Question

PARAGRAPH
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)

The average current in the steady state registered by the ammeter in the circuit will be

• A. proportional to $V_0^2$
• B. proportional to the potential $V_0$
• C. zero
• D. proportional to $V_0^{1/2}$

Answer 1

The correct option is A proportional to $V_0^2$

Let $u$ be the initial velocity of ball, $h$ be the distance between the plates and $t$ be the time taken by the ball to cover the distance between the plates i.e $h$
Using equation of motion,
Since $u=0$, $\begin{array}{}h=\frac{1}{2}a{t}^2& \text{(1)}\end{array}$
The balls experience force due to electric field given by,
$F=qE=\frac{2C{V}_o^2}{h}$
The acceleration will be, $a=\frac{F}{m}=\frac{2C{V}_o^2}{mh}$
Substituting this in equation $\left(1\right)$
$t=\sqrt{\frac{2h^{2}m}{2C{V}_o^2}}=\frac{h}{V_o}\sqrt{\frac{m}{C}}$
Average current,
$I_{avg}=\frac{q}{t}=\frac{C{V}_o}{\frac{h}{V_o}\sqrt{\frac{m}{C}}}$
$\Rightarrow I_{avg}\propto V_0^2$


Answer is option A.