Consider an evacuated cylindrical chamber of height h having rigid conducting plate 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 $$+V0$$ and the top plate at $$-V0$$. 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 correct?The average current in the steady state registered by the ammeter in the circuit will be

Question

Consider an evacuated cylindrical chamber of height h having rigid conducting plate 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  $$+V0$$ and the top plate at  $$-V0$$. 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 correct?The average current in the steady state registered by the ammeter in the circuit will be

Infinity Learn · Accepted Answer

When they collide and stick to upper plate, they come in contact to the potential of upper plate acquiring –ve charge, same amount as initial $$+ve$$ charge in a very short time internal. The motion of the balls will be periodic but not SHM.Acceleration,  $$a=Eqm=2V0hqm$$ time of journey,  $$t=2ha=2hhm2V0q=h2mV0q$$ Average current $$I=Nqt=NqV0qh2m$$⇒I ∝ $$q32V012$$ But $$q=CV0=4$$πε$$0rV0$$⇒q ∝ $$V0$$ From $$1$$,$$2$$   ⇒I ∝ $$V02$$

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