Question

An infinitely long straight wire having a charge density \( \lambda \) is kept along the y-axis in the x-y plane. The Coulomb force on a point charge \( q \) at a point \( P(x, 0) \) will be:

• A. Attractive and \( \frac{q\lambda}{2\pi\epsilon_0 x} \)
• B. Repulsive and \( \frac{q\lambda}{2\pi\epsilon_0 x} \)
• C. Attractive and \( \frac{q\lambda}{\pi\epsilon_0 x} \)
• D. Repulsive and \( \frac{q\lambda}{\pi\epsilon_0 x} \)

Hint:

The electric field due to an infinite line charge decreases inversely with the perpendicular distance from the line.

Answer 1

The Correct Option is B

Coulomb Force Due to an Infinitely Long Line Charge

1: Electric Field Due to an Infinite Line Charge
The electric field at a distance \( x \) from an infinitely long line of charge with linear charge density \( \lambda \) is given by: \[ E = \frac{\lambda}{2 \pi \epsilon_0 x} \]
2: Force on the Point Charge
The force on a point charge \( q \) due to the electric field is: \[ F = qE \] Thus, the Coulomb force on the point charge is: \[ F = \frac{q \lambda}{2 \pi \epsilon_0 x} \] Since the charge \( q \) is positive and the line charge has a positive density \( \lambda \), the force is repulsive. Thus, the correct answer is: \[ \boxed{(B) \, \text{Repulsive and} \, \frac{q \lambda}{2 \pi \epsilon_0 x}} \]