My Charged Insulating Marble: What Should Have Happened?

[jjustinn]

I have a charged insulating marble -- let's say I used an electrostatic generator to get it up to 1 coulomb, and it weighs 1g (don't make me say 9.8mN). I have it in an insulating "box" whose walls and ceiling are very far away.

I have it sitting still for a long time, so its E field is ~1/r^2, and since it's perfectly still, the outward forces in all directions balance out (and apparently aren't as strong as the rigid forces keeping the marble together), so it stays put. But then, my cat jumps on the table and jostles the box -- so the marble moves ever-so-slightly to one side (say, the x direction). So at that instant, the charge is slightly off-center, but it takes a finite amount of time for the field to update: so there is now a net force in the x direction (since E=1/x^2, and x<<1, it is even rather large), causing it to accelerate more in that direction, and the process repeats: this creates a current in the x direction, which in turn creates a B field in the Z direction (eventually), which would cause an acceleration in the y direction, so there is no "braking" force to be found. The marble accelerates to 0.9999C and blasts through the side of the box, nearly crippling the cat.

Now, what *should* have happened? Why don't we observe this with every perturbed charge? As far as I can tell, this is the accepted answer for classical calculations that include self-fields, even after renormalization (See http://en.wikipedia.org/wiki/Abraham–Lorentz_force, or Dirac's paper on classical self-fields).

 

[HallsofIvy]

You seem to be thinking that the "charged insulating marble" will be influenced by its own electric field. That is not true.

 

[Khashishi]

You have some sign error in your thought experiment. The self-interaction will tend to slow down the acceleration and give off Larmor radiation, not increase it.

 

[jjustinn]

You seem to be thinking that the "charged insulating marble" will be influenced by its own electric field. That is not true.

If that's the case, how does my marble know which electric fields are "its own" and therefore should be ignored? Not to mention explaining the Larmour equation (and other effects like the lamb shift) will be pretty hard without the self-field.
Now, it does appear that classical electrodynamics is totally useless for any calculations where the self-field is significant (eg not using "test charges") -- but I don't recall being taught that in school (though we always did use test charges), nor can I find anyone who will come out and say it.

You have some sign error in your thought experiment. The self-interaction will tend to slow down the acceleration and give off Larmor radiation, not increase it.

I'll have to reexamine the Larmour formula, but I think that's a different phenomena -- I can tell you right away that it depends on acceleration, and the effect I'm describing should even happen with constant velocity (though then the matter gets confused by wanting to look at it from the charge's frame).

Briefly, though, I just want to make clear that the signs are correct: first, we agree that if a my charged marble was instead a charged ball of sand, the grains would accelerate away from each other, right? This is basically the same effect. Since like charges repel like charges, the acceleration at x + dx (where the retarded center of the radial electric field is x) will in the same direction as the displacement, right? So we don't get braking, but acceleration.

If I'm mistaken, please show me how (and saying "that's not what happens" doesn't work, because everyone says that Maxwell's equations / the Lorentz equation are correct at the classical level -- which would seem to contradict this experiment).

 

[haruspex]

It cannot overtake its own field. You seem to have the marble moving faster than light.

 

[jjustinn]

It cannot overtake its own field. You seem to have the marble moving faster than light.

Actually, you've got it backwards -- for it NOT to be in its own field, it would have to be moving faster than light -- but that's a common misconception (that I once shared) presumably brought about by the fact this type of problem just isn't discussed.

If you're not convinced, look at it this way: a stationary charge is "in it's own" field, right? Eg it is surrounded by a sphere of radius ct (t=time it has been there and "on") with E=1/r^2? Since the field propagates out from the charge at speed c (and everything within that radius "is in the charge's field), it would have to move faster than c to NOT be in its "own field".

Or, a demonstration you can try: start whistling (or humming, singing, whatever). The sound of your whistle travels outward at Mach 1. According to your logic, to be able to "be affected by your whistling" (eg to hear it) you would need to be traveling faster than Mach 1 -- I don't know about you, but that hasn't been my experience, as demonstrated by the fact that people walking incredibly slowly in front of me seem perfectly able to hear the person talking on their phone.

Now, imagine that you're standing on a machine that will accelerate away from any sound it picks up on its very directional microphones (and let's say that the engine in it makes constant noise). Now in equilibrium, since the microphones are located symmetrically around the engine, it doesn't move -- unless that equilibrium is upset, in which case positive feedback starts.

 

[haruspex]

Actually, you've got it backwards -- for it NOT to be in its own field, it would have to be moving faster than light -- but that's a common misconception (that I once shared) presumably brought about by the fact this type of problem just isn't discussed.

If you're not convinced, look at it this way: a stationary charge is "in it's own" field, right? Eg it is surrounded by a sphere of radius ct (t=time it has been there and "on") with E=1/r^2? Since the field propagates out from the charge at speed c (and everything within that radius "is in the charge's field), it would have to move faster than c to NOT be in its "own field".

Or, a demonstration you can try: start whistling (or humming, singing, whatever). The sound of your whistle travels outward at Mach 1. According to your logic, to be able to "be affected by your whistling" (eg to hear it) you would need to be traveling faster than Mach 1 -- I don't know about you, but that hasn't been my experience, as demonstrated by the fact that people walking incredibly slowly in front of me seem perfectly able to hear the person talking on their phone.

Now, imagine that you're standing on a machine that will accelerate away from any sound it picks up on its very directional microphones (and let's say that the engine in it makes constant noise). Now in equilibrium, since the microphones are located symmetrically around the engine, it doesn't move -- unless that equilibrium is upset, in which case positive feedback starts.

No, the problem is that you have it overtaking the change in its field that results from its own motion.
In your whistling/talking analogy, that would require me to run so fast I hear what I said earlier.

 

[jjustinn]

No, the problem is that you have it overtaking the change in its field that results from its own motion.
In your whistling/talking analogy, that would require me to run so fast I hear what I said earlier.

Well-put (now THAT is a thought experiment!).

However, I think you're thinking of a point particle -- imagine, for instance, that you had a line of ears: you would hear "what you said earlier" at the "far ear"...but in this case, that analogy isn't very good (that's kind of what I was trying to get at with the ring of equidistant microphones, but that's pretty awkward).

Before digging further into the weeds here, you certainly accept the presence of self-field effects for accelerating/jerking objects, even though they're not moving at superluminal (or even relativistic) speeds, right? Granted, I made the assertion (perhaps foolishly, certainly without adequate investigation) that it worked with constant velocity - but it sounds like you're saying that those would only happen if it somehow "overtook" its field?

To make sure I had this right, I worked it out on grid paper -- and with a point particle, I think you're right, but with an extended object, I stand by my analysis. I moved the object one unit per time slice, and radiated the field at 3 units per time slice, so an object of radius 3.5units will end up at the first time slice with its far-right unit in its retarded field from t=0, while the field is 0 elsewhere: so it will be accelerated further in the same direction. Now v=c/3 is indeed a relativistic speed, but at the continuum limit, it holds for any non-point object at any speed (since there isn't that annulus of E=0 between the field that radiated out at t-1 and the new field generated at t).

Or, if you don't buy my hand-waving assertion about the continuum limit, take an object with a large-enough radius that it doesn't have to be moving at relativistic speeds for some part of it to be in the retarded field; say, r=2 light-seconds, v=1m/s. it's a bigass marble, but you shouldn't be able to violate conservation of energy no matter how big your object is (and I still think it works the same with a normal-sized marble).

 

[haruspex]

You seem to be rediscovering special relativity, i.e. the non-additivity of velocities at relativistic speeds.
In the frame of reference of a charged body moving at uniform velocity relative to you, its own field is stationary. Your problem arises because you take the speed of propagation of the field, as observed by you, and subtract the speed of the body, as observed by you.

 

[jjustinn]

You seem to be rediscovering special relativity, i.e. the non-additivity of velocities at relativistic speeds.
In the frame of reference of a charged body moving at uniform velocity relative to you, its own field is stationary. Your problem arises because you take the speed of propagation of the field, as observed by you, and subtract the speed of the body, as observed by you.

Right -- hence my hindsighted regret at moving from "an isolated stationary marble that got bumped" to one moving at constant velocity. But even in the constant-velocity case, granting that in the marble's own frame, it's in equilibrium, but that doesn't explain how the field transforms to perfectly cancel in our fame (I had a similar thread here on this specific topic a while back, and after scouring the internets, I found a book that showed that a rigid sphere with constant velocity had its field transformed so that it just canceled -- so it wasn't accelerated, but not because its field "outran" it, but because geometry conspired to make it just so).

 

[jjustinn]

Looking for more answers, I came upon a paper (http://www.philosophy.umd.edu/Faculty/mfrisch/papers/inconsistency.pdf) basically making the same claim (eg that the Maxwell-Lorentz equations taken literally lead to inconsistencies either directly -- by using point particles -- or indirectly, through the unbounded acceleration of the Dirac-Lorentz equation, which is just a carefully-derived equation of motion based of the Maxwell/Lorentz system), and a rebuttal (http://philsci-archive.pitt.edu/2843/1/Frisch.pdf) -- which claims (citing non-free papers as evidence) that the Maxwell/Lorentz system is only inconsistent if you DO ignore the self-field (eg contrary to what I was arguing).