paper-machine mentioned on #lesswrong the other day that “the electric field is defined counterfactually, i.e., “this at this point is related to the force that a test point particle would experience if it were there”. However, there is no such thing as a test particle, as each charge creates its own field, distorting the one that has to be measured. Moreover, there is no such thing as a point charge, because the electric field, inversely proportional to the square of the distance, grows without bounds near the point particle.
I remember that when I first learn this definition in high school, it bothered me for a short while, and then I got used to it, only noticing this counter-factuality very occasionally. So I will try to trace how it came about.
First, clearly the electric field is measurable, but not very directly. The relevant device measures either the displacement of a charged object, or the force required to keep it in place, depending on the design. In the former case the displacement is due to the force the field exerts on the charge. So, in both cases what is really measure is the force, and the field is inferred from it by calibration.
The jump from the Coulomb force between two charged objects to the mediating electric field generated by one of them and sensed by the other is the leap of faith Michael Faraday made without ever directly measuring the “undisturbed” field. I should really look through his biography and see what thought process he followed.
An insight like that is rare, and I am wondering about other examples where a counterfactual definition produces the most useful and accurate model of some physical phenomenon. The concept of force is probably one of those. Aristotle described “unnatural” motion as “forced”, and it would have worked well, if not for the invisible force of the Earth’s gravity, and probably air resistance. Otherwise Galilean relativity (unforced motion is along a straight line with constant velocity) would have been very intuitive. As it was, however, the world had to wait for Galileo and Newton to rather counterfactually define force as the rate of change of momentum.
I will try to find some more interesting examples of counterfactual definitions next.