Two long straight parallel wires are carrying charges \( \lambda_1 \) and \( \lambda_2 \) per unit length, respectively. The separation between their axes is \( d \). The magnitude of the force exerted on unit length of one wire due to the charge on the other wire is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( f = \displaystyle\frac{\lambda_1\lambda_2}{2\pi\varepsilon_0 d} \)
First find the electric field of a long line charge with Gauss's law, then the force on a length of the second wire.
The field of the first wire (charge \( \lambda_1 \) per unit length) at a distance \( d \) is
\( \displaystyle E = \frac{\lambda_1}{2\pi\varepsilon_0 d} \)
A length \( l \) of the second wire carries charge \( \lambda_2 l \), so the force on it is \( F = \lambda_2 l\,E \). The force per unit length is
\( \displaystyle f = \frac{F}{l} = \lambda_2 E = \frac{\lambda_1\lambda_2}{2\pi\varepsilon_0 d} \)