Yesterday in my class in finite element methods my professor took a couple of minutes to talk about a really simple derivation for both the Rayleigh limit and the Taylor limit. These limits basically are a balance between surface tension (holding the droplet in its spherical shape) and electrostatic repulsion. When the Rayleigh limit of charge is approached, the droplet tends to break up because the surface tension force is no longer strong enough to hold it together. There’s a ridiculously simple derivation of this equation that … well … should be obvious. But still, this might be simple enough for someone who reads vdov.net to actually read. The simple balance is surface tension against electric field.

Where is the surface tension, is the radius of the sphere, is the permittivity and is the electric field. The following derivation follows simply by application of a little algebraic gymnastics and Maxwell’s equations/stress tensor (where is the charge density and is the total number of charges).

Therefore,

This is remarkably similar to the actual Rayleigh limit (the only difference being a multiplier coming from the fact that this is in fact a sphere), which is given by: