Introduction:
In this scenario, we have two charges, each equal to q, placed at x = -a and x = a on the x-axis. A particle of mass m and charge q0 = q/2 is placed at the origin. We need to determine the motion of the particle when it is given a small displacement y.

Explanation:
To analyze the motion of the particle, we need to consider the forces acting on it.

Electric Force:
The electric force between two charges is given by Coulomb's law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

In this case, the particle at the origin experiences electric forces from the charges at x = -a and x = a. The force exerted by the charge at x = -a will be attractive, while the force exerted by the charge at x = a will be repulsive.

Net Electric Force:
To find the net electric force on the particle, we need to consider the vector sum of the forces due to the two charges. Since the charges are located on the x-axis, the forces will act along the x-axis as well.

The force due to the charge at x = -a can be calculated using Coulomb's law:

F1 = k * (q * q0) / (-a)^2

The force due to the charge at x = a can also be calculated using Coulomb's law:

F2 = k * (q * q0) / a^2

Since the charges have opposite signs, the forces will have opposite directions. Therefore, the net force on the particle will be the vector difference of the two forces:

Fnet = F1 - F2

Equation of Motion:
Using Newton's second law, we can relate the net force to the acceleration of the particle:

Fnet = m * a

Substituting the expression for Fnet, we get:

F1 - F2 = m * a

Simplifying the equation, we find:

k * (q * q0) / (-a)^2 - k * (q * q0) / a^2 = m * a

Multiplying both sides by a^2, we get:

k * (q * q0) / a - k * (q * q0) / (-a) = m * a^3

Simplifying further, we have:

2 * k * (q * q0) / a = m * a^3

Conclusion:
By solving the equation of motion, we can determine the acceleration of the particle when it is given a small displacement y. The direction of the acceleration will depend on the relative magnitudes of the charges and the mass of the particle.