Let's say we have a sphere of charge of radius R, volume V, with total charge Q at t=0, so that we can express this as [tex] \rho ( 0 ) [/tex] = Q/V. Now, if we were to "let go" of this clump of charge, the electrons would fly off due to the mutual repulsion. My question is how to model this, ie, how to write this as [tex]\rho ( t )[/tex]?(adsbygoogle = window.adsbygoogle || []).push({});

I can write the equations of motion for an electron originally at [tex]r_{\circ}[/tex] < R, but I want to deal with the charge density as my fundamental object here. I should also mention that for simplicity's sake, I'm only considering electrostatics here, so no B field.

What I'm missing is a way to express equations of motion for a charge density. I tried to guess the Lagrangian for a charge density by setting the charge density equal to a delta function [tex] \rho (x') = q \delta (x' - x(t) ) [/tex], as a way of modelling a particle of charge q with position x(t).

I feel like this may even be a fluid dynamics problem, with the pressure corresponding to the electrostatic pressure. Would I then just apply Euler's equations of fluid dynamics?

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# Equation of motion for charge density

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