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
- Coulomb's law of electrostatics
- relativistic length contraction and time dilation in Σ
- the principle of relativity
- 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?
