Magnetic effects without magnetism

Many textbooks on electromagnetism describe how some magnetic field effects turn into purely electric field effects under relativistic transformations between inertial coordinate systems. This doesn't mean, however, that the concept of the magnetic field is redundant in relativistic electrodynamics: magnetic fields are still said to be present in any inertial coordinate system through which electric charges move.

Nevertheless, it seems legitimate to me to ask whether there is scope for a theory of electricity that does not use the concepts of magnetism or the magnetic field at all.

The starting point for such an exploration could be an inertial system Σ in which light, or electromagnetic radiation, or perhaps "electric radiation", propagates in symmetrical conditions in all directions. I will assume that, locally, there is such a system at any point in space.

In Σ, clocks can be synchronized using Einstein's clock adjustment procedure so that every event is associated with a set of time and space coordinates, and events with equal time coordinates are simultaneous.

The forces between charged particles moving at constant velocity in Σ only depend on the amount of charge carried by those particles; their positions and velocities; the electric field constant from Coulomb's law of electrostatics; and the speed c at which electric field disturbances travel in Σ.

This suggests to me that it should be possible to describe and explain every "electromagnetic" phenomenon in Σ, between charges moving at constant velocities, purely in terms of Coulomb's law of electrostatics combined with the fact that electric field disturbances in Σ travel at c.

To consider this in greater detail, suppose two charges q1 and q2 are travelling at constant velocities u and v through Σ and suppose I would like to determine the force on q2 as measured by a co-moving Newton meter.

For the special case of q1 and q2 travelling in one and the same plane, we can choose our coordinate axes so that q2 is located at (0,0,0,0) in Σ, v points in the same direction as the x-axis, and u lies in the (x,y) plane. If r is the vector from q2 to q1 we have the following situation (Figure 1):



In classical relativistic electrodynamics, it's a relatively simple matter to determine the force on q2 as measured by a co-moving Newton meter by applying the classical concepts of the electric and magnetic fields: q1 is said to be surrounded by an electric field E and a magnetic field B, which can be determined using the formulas for E and B for a moving point charge. The force on q2 in Σ is then F = q2 (E + v x B).

The relativistic force transformation formula (for example as given in Rindler's Relativity, p. 124) can then be used to determine the force on q2 as measured in a co-moving inertial system S using a Newton meter.

The end result, according to my calculations, is

where α = ∢ (u, r) and θ = ∢ (v, r).

But it's also possible to calculate F without reference to the concept of a magnetic field or even the concept of the force on a moving charge.

To do this, we first need to define time coordinates in S using Einstein's clock adjustment procedure. Next, using standard relativistic transformations, u, r and t = 0 at r in Σ need to be transformed into u', r' and t ' at r' in S. Using these quantities, we can work out where q1 was in S at t ' = 0. Assuming the validity of the principle of relativity, we can then use the formula for the electric field of a moving point charge to determine the electric force on q2 in S via F = qE.

These calculations are considerably more laborious than the previous ones - but I have carried them out step by step and, as expected, they yield exactly the same result for F as the first method.

Now, the electric field of a moving point charge can be derived from Coulomb's law of electrostatics and relativistic length contraction (for example by means of a slight extension of the arguments presented in this document by Daniel V. Schroeder), and the relativistic transformations of time, space and velocity can be obtained from length contraction and time dilation in Σ.

As a result, the force on q2 in Figure 1 as measured by a co-moving Newton meter can be calculated using nothing but

1. Coulomb's law of electrostatics
2. relativistic length contraction and time dilation in Σ
3. the principle of relativity
4. the concept of the electric field

Without any reference to magnetism.

Is the "electric field" a more powerful concept than my previous two posts have suggested, after all?

The concept of the "electric field" (2)

Many of W. Geraint V. Rosser's formulations in his Interpretation of Classical Electromagnetism show that, for him, the electric field E is just a mathematical concept which doesn't correspond to any physical reality.

He first introduces E as an abbreviation for the expression q1R1 / 4πε0R13 in Coulomb's law for the force on a test charge q in the presence of another charge q1 separated from q by the vector R1.

He adds (page 5): "It is then said [my emphasis] that the charge q1… gives rise to an electric field… at the position of the test charge q".

According to Rosser, this is thus just a manner of speaking, from which we shouldn't conclude that the "electric field" has a physical existence in the same way that "charge" or "charged particles" have a physical existence.

Rosser goes on to introduce the magnetic field in a similarly mathematical manner.

The goal of classical electromagnetism, according to Rosser, is to determine the force on a test charge q moving at the velocity u in the presence of known charge and current distributions in the vicinity of q.

This is achieved by first working out the mathematical entities E and B from the known charge and current distributions and then determining F via F = q(E + uxB).

But how then does classical electromagnetism answer the question of how charged particles in different locations communicate changes in their location or states of motion to each other?

Rosser repeatedly refers to an "information collecting sphere" which continually collapses at the speed of light from infinity towards a test charge q and which collects information on the charge and current distributions it encounters. This information can then be used to calculate electric effects in the location of q (page 62).

However, Rosser stresses that this is an "imaginary" sphere introduced "for purposes of exposition only".

Indeed, as a physical model, the information collecting sphere seems to raise more questions than it answers: where does it come from, how does it collect and store information, and how does that information act on q?

Rosser circumvents these questions by presenting the information collecting sphere as nothing but a mathematical tool without any physical significance.

"The attitude we have tried to cultivate is that, in the context of classical electromagnetism, there is no need to say anything about what may or may not happen in the empty space between the charge and current distributions and the field point," he says (page 88).

Rosser's proposition that the concepts of the electric and magnetic fields are physically vacuous seems perfectly plausible to me, based on the way in which these concepts are usually developed in classical electromagnetism. It becomes even more compelling if we accept the principle of relativity (the principle that the laws of physics should have the same form in all inertial frames of reference) and consider how two co-moving charges interact with each other as seen from different frames of reference.

Let Σ be a first frame of reference in which two charges move at the same velocity u.

In this frame of reference, the two charges are surrounded by an electric and by a magnetic field, and the force they exert on each other is a function of the magnitude and direction of these fields as well as of u and the magnitude of the charges.

Now consider a second frame of reference S moving at u relative to Σ. The two charges are stationary relative to this frame and therefore only surrounded by electric fields. The force they exert on each other is then purely a function of these electric fields and the magnitude of the charges.

The force acting on an electric charge in the presence of another electric charge can thus in this case be explained in completely different ways using the concepts of E and B. But the application of this force constitutes a single event for which it should be possible to give a single explanation which holds in any frame of reference. It is clear that the concepts of E and B cannot furnish such an explanation.

And that's not a problem, according to Rosser's interpretation of classical electromagnetism, since E and B merely serve as convenient mathematical concepts which do not correspond to any particular physical reality.

Rosser's interpretation of classical electromagnetism is very clear and perfectly coherent, but it's also very frustrating for somebody who wants to understand just how charged particles communicate changes in their location and their states of motion to each other.

This all the more since I believe an understanding of these processes would help to shed light on the cause of relativistic length contraction - a fundamental building block in the special theory of relativity.

That is why I have started to develop my sphere model of electricity, in which local accelerations of an electric charge lead to electric disturbances which travel outwards through a series of electric spheres surrounding the charge. These disturbances modify the way in which the charge acts on other charges.

As it stands, my model may give a reasonably plausible explanation of electric length contraction (as I have argued here and here, with some provisos set out here), but it's far removed from explaining the full range of electric interactions between charges in any state of motion.

Sadly, it seems that classical electromagnetism offers me little guidance on how I might be able to expand or modify my model so that it can explain a broader range of phenomena.