Trajectory of charged particle in uniform gravitational potential

[jjustinn]

It sounds like an easy-enough problem, but even writing down the ridiculously nonlinear equation that would need to be solved is making my face hurt.

I'm talking classical/Newtonian gravitation, action-at-a-distance, constant-force. It could really be any external non-EM conserved force; gravity just seemed like the easiest.

However, the charge is NOT a test charge: e.g. its own self-field reactions are important.

Let's say it's a spherical marble, 1cm diameter, weighing 5g and charged to 1C, released from rest in a vacuum 50m above the surface.

Now I seem to recall that there is no "radiative braking" on a charge with uniform acceleration -- which is what the trajectory would be if there was no radiative braking...so you see why my face hurts.

Has anyone tackled this (or something similar) in he past? It seems like it should be about the simplest possible situation -- ye olde one-body problem -- but I can't find any references to it.

Thanks,
Justin

 

[haruspex]

Now I seem to recall that there is no "radiative braking" on a charge with uniform acceleration

My understanding is that a net acceleration of charge always has associated radiation, whether absorption or emission.

 

[cragar]

the question you ask is not simple at all. You want to know if a charged object in free-fall will radiate. Well we have Einsteins equivalence principle that says being in free fall is like floating in free-space. Try looking up radiation from objects in free-fall.

 

[jjustinn]

the question you ask is not simple at all. You want to know if a charged object in free-fall will radiate. Well we have Einsteins equivalence principle that says being in free fall is like floating in free-space. Try looking up radiation from objects in free-fall.

That's why I was hesitant to say "gravity" and tried to make clear that I was only talking about non-relativistic gravity; if you prefer, you can think of it as an unknown constant-potential field whose coupling has nothing to do with either force or mass; for instance, the great invisible pink unicorn stamps upon the particle so that it exerts a constant force upon it.

The problem you're talking about *is* very interesting, and there seems to be a lot of lively debate on the topic...which, unfortunately, makes finding discussion of the "simpler" problem nearly impossible ;).

My understanding is that a net acceleration of charge always has associated radiation, whether absorption or emission.

Right -- but just because there is power being radiated (Larmou formula-power proportional to acceleration) doesn't mean there's a net force on the particle (Abraham-Lorentz: force proportional to jerk). This link explains that part (http://books.google.com/books?id=Lh...orce on uniformly accelerating charge&f=false), but of course doesn't go into the consequences of this with respect to an unconstrained charge experiencing a net force; apparently the net energy flux out (Larmour formula) is exactly balanced by the energy flux in (work on the particle?)...

 

[sardar]

Dear Justin,

Thank you for your inquiry about the trajectory of a charged particle in a uniform gravitational potential. While this problem may seem simple at first, the inclusion of the particle's own self-field reactions adds complexity and makes it a challenging one to solve.

In classical/Newtonian gravitation, the force on a charged particle in a uniform gravitational field is given by the equation F = qE, where q is the charge of the particle and E is the electric field. However, in this case, the charge is not a test charge and its own self-field reactions must also be taken into account.

To solve this problem, we would need to use the equations of motion for a charged particle in an electric field, as well as the equations for the self-field reactions. The resulting equation would be nonlinear and difficult to solve analytically. It may be possible to use numerical methods to approximate the trajectory of the particle, but even then, the calculations would be complex and time-consuming.

There may be some simplifications that can be made, such as assuming the particle is a point charge or neglecting its self-field reactions. However, these simplifications may not accurately represent the real-world scenario.

In terms of previous research on this topic, there have been studies on the motion of charged particles in a uniform gravitational field, but these often neglect the self-field reactions. It is possible that some research has been done on this specific scenario, but I was unable to find any references to it. Perhaps you could try reaching out to experts in the field or conducting further literature searches to see if there are any relevant studies.

I hope this information helps in your understanding of the trajectory of a charged particle in a uniform gravitational potential. It is indeed a complex problem, and further research may be needed to fully explore all aspects of it.

Best regards,