Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius $r < < h$ . Now a high voltage source (HV) is connected across the conducting plates such that the bottom plate is at $+ V_{0}$ and the top plate at $- V_{0}$ . Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)
Which one of the following statements is INCORRECT?

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The balls will execute simple harmonic motion between the two plates

The time taken by the spheres to move from bottom to top is proportional to $V_{0}^{- 1}$

The average current between the plates is proportional to $V_{0}^{2}$

The balls will bounce back to the bottom plate carrying the opposite charge they went up with

answer is A.

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Detailed Solution

$E = \frac{2 V_{0}}{h}$

When sphere will be at bottom, charge is $q = K V_{0}$
When it will touch the top plate, charge would be $q' = K V_{0}$
Current $i = \frac{q}{T}$ , where T is time taken by spheres to move from bottom to top
$h = \frac{1}{2} (\frac{2 V_{0} q}{h m}) T^{2} \Rightarrow h = (\frac{K V_{0}^{2}}{M H}) T^{2} T^{2} = \frac{m h^{2}}{K V_{0}^{2}} \Rightarrow T = (\sqrt{\frac{m}{k}}) \frac{H}{V_{0}} i = \frac{K V_{0}}{(\sqrt{\frac{m}{k}} \frac{H}{V_{0}})} \Rightarrow I α V_{0}^{2}$