Step-by-step Solution:

1. Analyze the Initial Setup:

   • We have a cylindrical chamber with two conducting plates at the top and bottom, separated by a height h.
   • A potential difference V is applied across the plates. Let's assume the top plate is at a positive potential +V and the bottom plate is grounded (at 0 potential). This creates a uniform electric field E directed downwards, with magnitude E = V/h (ignoring edge effects).
   • Conducting spherical balls are placed on the bottom plate.

2. Charging of the Balls at the Bottom Plate:

   • The balls are conducting and are in direct contact with the bottom plate, which is at 0 potential. Therefore, the balls will also be at 0 potential.
   • The downward electric field E polarizes the conducting balls. It attracts negative charges (electrons) to the top surface of the balls and repels positive charges to the bottom surface.
   • Since the balls are in contact with the bottom plate, the repelled positive charge can flow away into the plate (or electrons can be drawn from the plate to neutralize the induced positive charge and add to the induced negative charge). The result is that each ball acquires a net negative charge, let's call it q.

3. Motion Towards the Top Plate:

   • Once a ball has a net negative charge q, it experiences an electrostatic force $F_e = qE$.
   • Since q is negative and the electric field E is directed downwards, the force $F_e$ is directed upwards ($F_e = |q|E$).
   • There is also a downward gravitational force $F_g = mg$, where m is the mass of the ball.
   • The problem states the balls are made of a "light weight" material, which implies that the upward electrostatic force is greater than the downward gravitational force: |q|E > mg.
   • Under the net upward force $F_{net, up} = |q|E - mg$, the ball accelerates and moves towards the top plate.

4. Interaction with the Top Plate:

   • The ball, carrying a negative charge q, reaches and collides with the top plate.
   • The top plate is a conductor at a positive potential +V.
   • Upon contact, charge is redistributed between the ball and the top plate until the ball also reaches the potential +V.
   • To reach a positive potential, the ball must lose its excess electrons and become positively charged. Let's call its new charge q'.

5. Motion Towards the Bottom Plate (Bouncing Back):

   • Now the ball has a net positive charge q'.
   • The electric field E is still directed downwards.
   • The electrostatic force on the ball is now $F'_e = q'E$. Since q' is positive, this force is also directed downwards.
   • The total downward force on the ball is $F_{net, down} = q'E + mg$.
   • This net downward force causes the ball to accelerate and move back towards the bottom plate.

6. Evaluating the Options:

   • A: The balls will execute simple harmonic motion between the two plates. The force on the ball during its transit is constant (|q|E - mg on the way up, q'E + mg on the way down). Simple harmonic motion requires a restoring force proportional to displacement (F = -kx), which is not the case here. So, A is incorrect.
   • B: The balls will bounce back to the bottom plate carrying the same charge they went up with. The ball goes up with a negative charge q. It acquires a positive charge q' at the top plate before bouncing back. The charges are different. So, B is incorrect.
   • C: The balls will stick to the top plate and remain there. After touching the top plate, the ball acquires a positive charge q', and there is a net downward force (q'E + mg) acting on it. This force will push it away from the top plate, not make it stick. So, C is incorrect.
   • D: The balls will bounce back to the bottom plate carrying the opposite charge they went up with. The ball travels up with a negative charge (q). After contacting the top plate, it acquires a positive charge (q') for its journey back down. The charge it carries on its return trip (q') is opposite in sign to the charge it had on its way up (q). This statement is consistent with our analysis. So, D is correct.

This process will repeat, causing the balls to oscillate between the plates, acting as charge carriers. This is the principle behind devices like Franklin's bells.