How do two charged particles move in an electromagnetic field?

[ConradDJ]

A simple question: if we have two particles with opposite charge, and nothing else going on to affect the field, then how do they move?

If they start out stationary with respect to each other, then they accelerate toward each other, and this current creates a magnetic field. That would slow the acceleration, but not change the direction of the particles' motion.

What about the general case in which they start off with some relative angular velocity?... so there are two currents, and I suppose the magnetic fields must deflect the particles in a direction perpendicular to their relative motion... but my intuition seems to break down here.

Does anyone know where I can look for a diagram of this situation?

Many thanks -- Conrad

 

[ConradDJ]

Here's what I'm trying to understand --

Newtonian gravity is very simply formulated as a relationship between two bodies. If you add a third body, the situation gets chaotic, and I understand there's no analytic solution to the equations.

Classical electrodynamics seems rarely to be formulated that way, in terms of two-body interaction. It's evidently much simpler to describe it in terms of a single charged particle or a current moving in a field.

So I'm trying to find out how difficult the 2-body problem is in electrodynamics. Obviously the equation will be more complex than Newton's... but is it straightforward to derive this from Maxwell's equations?

If anyone can point me to a discussion of this, I'd appreciate it.

 

[Bob_for_short]

I think you can write two similar equations: each for one particle in the field of another one plus the filed equations with sources due to these two particles. If you neglect the radiative friction, it is sufficient in your case. In non relativistic case it will a potential interaction of two charges with small corrections due to magnetic forces. See Landau-Lifgarbagez 'Theory of Field' or so.

 

[ConradDJ]

Bob -- thank you for getting me oriented.

But now since you were the one who responded, I have to ask a further question, having in mind your very interesting "reformulation" of QED in the Independent Research forum... Does this two-particle problem look significantly different if described in terms of the Center of Inertia of a compound system?

In the context of classical electrodynamics, is that just a matter of writing the same set of equations in a center-of-inertia reference-frame? Or does the "compound system" perspective play an interesting role here as well?

Conrad

 

[Bob_for_short]

The center of inertia and relative coordinates (R and r) are convenient because the corresponding equations are decoupled (independent of each other):

M_totd²R/dt²=0

µd²r/dt²=q₁q₂r/r³

These equations are written in an arbitrary reference frame, so generally R(t) = R(0) + Vt. In the center of inertia reference frame the evolution of R is the most trivial: R = 0, but it does not influence the second equation anyway.

In the approximation where only electrostatic and magnetic (instant) interactions are taken into account, this separation of variables should seemingly hold. I do not have any book at hand right now, so I cannot make sure and type it for you. I think the first equation will not change (probably the definitions of R and M_tot will obtain some relativistic and interaction corrections) and the second equation will contain in addition a magnetic force term.

In case where the interaction potential retardation and radiation are essential (they are additional variables, degrees of freedom that carry energy-momentum) the equation system should include the field equations too, that's for sure. As the number of equation increases and the equations get more complicated, the advantage of using R and r (instead of r₁ and r₂) may disappear.