I^2}{d}\) and is repulsive

Hint:

Use the field of a long wire, mu0 I by 2 pi d, and then F = I L B. Currents in the same direction attract.

Answer 1

The Correct Option is A

Step 1: Understanding the question:
Two long parallel wires A and B carry the same current in the same direction. We want the force per unit length on B and whether it attracts or repels.

Step 2: Field of A at the place of B:
A long straight wire carrying current \(I\) makes a field \(B_A = \dfrac{\mu_0 I}{2\pi d}\) at distance \(d\). By the right hand rule, this field is perpendicular to the plane of the two wires.

Step 3: Force on B:
The force on length \(L\) of B is \(F = I\,L\,B_A\), so the force per unit length is
\[ \frac{F}{L} = I B_A = \frac{\mu_0}{2\pi}\frac{I^{2}}{d} \]
Applying \(\vec{F} = I\vec{L}\times\vec{B}\) shows that this force points towards A. Parallel currents in the same direction attract.

Step 4: Check the options:
(A) \(\dfrac{\mu_0}{2\pi}\dfrac{I^{2}}{d}\), attractive: correct.
(B) same size but repulsive: wrong, same direction currents attract.
(C) and (D) have \(\mu_0/4\pi\): wrong, the correct factor is \(\mu_0/2\pi\).

Final Answer:
The force per unit length is \(\dfrac{\mu_0}{2\pi}\dfrac{I^{2}}{d}\) and it is attractive, option (A).