In W. G. V. Rosser's Interpretation of Classical Electromagnetism, the "electric field" is first and foremost a mathematical entity which does not correspond to any particular physical reality.
The only electromagnetic concept which, for Rosser, corresponds to a distinct aspect of reality is that of "electric charge", which is seen as the cause of "forces" between charged particles, which manifest themselves in changes in the states of motion of those particles.
In Rosser's interpretation, the "electric field" is essentially a convenient mathematical tool to help work out the force that acts on a charge q at a time t on the basis of our knowledge of the positions and states of motion of a given set of charges near q over a certain period time leading up to t.
To see all this in greater detail, let me first look at Rosser's definition of the "electric field E at a fixed point in empty space". This field, Rosser says, "is defined in terms of the force Felec acting on a stationary test charge of magnitude q placed at the field point using the relation:
E = Limit (Felec/q)
in the limit when the magnitude of the test charge q tends to zero."
Rosser explains the need to let q tend to zero by pointing out that a large charge q would lead to changes in the configurations of surrounding charges, which would in turn change the force acting on q. However, it seems to me that such changes would only occur after a certain time, so I'm not convinced the limiting condition is essential, provided empirically the ratio of the measured force to q does not depend on the size of q.
Leaving this issue aside, Rosser's definition only works if we have already defined "force". Rosser does this on page 1 of his book, where he defines the force F on a particle of rest mass mo that is moving at the velocity u as
F = d/dt (mou/(1-u2/c2)1/2)
where c is the speed of light.
This definition might appear to pose a problem if we want to derive electric length contraction and time dilation by the relativistic factor from the equations of electromagnetism, since it already includes the very relativistic factor we are supposed to derive. However, since the electric field is defined purely with reference to the force on a stationary test charge, u = 0 and Rosser's force formula reduces to the classical F = ma.
Things get more complicated when Rosser applies the concept of the electric field to moving charges. "It is assumed in classical electromagnetism", Rosser says, "that, if the charge q is moving and accelerating in an electric field, the electric force on the moving charge is still given by
Felec = qE."
Clearly, in this situation we need a concept of force that applies to moving particles, and Rosser suggests that the relativistic definition of force should be used. However, empirically, using this definition of force, the total force on a test charge q moving at u in the presence of other moving charges is in general not given by
F = qE
but by
F = qE + quxB
for a suitably defined field B.
So, is Felec = qE for a moving test charge a law, or is it rather a definition of a new concept of "electric force", which is deemed to act on a charged particle in any state of motion, as distinct from the total "electromagnetic force" that acts on that particle?
The latter is indeed how classical electromagnetism seems to proceed.
Let me summarize what is a matter of definition and what is a matter of empirical fact in the development of the concept of an electric field in Rosser's interpretation of classical electromagnetism.
1) The electric field E in a field point P at a given time t is defined as the total force on a small stationary electric test charge q in P at the time t in the presence of other charges in any state of motion, divided by q. "Force" can in this context be defined in accordance with the classical formula F = ma.
2) It is then found empirically that the force on a stationary charge q of any magnitude in the presence of other charges in any state of motion is F = qE.
3) The electric force Felec on an electric test charge q in any state of motion in the presence of other charges in any state of motion can then be defined as Felec = qE.
4) It is found empirically that, in the presence of other charges which are all stationary, the total force F on an electric test charge q in any state of motion is given by F = Felec if F is defined using the relativistic definition of force.
5) It is found empirically that, in the presence of other charges in any state of motion, the total force F on an electric test charge q in any state of motion is in general not given by F = Felec if F is defined using the relativistic definition of force.
This analysis raises questions about the physical significance of the concept of the "electric field" in classical electromagnetism. As long as we only consider electric forces on stationary charges, the significance of the concept seems fairly clear: the force on such a charge is always qE, regardless of the size of q, even though we have defined E with reference to small test charges only.
However, the picture gets much murkier once we consider effects on moving charges.
To start with, if we want to maintain the classical definition of force F = ma, it turns out that qE is not equal to that force for a moving charge q even if the surrounding charges are all stationary. This presents us with a choice.
1) We could consider that the concept of E is not useful in this situation as the law that links E to F in the stationary case does not accurately reflect the observed phenomena for a moving charge q; or
2) We maintain the law that links E and F in the stationary case by adjusting the classical concept of force so that the law applies to moving charges q, too.
Classical electromagnetism opts for the latter possibility. But there is an additional difficulty when the charges around the moving charge q are in motion, too. For then it is found empirically that in general the law that links E and F doesn't even apply if the new, relativistic concept of force is used. To make the law fit the observed phenomena, a new concept, the "magnetic field B", is introduced and incorporated into the force law, which becomes F = qE + quxB.
In light of all this, is there any sense in which E can be identified with a physical reality which can be regarded as the cause of forces on moving charges or of changes in their motion?
I will discuss Rosser's answer to this question in my next post.
