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 $r \ll h$. 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)
(image)
($1$) Which one of the following statements is correct?
($A$) The balls will stick to the top plate and remain there
($B$) The balls will bounce back to the bottom plate carrying the same charge they went up with
($C$) The balls will bounce back to the bottom plate carrying the opposite charge they went up with
($D$) The balls will execute simple harmonic motion between the two plates
($2$) The average current in the steady state registered by the ammeter in the circuit will be
($A$) zero
($B$) proportional to the potential $V_0$
($C$) proportional to $V_0^{1 / 2}$
($D$) proportional to $V_0^2$
Give the answer quetion ($1$) and ($2$)
IIT 2016, Advanced
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$1.$ The correct option is A The balls will bounce back to the bottom plate carrying the opposite charge they went up with.
The ball gets positively charged when touches the bottom plate and will experience a constant force due to electric field generated due to the potential difference between the plates which is directed upwards. The balls get accelerated upwards under the influence of this force.
After hitting the top plate, the balls will get negatively charged and will now get attracted to the bottom plate which is positively charged. The motion of the balls will be periodic but not $SHM$.
$2.$ $I_{a v} \propto \frac{Q}{t}$
$\text { Here } Q \propto V _0$
$\text { Also } \quad S = ut +\frac{1}{2} at ^2$
$h=\frac{1}{2} \frac{ QE }{ m } t ^2=\frac{1}{2}\left(\frac{ Q \times 2 V_0}{ mh }\right) \times t ^2$
$\therefore \quad t \propto \frac{1}{V_0} \quad-\text { (iii) } \quad\left[\because Q \propto V _0\right]$
$I _{ av } \propto \frac{ V _0}{1 / V _0}= I _{ av } \propto V _0^2$
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