Two identical positive charges Q each are fixed at a distance of ‘2a’ apart from each other. Another point charge q₀ with mass ‘m’ is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge q₀ executes SHM. The time period of oscillation of charge q₀ will be :

Step-by-Step Explanation:

Force on q₀ at Equilibrium Position:

At the midpoint, the charge q₀ experiences forces due to both charges Q. Since these forces are equal in magnitude but opposite in direction, they cancel each other out, and the net force on q₀ is zero at this equilibrium position.

Displacement from Equilibrium and Restoring Force:

If q₀ is displaced by a small distance x along the line joining the charges, it will experience a net restoring force that pulls it back towards the midpoint. This restoring force is proportional to the displacement x, a characteristic of simple harmonic motion (SHM).

Calculation of Restoring Force:

When q₀ is displaced slightly by x, the forces from the charges Q create a net force that acts as a restoring force, and for small x, it follows Hooke's law.

Time Period of SHM:

For SHM, the time period T is given by:

T = 2π √(m / k_eff)

After calculations, the effective time period for the charge q₀ executing SHM is:

T = 2π √((4 π ε₀ m a³) / (Q q₀))

Final Answer:

The time period of oscillation of the charge q₀ is:

T = 2π √((4 π ε₀ m a³) / (Q q₀))

Therefore, the correct option is A