Speed of propogation of electrostatic field

[csmcmillion]

(Not sure if this should go in SR/GR forum)

I watched a Susskind lecture last week in which he was discussing the fundamental law of charge conservation. He used the example of an electrostatic field surrounding a point charge, and stated that, were we able to suddenly make the charge disappear, we could send information instantly b/c the field propagates instantly. I think I must have misunderstood him. I understand disturbances in the field (photons/EM waves) travel at c, but is it a settled matter that the field itself propagates instantly?

 

[Meir Achuz]

He might have meant to demonstrate that the static equations cannot be applied if there is time dependence.

 

[csmcmillion]

He might have meant to demonstrate that the static equations cannot be applied if there is time dependence.

I don't know. I found this in the WP article on Speed of Gravity:

"The speed of physical changes in a gravitational or electromagnetic field should not be confused for "changes" in the behavior of static fields that are due to pure observer-effects. These changes in direction of a static field, because of relativistic considerations, are the same for an observer when a distant charge is moving, as when an observer (instead) decides to move with respect to a distant charge. Thus, constant motion of an observer with regard to a static charge and its extended static field (either a gravitational or electric field) does not change the field. For static fields, such as the electrostatic field connected with electric charge, or the gravitational field connected to a massive object, the field extends to infinity, and does not propagate. Motion of an observer does not cause the direction of such a field to change, and by symmetrical considerations, changing the observer frame so that the charge appears to be moving at a constant rate, also does not cause the direction of its field to change, but requires that it continue to "point" in the direct of the charge, at all distances from the charge.

"The consequence of this, is that static fields (either electric or gravitational) always point directly to the actual position of the bodies that they are connected to, without any delay that is due to any "signal" traveling (or propagating) from the charge, over a distance to an observer. This remains true if the charged bodies and their observers are made to "move" (or not), by simply changing reference frames. This fact sometimes causes confusion about the "speed" of such static fields, which sometimes appear to change infinitely quickly when the changes in the field are mere artifacts of the motion of the observer, or of observation.

"In such cases, nothing actually changes infinitely quickly, save the point of view of an observer of the field. For example, when an observer begins to move with respect to a static field that already extends over light years, it appears as though "immediately" the entire field, along with its source, has begun moving at the speed of the observer. This, of course, includes the extended parts of the field. However, this "change" in the apparent behavior of the field source, along with its distant field, does not represent any sort of propagation that is faster than light.


I don't totally understand it.

 

[Bill_K]

There's a basis behind Susskind's remark. In the standard method of quantizing the electromagnetic field, the Coulomb interaction may be expressed in either of two ways: a) as a current-current interaction which is manifestly covariant, and b) as an instantaneous interaction between charges. But thanks to charge conservation, they are equivalent.

We cannot make a charge suddenly disappear. So stating what would happen if you did make one disappear is meaningless! Having assumed something which is impossible, you may conclude anything you like. I'm afraid Susskind is just trying to be a bit sensational with this.

 

[A.T.]

Even if we could make the charge disappear, then it still seems to me that this would constitute a change in the field which propagates at c, and not instantaneously.

 

[Per Oni]

I don't totally understand it.

I think what that article is saying is:

Say you are drawing a static charge in the middle of a sheet of paper. So a dot with many field lines spreading out, starting at the dot and ending at the paper edge. Now put your pen at the edge of the sheet and move it sideways or any other ways you want. The field lines are always pointing directly to your pen, there’s no delay because the lines were there in the first place.
By symmetry considerations this must therefore also be true for the case of your pen representing a charge as seen from the dot.

The the angle changes but will at all times be pointing instantly towards the charge.

 

[csmcmillion]

I think what that article is saying is:

Say you are drawing a static charge in the middle of a sheet of paper. So a dot with many field lines spreading out, starting at the dot and ending at the paper edge. Now put your pen at the edge of the sheet and move it sideways or any other ways you want. The field lines are always pointing directly to your pen, there’s no delay because the lines were there in the first place.
By symmetry considerations this must therefore also be true for the case of your pen representing a charge as seen from the dot.

The the angle changes but will at all times be pointing instantly towards the charge.

That makes lots of sense. However, it doesn't explain what I think Susskind was saying. If I suddenly erase the point charge, the field lines an arbitrary distance will cease to exist. The field (or lack thereof) cannot (IMHO) propagate instantaneously. (scratching head)

 

[Per Oni]

The current becomes: I = dq / dt so here: I = e / 0 which is anything you want it to be.

I would stop scratching it’s not worthit.

 

[Born2bwire]

That makes lots of sense. However, it doesn't explain what I think Susskind was saying. If I suddenly erase the point charge, the field lines an arbitrary distance will cease to exist. The field (or lack thereof) cannot (IMHO) propagate instantaneously. (scratching head)

Perhaps another way to look at this is in terms of the potentials. You can describe the electromagnetic field using the scalar and vector potential. However, because there is a degree of freedom in how you define the potentials, you have to make a gauge choice to make the potential unique. One of the gauge choices, the Coulomb gauge, makes the scalar potential an instantaneous potential. Thus, the scalar potential behaves just like it does in the electrostatic case. This would give the appearance of violating causality but we find that the vector potential, which also contributes to the electric field for the time varying case, gives rise to the retardation in the final electric field. This is of course just a restatement of what Bill_K posted earlier.

There are other ways of seeing this as well I think. Take for example Gauss' Law which states that the enclosed charge of a given volume is directly proportional to the displacement field flux. This holds for both the static and time-varying cases and so we see that if we were to instantaneously destroy a charge then we would be able to see this in the displacement field flux at any given distance (and by judicious choice of symmetry we can make the flux directly proportional to the actual field). But this, again, violates charge conservation because the only way to remove the given charge would be to add an equal but opposite amount of charge. But this means that an opposite charge must enter our Gaussian surface and thus we would see the enclosed charge change to zero as a function of the size of our suface (which in this way preserves special relativity).

So there are ways of seeing how his statement can come about but there is the underlying caveat that it cannot actually happen.

 

[T1987a2T]

The theory explains the function in One Dimensional Harmonizing Oscillators (O.D.H.O's) Functions. The State of Fields determines the decaying rate from it's Mo (resting state) to it's potential (ρ) in a Tmass and Bmass function. In the question the formula seeks the interaction of EMF Functions (Electro-Magnetic Fields/Forces)of J. The (h) constant @ O of Top Mass and Bottom Mass is presurfistic of Natural Phenomena Occurrence of Tmass and Bmass. The (J) distribution between Tmass and Bmass is chaotic and unpredictable, but the function remains the same in a Natural State but has the (ρ) in a manipulated artificial environment.