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 lt lt 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 be 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 correct ?

Options

A. (a)proportional to `V_0^2`
B. (b)proportional to the potential `V_0`
C. (c)zero
D. (d)proportional to `V_0^(1//2)`

Explanation

A the balls keep on carrying charge from one plate to another, current will keep on flowing even in steady state. When at bottom plate, if all balls attain charge q,
`(kq)/r=V_0 , (k=1/(4pie_0)) implies q=(V_0r)/k`
Inside cylinder electric field `E=[V_0-(V_0)]h=2V_0h`
`implies` Acceleration of each ball `a=(qe)/m=(2hr)/(km). V_0^2`
`implies` Time taken by balls to each other plate `t=sqrt((2h)/a) =sqrt((2h.km)/(2hrv_0^2))=1/V_0sqrt((km)/r)`
If there are n balls , then
Average current `j_(av)=(nq)/t=n times (V_0r)/k +V_0sqrt(r/(km)) implies j_(av) prop v_0^2`