Given:
Charge of particles A and B : q = Q
Separation between A and B : d

Here x is the displacement of particle C along AB.
Distance between A and C is :
Distance between B and C is :
Formula used:
For part (a), we will be using Coulomb’s Law,
Where F is the electrostatic force on B due to A, k is a constant .
k = = 9× 10⁹ Nm²C^-2 and r is the distance between the two charges.
q₁=q₂=Q.
now, the net force acting on C is:





F_net is the net electric force experience by particle C of charge q.
(b)
When x≪d,
x²≪(d/2)²
Thus x² can be neglected.
We get,

∴ F_net α x

Thus, when x≪d force is proportional to x

(c)
The condition for Simple harmonic Motion of a particle is:
F'=mω²x
Here, m is the mass of the particle C, ω is the angular frequency.
Let F_net = F’
Thus comparing two equations of F’, we get
We know that Where t is the Time Period.


Hence time period when particle is displaced along AB is