A Coulomb law for moving charges

Coulomb's law expresses the force between two charges q1 and q2 that are stationary in an inertial frame of reference Σ as a function of the distance r between them:

(Equation 1)

Can this equation be generalized to the case of two charges q1 and q2 which are stationary relative to each other but which are travelling at v through an inertial frame of reference Σ in which light propagates in symmetrical conditions and clocks have been Einstein-adjusted?

What form does Coulomb's law take in that case, and can the shape of that law be explained in terms of a simple model of how electric field disturbances propagate, without invoking length contraction, time dilation or magnetism?

Recall that in my sphere model of electricity, as discussed here, charges that are stationary in Σ are deemed to be surrounded by concentric electrical spheres with a particular sphere density λΣ = nΣ /r for a line of length r that starts in the origin.

(Figure 1)

When the charge is locally accelerated to a speed v in Σ and then continues to move at that speed, the information about the acceleration spreads through the spheres at the speed c, leading to the following situation after a time t has passed:

(Figure 2)

I will now consider the case of two charges q1 and q2 moving at v through Σ and separated by the distance r. Let n1 be the number of spheres surrounding q1 that cut through r ; n2 the number of spheres surrounding q2 that cut through r ; and β the angle at which the spheres that surround q1 and q2 cut through r :

(Figure 3)

The fact that the two angles marked β are the same can be seen by prolonging the line P2P3 in Figure 2 downwards until it meets the circle on which P3 lies and realizing that the resulting triangle formed by that point, P1 and P3 is isosceles.

My contention is that, in the situation shown in Figure 3, Coulomb's law for stationary charges in Σ must be modified as follows to obtain the force between q1 and q2 as measured by co-moving springs:

(Equation 2)

This Coulomb law for moving charges can alternatively be formulated in terms of sphere densities. If λ1 is the sphere density on r of the spheres around q1, and λ2 is the sphere density on r of the spheres around q2, then:

(Equation 3)

[PS: As indicated here, a more general analysis shows that the two densities that need to be considered are the density of q1 spheres over r (a) along the line connecting the two charges on the side facing towards q2 and (b) along the line connecting the two charges on the side facing away from q2 . The result in the case of two charges moving at the same velocity is the same, but in more general situations this is the approach that must be taken.]

Equation 2 constitutes a fundamental law in my sphere model of electricity as it is simple, plausible, empirically correct, and very powerful:

1) The law could hardly be any simpler. It is nicely symmetrical and, in addition to the concept of electric charge, Equation 2 merely includes two basic sphere model parameters: the number of spheres that cut through r and the angle at which they cut through r. The sine of that angle and the sphere numbers are simple factors in Equation 2.

2) Equation 2 is plausible as it represents a plausible generalization of Coulomb's law for charges that are stationary in Σ.

I have already set out here why the addition of the factor nΣ2/n1n2 is a natural generalization of Coulomb's law to moving charges.

As for the factor 1/sin2β, it can be seen in Figure 3 that r crosses each sphere shell along a path which is longer by the factor 1/sinβ than the shell's thickness, and this is true for both charges.

In the case of charges that are stationary in Σ, on the other hand, the path taken by r through each shell corresponds to the shell's thickness. Apparently what matters is each shell's local thickness, so r in the Coulomb law for moving charges must be normalized by multiplying it by sinβ.

3) The proposed law is empirically correct in the sense that it gives the same force as the tried and tested classical theories of electromagnetism and special relativity. This can be seen by determining the sphere density on a straight line emanating from a moving charge as a function of v, c and the angle α between that line and the vector v, by applying the cosine rule to the triangle P1P2P3 in Figure 2:

The term sinβ can also be expressed as a function of v, c and α, by applying the sine rule to the same triangle:

Inserting these expressions for the sphere density and sinβ into Equation 3 yields:

(Equation 4)

This equation is nothing but the result for the force between q1 and q2 as measured by co-moving springs obtained in classical theory, by applying Coulomb's law for stationary charges to q1 and q2 in the moving system and taking into account relativistic length contraction.

Therefore, if classical theory is empirically correct, then so is my Coulomb law for moving charges.

4) Equation 2 is very powerful because it has momentous consequences. If it is accepted as fundamental, Equation 4 follows and with it electric length contraction by the relativistic factor in the direction of movement, and in that direction only.

And that's not all. My sphere model implies the independence of the speed of light from the movement of its source, in other words two light signals emitted from the same location in the same direction will always travel side by side regardless of the relative speed of the sources of the light. Together with length contraction, this implies time dilation by the relativistic factor if light clocks of any orientation are used.

And that's still not all because length contraction and time dilation by the relativistic factor together imply all of special relativity, including the constancy of the two-way speed of light in all inertial frames of reference and including the Lorentz transformations if clocks in all inertial frames of reference are Einstein-adjusted.

All of that follows from the simple, innocent-looking Equations 2 and 3. But some big questions remain.

Can I be sure that it is possible to generalize from "electric length contraction" and "light clock time dilation" by the relativistic factors to length contraction and time dilation in general?

Can the sphere model of electricity be generalized to describe and explain the interaction of charged particles in arbitrary states of motion relative to each other?

Can it be used to explain the full range of "magnetic" phenomena?

Can Equations 2 and 3 be explained in more fundamental terms than the somewhat ad hoc way in which I have introduced them?

What role, if any, might be played by models of how charged particles exchange information with each other?

I suspect I may only be able to answer these questions once I have gained a deeper understanding of existing models of electricity and magnetism, not to mention other areas of physics.

I intend to make a start by studying and reviewing W. Geraint V. Rosser's Interpretation of Classical Electromagnetism (1997).

Any suggestions for other relevant literature, whether in the comment section or by email, would be gratefully received.

Dialogue on the cause of length contraction

After the excitement of my previous post on the cause of relativistic length contraction, let me pause for a moment to discuss my revolutionary discovery with a hypothetical reader.

Reader: OK then, here's what I think: your supposed "revolutionary discovery" is quite redundant. Elementary textbooks make it clear that length contraction is a consequence of the relativity of simultaneity. For example, Leo Sartori (1996) states that length contraction "can be linked to the relativity of simultaneity" (p. 83). As he explains, "if ground observers measure the position of the front and rear ends of a moving train simultaneously according to their clocks, these measurements take place at different times according to train clocks. According to train observers, therefore, the length measured by ground observers is incorrect." No further explanation is required.

Me: With all due respect, any textbooks that interpret length contraction as a consequence of the "relativity of simultaneity" are mistaken. To see this, it is important to remember that the "relativity of simultaneity" is a result of the decision to use Einstein's clock adjustment procedure to adjust clocks in all inertial frames of reference. It would be equally possible to adjust clocks in a manner which makes "simultaneity" absolute, for example by using Einstein's clock adjustment procedure in just one inertial frame of reference and adjusting all other clocks to zero when they pass a clock in that frame showing zero. There would then be no "relativity of simultaneity". But all moving objects in the frame of reference in which clocks have been Einstein-adjusted would still be length-contracted. Which proves my point.

The truth is that length contraction in all inertial frames of reference in which clocks have been Einstein-adjusted is a fundamental empirical fact for which special relativity has no explanation. Unless the constancy of the two-way speed of light is accepted as a fundamental empirical fact, in which case length contraction follows. But that makes no sense because speed is derived from length and time measurements and therefore length contraction and time dilation are more fundamental than the constancy of the two-way speed of light.

Reader: I see that you enjoy picking holes in other people's arguments. Let me pick one in yours. You base your explanation of length contraction on the concept of force. There's a problem with that because the definition of force via Newton's laws of mechanics involves the concept of mass and requires the existence of time coordinates in any frame of reference in which forces are said to act. In your derivation of length contraction, however, you have not specified a clock adjustment procedure in the moving frame of reference, so you have not defined any time coordinates in that frame, let alone explained the concept of mass. Therefore, your reference to a force acting between two moving charges simply makes no sense.

Me: Point taken, I should have said a bit more about the concept of force underlying my previous post. What I had in mind was not so much Newton's laws but a static concept of force, in which, for example, a particular "force" is said to act on a spring if the spring is extended by a particular amount. Two such springs that are stationary in a frame of reference Σ in which light propagates in symmetrical conditions and clocks have been Einstein-adjusted can then be used to find experimentally that two charges exert forces on each other which are proportional to the amount of charge attached to either spring and inversely proportional to the square of the distance between the charges. This is nothing but the Coulomb law for stationary charges.

The same kind of experiments can be performed in any spatial inertial frame of reference S that moves relative to Σ, and the Coulomb law will be found to be valid in that frame, too. All of that can be done without defining any time coordinates in the moving frame and without defining mass. But these experimental findings do not tell us anything about length contraction. Indeed, they are compatible with any length contraction factor. And this is where my sphere model comes in.

Using this model, the Coulomb law can be formulated so that forces between charges that are stationary in S can be expressed as a function of their distance as measured in Σ. It would be found experimentally that this version of the Coulomb law predicts the correct force between such charges as measured in S using springs. This formulation of the Coulomb law can then be used to derive the length contraction factor in Σ in the manner shown in my previous post.

Reader: Even if you can resolve the force issue by relying on a very limited concept of force, it remains that your model is pretty tautological. It merely "explains" electric length contraction because you have adapted it to that purpose. You manipulate the Coulomb law in exactly such a way that you get the desired result. I'm sorry, but all of this does not add up to much of an explanation at all.

Me: Your objection is not unfounded but I suspect it could be applied to any model of the physical world. For we surely strive to design any such model in a manner which makes it correspond to what we find empirically. The real question, it seems to me, is whether we can get a model to explain a very broad range of phenomena while keeping it simple and plausible. In the case of the sphere model of electricity, initially I developed it not in order to explain length contraction but simply to explain the independence of the speed of light from its source. If anything, I came up with the sphere model despite the fact that at first sight it seemed quite incapable of explaining length contraction. It thus came as a surprise to me to find that length contraction can be explained quite naturally in terms of that model. The independence of the speed of light from its source and length contraction may not add up to a very broad range of phenomena, but I feel I've made a promising start! At the same time, I feel my model is reasonably plausible and astonishingly simple.

It is true that the plausibility or simplicity of any particular physical model is to some extent a matter of opinion. As Kevin Brown suggests in his Reflections on Relativity (2010),most physicists appear to accept a four-dimensional Minkowskian spacetime incorporating length contraction and time dilation as a sufficiently simple and plausible basic model that does not require any further explanation (p. 48). Fair enough. To me, however, length contraction and time dilation by the relativistic factor call for an explanation, more specifically an explanation that must somehow turn on how force field disturbances propagate. And that's what I'm trying to find, if only to satisfy my own curiosity.

Reader: That may be so, and good luck with it, but it seems to me that you are miles away from your goal since your explanation at best only explains "electric length contraction" and not "length contraction of everything".

Me: Absolutely, this is just a first step. But the very fact that everything is length-contracted by the relativistic factor suggests to me that the speed of light is also the speed at which any other field disturbances propagate, and that an explanation of electric length contraction can potentially serve as a template for an explanation of length contraction more generally. It also suggests to me that there is a fundamental unity to the physical world without which the world could not in fact function in the way it does.

Reader: You admit, then, that at the moment you can only - at best - explain "electric length contraction". Well, if that's all then it's no more than what Hendrik Lorentz did more than 100 years ago. Today this kind of "electric" length contraction, in other words the contraction of electric fields by the relativistic factor, is derived from Maxwell's equations in any textbook on classical electromagnetism. Once again, your "revolutionary" model, if it has any explanatory power at all, adds nothing to what has been known for over a century.

Me: Yes and no. Let me refer you back to my criteria for what makes for a good physical model: its applicability to a wide range of phenomena, its plausibility, and its simplicity. I'd suggest that the way electric length contraction is derived in classical electromagnetism falls foul of the third requirement: simplicity. Indeed, so long and complex is the chain of reasoning - involving the development of far-from-simple concepts such as electric and magnetic fields, a series of assumptions in the development of Maxwell's equations, and a range of advanced mathematical techniques - that in the end it is totally unclear what the explanation for electric length contraction really is. Indeed, I harbour a suspicion that length contraction and time dilation - or equivalently the constancy of the two-way speed of light - are somehow built into the premises of classical electromagnetism without explicitly acknowledging it.

Especially the "magnetic field" seems suspect to me. The basic equations of magnetism involve essentially the electric field constant and the speed of light, so magnetism seems to be no more than an electric phenomenon that comes about as a result of the fact that electric field disturbances in Σ propagate at c.

Clarifying all this would require a detailed examination of the logical structure of classical electromagnetism. As Kevin Brown's book Reflections on Relativity shows, fully understanding and, potentially, reformulating classical electromagnetism has occupied the minds of many illustrious physicists over the decades, so this will be no easy task. In truth, every time I leaf through that book I am reminded of this: if understanding physics, let alone the physical world, is to climb a mountain, then all I have achieved in this blog so far is to step on a molehill.