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.
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