Interaction of a Charged Particle with an Infinite Sheet of Charge

Introduction:
When a particle of charge q and mass m is released near an infinite sheet of charge with charge density σ, it experiences an electric force due to the electric field created by the sheet. This force causes the particle to accelerate and acquire a velocity v over a distance AB of length l.

Electric Field and Surface Charge Density:
The electric field created by the infinite sheet of charge can be calculated using Gauss's Law. The electric field is uniform and perpendicular to the sheet. The magnitude of the electric field E is given by E = σ/2ε₀, where ε₀ is the permittivity of free space.

The surface charge density σ represents the amount of charge per unit area on the sheet. It is defined as the charge Q divided by the area A of the sheet, i.e., σ = Q/A. The unit of surface charge density is C/m².

Force and Acceleration:
When the charged particle is released near the sheet, it experiences an electric force due to the electric field of magnitude F = qE. The direction of the force depends on the sign of the charge q. If q is positive, the force is in the direction of the electric field, and if q is negative, the force is in the opposite direction.

According to Newton's second law, F = ma, where m is the mass of the particle and a is its acceleration. Thus, the acceleration of the particle can be calculated as a = F/m = qE/m.

Velocity Acquisition:
The particle accelerates due to the electric force and acquires a velocity v over a distance AB of length l. The relationship between velocity, acceleration, and distance is given by the kinematic equation v² = u² + 2as, where u is the initial velocity of the particle (which is zero in this case).

Solving for v, we get v = √(2as), where s is the distance traveled by the particle. Substituting the value of acceleration from the previous equation, we obtain v = √(2qE/m) * s.

Conclusion:
In summary, when a particle of charge q and mass m is released near an infinite sheet of charge with charge density σ, it experiences an electric force due to the electric field created by the sheet. This force causes the particle to accelerate and acquire a velocity v over a distance AB of length l. The surface charge density σ represents the amount of charge per unit area on the sheet, and the electric field created by the sheet is uniform and perpendicular to the sheet. The magnitude of the electric field is given by E = σ/2ε₀, and the acceleration of the particle is given by a = qE/m. The particle's velocity can be calculated using the kinematic equation v = √(2qE/m) * s.