An update on my findings

An update on my findings regarding the forces between uniformly moving charges is now available in a new article I have posted online: ‘A new formula for the forces between constantly moving charges’.

The article presents the magnitude of the force exerted by a charge q₁ moving at u on a charge q₂ moving at v in an inertial frame of reference Σ, as measured in the rest frame of q₂. The direction of that force in Σ is also provided. This is a significant update on my first article on the sphere model of electricity (published by Physics Essays) in that it presents that force just in terms of general vector quantities. The new formulation could already be found on my blog, but without much explanation.

This update presents the theory very briefly and it is relatively easy to understand. But it does so without providing an explanation of how the new formula comes about. For this, I refer readers to my first article.

The journal Foundations of Physics has declined to even review the draft of my new article. Its reasoning is simple and straightforward:

“The author(s) of this manuscript fails to make clear how their work relates to current discussions in the foundations of physics. Regrettably, this fact places the current submission outside the scope of Foundations of Physics.”

In other words, if your work is absolutely new and does not relate to “current discussions”, you need not apply for publication. Your work will not even be looked at by a reviewer. It will not even be assessed for correctness or usefulness.

This is strange. How does this journal hope to publish meaningful contributions to the foundations of physics if it only allows contributions that relate to “current discussions”? I can only conclude that Foundations of Physics is an extremely conservative publication: any suggested publication of something completely new is rejected out of hand.

A few years ago, when I submitted my first article on the sphere model of electricity to the same journal, it was at least assessed by a reviewer. However, the reviewer didn’t seem to have read the entire article, and his assessment included the suggestion that the paper was “very much too long for what it contains”. So it was rejected. But the much shorter paper submitted now hasn’t fared any better.

Foundations of Physics is not alone in its principle of staunch conservatism in its publishing policy.

For example, on the submission guidance page of the International Journal of Theoretical Physics I was forwarded, it says that the journal “will generally not consider” papers “limited to proposing new physical theories or questioning the established foundations of physics”. In other words, if you believe you can offer an improvement to those foundations, you need not apply!

I have now begun to look at the case of accelerated charges, and I’m hoping I will be able to post an introduction to my results sometime soon – on this blog.

'Dialogue concerning magnetic forces' published

The electromagnetic forces between moving charges are fully understood, right? Wrong! In a paper entitled 'Dialogue concerning magnetic forces', published by the journal Studies in History and Philosophy of Modern Physics (Share Link), I argue that we need to rethink the forces between moving charges if we are to make sense of the physical world. The Share Link provides free access to the full article as published by the journal, until 12 January 2020.

The 'Dialogue' (PDF of accepted manuscript) completes a series of writings which together provide a full statement on a new model of electricity that has the potential to radically overhaul our understanding of electrodynamics and special relativity:

• The article 'A new model to explain the forces between moving charges' (PDF) published in 2018 by the journal Physics Essays (copyright: Physics Essays Publication) contains a complete explanation of the 'sphere model' of electricity, including my motivation for developing it, its implications and its mathematical formulation.
• A supplementary PDF document contains additional technical details with supporting graphics.
• A blog post presents a simplified view of the sphere model with lots of images and animations and without any equations.
• I have also given an account of how I arrived at the key sphere model equation, an aspect which is normally hidden from view but which can help to understand the model.
• Another blog post considers whether the model will revolutionise our understanding of electrodynamics and special relativity.
• Finally, the 'Dialogue' (PDF of accepted manuscript) explains my motivation for developing the sphere model over eight long years.

My blog on special relativity provides the broader context for the sphere model. Highlights include:

• How to use sound to do things with time
• Humpty Dumpty and the simultaneity syndrome in modern physics
• The limits of convention in relativity
• Convention in relativity - my findings

Links to all my blog posts can be found in chronological order on the physics page of my personal website. Happy reading!

How I arrived at the sphere model equation

So, how did I arrive at the sphere model equation describing the forces between two uniformly moving point charges as measured by co-moving spring balances? I started from Coulomb's law of electrostatics:

      (1)

The law describes the force between two stationary charges q₁ and q₂ separated by the distance r. We know from classical electromagnetism and special relativity that the forces between two charges q₁ and q₂ moving at constant velocities u and v, as measured by co-moving spring balances, look rather different from this: they depend on those velocities, the distance between the charges and the speed of light in a rather complicated way. Classical electromagnetism describes the force exerted by q₁ on q₂ in this more general case by means of two fields: the electric field E associated with q₁ and the magnetic field B associated with q₁. These need to be inserted into the Lorentz force law

    (2)

and a relativistic force transformation needs to be applied to obtain the force on q₂ as measured by a co-moving spring balance. I was convinced, however, that it should be possible to explain the forces between moving charges as measured by co-moving spring balances much more simply, in terms of a single 'electric' field surrounding any charge in any state of motion. The first question I needed to answer was what shape this single 'electric' field might take.

Step 1

I assumed that in any point in space and time there is a uniformly moving frame of reference S in which the electric field of any charge at rest - i.e. the electric properties of the space surrounding such a charge - is isotropic (the same in every direction). To me that meant that the field can be represented as a succession of concentric and equally spaced sphere surfaces. In S, any electric field disturbances resulting from the acceleration of a charge at rest would thus propagate in isotropic conditions. It would therefore be possible to synchronize clocks at rest in S using Einstein's clock adjustment procedure, and as a result the speed c at which electric disturbances propagate in S would be the same in every direction.

Note that this set of assumptions is much weaker than those made in classical theory, where it is somewhat implausibly assumed that the electric properties of the space surrounding charges at rest are isotropic in any uniformly moving frame of reference. What is more, in the sphere model there is no need to accept the constancy of the two-way speed of light in every uniformly moving frame of reference as a fundamental law of nature that cannot be explained any further, as is done in classical theory.

With all this in place, it was clear that accelerating a charge from rest in S to a particular velocity u would distort the sphere surface configuration surrounding it as follows:

Fig. 1

The sphere fields of two charges q₁ and q₂ travelling at constant velocities u and v in S and separated by the distance r can consequently be represented as follows:

Fig. 2

My hypothesis was now the following: the past accelerations of any two charges q₁ and q₂ moving at u and v in S bring about changes in the sphere field configuration surrounding the charges, and corresponding changes in the flow of information between them, which result in changes in the forces between those charges, as measured by co-moving spring balances, compared to the static case. More specifically, I hypothesised that it should be possible to describe those changes in the sphere field configuration, and corresponding changes in the flow of information, by a series of sphere field parameters x₁, x₂, x₃… and a direction vector y so that (1) would become

    (3)

Let me pause here for a moment to reflect on the significance of this hypothesis: if true, it would mean that the forces between uniformly moving charges as measured by co-moving spring balances could be explained with reference to a single field, the 'sphere field', and without the need to perform any relativistic transformation. The sphere model would thus be able to explain the mysterious appearance of 'magnetic forces' in classical theory. What is more, given the distortions in the sphere fields of charges moving in S, it would imply that locally Einstein clock adjustment would synchronize clocks only in S and not in any other uniformly moving frame of reference. This would instantly put paid to some rather outlandish ideas that are floating around in the literature based on misinterpretations of special relativity, such as the idea that we live in a 'block universe'. As if that wasn't enough, the sphere model would explain how the electric forces between moving charges are transformed in a manner that is consistent with length contraction. It would thus provide a partial explanation of length contraction.

But how might I be able to confirm my hypothesis? I decided to pursue a two-pronged approach.

Step 2

I assumed that the classical results for the forces between uniformly moving charges as measured by co-moving spring balances are empirically correct. This enabled me to work out what I had to aim for in my search for the sphere model parameters  x₁, x₂, x₃… and the direction vector y. For simplicity, I decided to define a cartesian coordinate system in which the charge q₂ is located at the origin and moves in the direction of the x-axis, while the charge q₁ is located in the xy-plane, as follows:

Fig. 3

It is also convenient to define an additional angle α as follows:

This is not an independent parameter and the following relationship holds:

In terms of this coordinate system, using the formulae for the electric and magnetic fields of moving point charges, the Lorentz force law, and the transformations of special relativity, classical theory tells us that the magnitude of the force exerted by q₁ on q₂ as measured by a co-moving spring balance is as follows:

(4)

And the direction of that force as seen from S (i.e. the direction in which q₂ is accelerated in S) is parallel to the vector

     (5)

The task of finding sphere model parameters that would produce this magnitude and this direction seemed formidable. One of the difficulties I saw was that the magnitude was not particularly factorised. Specifically, I saw little hope of finding a single sphere model factor corresponding to the square root term in (4). The first thing I did, therefore, was to play around a bit with (4). I soon found that it could be expressed as follows:

(6)

This looked much more promising. Note how there are now several terms in the equation which are of the form a² - b². This factorises nicely into (a + b)(a - b). My task now looked a little bit more manageable but still quite daunting. How should I go about finding sphere model parameters that would produce (5) and (6)?

Step 3

I started by simply having a look to see which aspects of the sphere field configuration in Figure 2 were different from the sphere field configuration in the static case. The first that struck me was the q₁ sphere surface density along the line connecting the charges. This was clearly different from the density that pertains in the static case (whatever precisely the magnitude of that density might be).

I had soon gathered a few pertinent results: a) the q₁ sphere surface density along the line connecting the charges is constant on the 'near side' of q₁ (the side on which q₂ is located) and on the 'far side' of q₁ (the side on which q₂ is not located); b) as can be seen in Figure 4, information that travels from q₁ to q₂ from the location of q₁ at the 'retarded time' (t₀ in Figure 4), such that it arrives at q₂ at the 'current time' (t₁ in Figure 4), travels through the q₁ sphere field along the line connecting the charges at the current time, in other words the sphere surface densities along that line in its current position are potentially relevant bits of information transmitted from q₁ to q₂; c) if I define the 'density factors' as the factors by which these densities are different from the density in the static case, and if I multiply the near- and far-side density factors, then I obtain one of the terms in (6), namely (1 - u²/c²). It looked like I had found my first sphere model parameter x₁!

Fig. 4

A second parameter was soon to follow: I noticed that the angle at which the connecting line cuts through neighbouring q₁ sphere surfaces is different from the 90-degree angle in the static case, and consequently the path taken by electric information from one sphere surface to the next is longer than the local perpendicular distance between the sphere surfaces. The factor by which it is longer is the same on the near side and the far side. The product of the two turned out to be another factor in (6), namely 1/(1 - u²sin² α/c²).

But the hard bit was yet to come. It was the remaining square root term in (6). It did not seem likely that just taking a long hard look at the sphere field configuration in Figure 2 would enable me to detect that factor in that configuration. I suspected that the square root term had something to do with the frequency at which electric information coming from q₁ arrives at q₂. That frequency could be expected to depend on both u and v. I started to experiment with a number of sphere model frequency factors, initially for special cases and in two dimensions only. It was only when it occurred to me that I might have to look at 'near-side' and 'far-side' frequencies that I was able to make decisive progress. Eventually I discovered that I had to form the geometric mean of near-side and far-side information transmission rates, which depend on the q₁ and q₂ sphere surface configurations, to obtain the desired term.

It thus turned out that each of the factors I had identified - the density factor, the angle factor and the frequency factor - could be regarded as the geometric mean of near-side and far-side density, angle and frequency factors. Is that plausible? In principle I think it is: it seems plausible that the information that is transmitted from q₁ to q₂ concerns a whole area around the mathematical point on which q₁ is centred. That information will have travelled along the line connecting the two charges, so again it is plausible that it includes near-side and far-side information on parameters along that line. The fact that what matters is the geometric mean of near- and far-side factors, rather than some other kind of combination of the two, is perhaps less immediately plausible. No doubt one day somebody will find an explanation for it.

At this point I felt like celebrating. I had succeeded in identifying just three simple sphere field parameters by which (1) had to be modified to obtain the magnitude of the force between two charges moving uniformly in S. But then I remembered that I had not finished the job. I had yet to explain, in terms of the sphere model, why the direction of the force as seen from S is parallel to (5)!

Step 4

Again, the task of obtaining the correct direction vector looked daunting, but this time I succeeded more quickly than I had anticipated. I had a good idea where to look: it would be plausible for the direction vector to depend on the direction from which q₁ information arrives  at q₂. However, when I calculated the 'near-side direction vector' n₁^< (see Figure 5), which describes that direction in S, it wasn't parallel to (5). The near-side direction vector is the vector pointing from the 'retarded' position of q₁ at the centre of the q₁ sphere surface on which q₂ is located to the current position of q₂. I then figured that perhaps I had to include the 'far-side direction vector' n₁^>, in other words the vector pointing from the retarded position of q₁ to the far-side intersection of the line connecting q₁ and q₂ at the current time with the q₁ sphere surface centred on the retarded position. Still that didn't work.

Fig. 5

It was only when I started to look at the situation from the point of view of q₂ that I struck gold. I found that I had to consider the near-side and far-side directions from which q₁ information arrives at q₂ from the point of view of q₂. Those two directions are given by n₁^< - v/c and n₁^> - v/c. Simply adding up those two vectors still didn't do the trick, however. I found that I had to modulate those two vectors by particular factors. Just by looking at the x- and y-components, I was able to calculate the factors that are required to obtain the correct result. They turned out to be q₂ sphere field parameters that I had previously worked out for q₁: the q₂ angle factors! What is more: the z-component worked with those same factors. This could be no accident! I felt vindicated. My intuition had been right. All aspects of the forces between uniformly moving charges as measured by co-moving spring balances could be expressed simply in terms of a few basic sphere model parameters. The end result was as follows:

     (7)

Here d₁, e₁ and f₁ are the q₁ density, angle and frequency factors, and e₂^< and e₂^> are the near- and far-side q₂ angle factors.

To my mind, this is an amazing result. It means that the forces between uniformly moving charges as measured by co-moving spring balances can be expressed simply in terms of a few sphere model parameters, without reference to any 'magnetic field' and without the need to perform any relativistic transformations. Equation (7) can also be written purely in terms of u, v, n₁^< and n₁^>, as follows:

      (8)

However, equation (7) shows more clearly how this force comes about, namely as a result of changes in the sphere field configuration compared to the static case described by Coulomb's law. And that concludes my account of how I arrived at the sphere model equation describing the forces between two uniformly moving point charges as measured by co-moving spring balances.

One last thing. The need to systematically take into account near- and far-side sphere field properties is a remarkable feature of the sphere model. I have tried to explain it by assuming that information travelling from q₁ to q₂ in the direction of n₁^< includes full information on near- and far-side sphere field properties along the line connecting the two charges at the current time. There is another possibility. It is that in S the information travels well and truly in the near- and far-side directions given by  n₁^< and n₁^> and is exchanged instantly over the q₁ sphere surface on which q₂ is located at the current time, t₁, as shown in Figure 6 below.

Fig.6

The idea would be that, while energy in the form of sphere field disturbances can only be transmitted at c in S, since it has to travel outwards from one sphere surface to the next, information on interlinked near- and far-side sphere field properties can be exchanged instantly on one and the same sphere surface. The whole situation reminds me a bit of what I've read here and there about the phenomenon of quantum entanglement. I am no expert on quantum mechanics, so I don't know whether there is a connection. But it is intriguing to think that, in addition to solving the mystery of the magnetic field in classical electrodynamics, the sphere model might also shed some new light on quantum entanglement, or vice versa.

Are we on the brink of a revolution in our understanding of electrodynamics?

Do you often lie awake late at night worrying about the origin of magnetic forces between moving charges or the reason for the constancy of the two-way speed of light for any observer?

If you do, then the sphere model of electricity, accessible in full here and explained in simplified form here, is for you.

That's because the model sheds a good deal of light on the following questions:

• how magnetic effects between moving charges come about
• why the speed of light is independent from the speed of the source
• why moving bodies are length-contracted by the relativistic factor for any observer using Einstein-adjusted clocks
• why moving clocks run slow by the relativistic factor for any observer using Einstein-adjusted clocks
• why the two-way speed of light is constant for every observer
• why the one-way speed of light is constant for every observer using Einstein-adjusted clocks
• in which conditions Einstein's clock adjustment procedure synchronizes distant clocks
• what follows from all this for the correct interpretation of special relativity.

If you never lie awake late at night worrying about those questions, then now may be a good time to start doing so.

For the sphere model has the potential to radically overhaul our understanding of electrodynamics and special relativity.

Let me explain why.

Turn back the clock 150 years or so when the world had not yet heard about special relativity and Maxwell was putting the finishing touches to his theory of electromagnetism.

Magnetic fields were conceived of as vortices in an all-pervasive medium caused by the movement of charged particles. The rotary motions in the medium were thought to deflect other moving charges from their path.

The magnetic forces thus produced were believed to act in a direction perpendicular to the direction of motion of the charges on which they act.

The speed of propagation of electromagnetic disturbances was thought to be a result of the electric and magnetic properties of the hypothetical all-pervasive medium, also known as the aether.

Wind forward 50 years and this picture of the origin of the forces between moving charges was crumbling fast.

It turned out that there was no aether and no rotary motion, and that in general magnetic fields do not deflect charged particles in a direction perpendicular to their direction of motion in the first place.

Special relativity came to the rescue of the faltering theory by stipulating that moving bodies respond differently to transverse and longitudinal forces, to correct for the direction error.

But essentially electromagnetism was now devoid of any physical substance. All that was left of it was a mathematical shell without any explanatory power, propped up by crutches provided by special relativity.

Since then nobody has come up with a convincing alternative, so the theory has limped on from textbook to textbook and from generation to generation.

Let's now shed all this historical baggage for a moment. Let's make a fresh start by pretending we've never heard of magnetic fields or magnetic forces.

Let's pretend that all we know is Coulomb's law of electrostatics, the fact that electric disturbances propagate at c, and the fact that the forces between moving charges (as measured by co-moving spring balances) depend on that propagation speed as well as on the states of motion of those charges (including past states of motion).

How would we then go about explaining this state of affairs?

We could start from a simple assumption that is in line with Coulomb's law and that is much weaker than the assumptions made in classical theory: in every point in space and time there is an inertial frame of reference in which the electric properties surrounding stationary charges are isotropic (the same in every direction).

We could then ask: in such a frame of reference, how are the electric properties and the flow of information between charged particles modified as a result of acceleration to give rise to the forces we observe between moving charges?

This is the question I started to work on eight years ago or so, and it has led me inevitably and inexorably to the sphere model of electricity.

The sphere model is all about the way in which past accelerations modify the sphere arrangements surrounding charged particles and the flow of information between them.

The sphere model not only revolutionizes our understanding of the forces between uniformly moving charges but also sheds new light on special relativity: it implies that, in general, events with equal time coordinates as determined by Einstein-adjusted clocks are not simultaneous.

So, are we on the brink of a revolution in our understanding of electrodynamics?

The answer has to be 'no'.

The sphere model only covers the forces between uniformly moving charges. A fully-fledged model would have to explain the forces between charges in any state of motion.

What is more, the model at best only offers some intangible and difficult-to-understand conceptual advantages over classical electromagnetism. It makes no new predictions (at least not in its current state of development) and it doesn't enable us to build any new machines.

Why would any working physicists want to devote any of their precious time to it?

As for myself, I'm not even a working physicist. I've been working on the sphere model by snatching a few minutes here and there between paid work and family life.

Extending the model to moving charges in any state of motion promises to be mathematically considerably more challenging than developing it for uniformly moving charges.

I'm sure it can be done. All the ingredients are there. But doing it, if indeed I decide to have a go, might well take me another ten years. And I need a break.

The revolution will have to wait a bit longer.