Equation of motion for charge density

[Zadig]

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 $$\rho ( 0 )$$ = 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 $$\rho ( t )$$?

I can write the equations of motion for an electron originally at $$r_{\circ}$$ < 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 $$\rho (x') = q \delta (x' - x(t) )$$, 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?

 

[LawlQuals]

This is for a numerical model, correct?

I think fluid dynamics would be a little too crude for this kind of calculation. How about looking into the Klimontovich equation (kinetic theory), and modifying it appropriately for this situation ($$\vec{B} \equiv 0$$)? You would track each charged particle, and trace its trajectory by iterating solutions to the continuity equation. If your end interest is to model the charge density, you would need to model the scenario with Eulerian dynamics, and keep account of a dynamic volume so as to calculate the desired parameter. I am not super competent in numerical solutions, so my suggestions can only be vague at best.

The following may be useful: http://accelconf.web.cern.ch/Accelconf/e02/PAPERS/TUPRI113.pdf , but you may find more helpful literature online, that was just the first paper I located.

 

[Bob S]

The simplest model might be to assume spherical symmetry, and a total charge Q(r,t) inside radius r at time t. The radial outward force on an electron with charge e at r,t is then

F(r,t) = eQ(r,t)/4πε₀r²

No charge at radius r'>r will have any effect on F(r,t). At time zero, the total charge is Q₀ = (4/3)πρ₀R³, where ρ₀ is a uniform charge density inside R.

For the outermost electron shell of thickness dr, at r = R,

F(R,0) = ma = eQ₀/4πε₀R²

Since this outer shell will always be outside the rest of the charge, the integration of this to obtain a radial velocity v(t) and position r(t) for the outer shell should be simple. Knowing velocities and radial positions at time t of all shells should give the charge density ρ(r,t).

Fortunately, since the charge Q(r,t) inside a radius r scales as ~r³, and the denominator in F(r,t) = eQ(r,t)/4πε₀r² as r², the fastest moving shell is the outermost shell, and the inner shells are accelerated less.

Bob S