Question

The force per unit length between two parallel current carrying straight conductors separated by \(2d\) is given by the formula:

• A. \( \frac{\mu_0 i_1 i_2}{4\pi d} \)
• B. \( \frac{\mu_0 i_1 i_2}{8\pi d} \)
• C. \( \frac{\mu_0 i_1 i_2}{2\pi d} \)
• D. None of these

Hint:

Always substitute actual distance \(r\) in formula.

Answer 1

The Correct Option is A

To determine the force per unit length between two parallel current-carrying straight conductors, we use Ampère's force law. According to this law, the force per unit length \(f\) between two parallel conductors carrying currents \(i_1\) and \(i_2\), separated by a distance \(r\) in a vacuum, is given by:

\(f = \frac{\mu_0 i_1 i_2}{2\pi r}\)

where:

• \(\mu_0\) is the permeability of free space, with a value of \(4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}\).

In our question, the separation between the conductors is given as \(2d\). We substitute \(r = 2d\) into the formula:

\(f = \frac{\mu_0 i_1 i_2}{2 \pi (2d)} = \frac{\mu_0 i_1 i_2}{4 \pi d}\)

Thus, the force per unit length between the two conductors is:

\(\frac{\mu_0 i_1 i_2}{4 \pi d}\)

This matches the given correct answer: \(\frac{\mu_0 i_1 i_2}{4 \pi d}\).

Let's analyze why the other options are incorrect:

• \(\frac{\mu_0 i_1 i_2}{8 \pi d}\): This would be the case if we mistakenly considered \(4d\) instead of \(2d\).
• \(\frac{\mu_0 i_1 i_2}{2 \pi d}\): This assumes the distance between the conductors is \(d\) instead of \(2d\).

Hence, the option \(\frac{\mu_0 i_1 i_2}{4 \pi d}\) is correctly derived from the formula when the separation is \(2d\).