iii. Consider the situation in which is kept constant and is slowly increased. · OlympiadBench #1169
Physics · thermodynamics
Problem
iii. Consider the situation in which r is kept constant and φ is slowly increased. The droplet becomes unstable and breaks into pieces if the hydrostatic pressure inside the droplet becomes smaller than the atmospheric pressure. Find the critical potential φmax at which this will happen.
Given constants:
Problem T2. Kelvin water dropper
The following facts about the surface tension may turn out to be useful for this problem. For the molecules of a liquid, the positions at the liquid-air interface are less favourable as compared with the positions in the bulk of the liquid. This interface is described by the so-called surface energy, U=σS, where S is the surface area of the interface and σ is the surface tension coefficient of the liquid. Moreover, two fragments of the liquid surface pull each other with a force F=σl, where l is the length of a straight line separating the fragments.
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A long metallic pipe with internal diameter d is pointing directly downwards. Water is slowly dripping from a nozzle at its lower end, see fig. Water can be considered to be electrically conducting; its surface tension is σ and its density is ρ. A droplet of radius r hangs below the nozzle. The radius grows slowly in time until the droplet separates from the nozzle due to the free fall acceleration g. Always assume that d≪r.
Part A. Single pipe
Context question:
i. Find the radius rmax of a drop just before it separates from the nozzle.
Context answer:
rmax=34ρg3σd
Context question:
ii. Relative to the far-away surroundings, the pipe's electrostatic potential is φ. Find the charge Q of a drop when its radius is r.
Context answer:
Q=4πε0φr