How do we know the electric field has infinite range?

Question

This question received answers I deem completely irrelevant to its true intent, so I'll ask it differently: how do we know? As OP noted, field lines are a theoretical concept - but what empirical evidence do we have that electric field lines do in fact have limitless range? We need not even go that far; answering the original requires answering the following:

How quickly does the electric field propagate? That is, if we (a) added or removed a charge from the Universe, or (b) moved the charge, when would other charges "feel" its effect?

1. If instantly, then we transmitted information through space instantly; this violates relativity

2. If not instantly, then the field propagates with finite speed; this necessitates the concept of a field "carrier", whatever it might be (i.e. "some physical thing that affects other physical things")

In case 2, the carrier cannot be infinite in quantity, as there isn't anything infinite within a finite closed system (e.g. a volume spanned by field after some initial time). However; infinite range demands an infinite carrier - else, the larger the radius, the less of the surface area spanned actually experiences any field. So, for a very long range, only tiny patches of regions will experience anything from the charge; the vast majority of matter at the radius will experience exactly zero E-field. Thus: infinite range is impossible.

Am I wrong?

Answer 1

Even though your question is phrased a bit aggressively, it does demonstrate an important point. Yes, you are correct that the photons emitted from a source, which make up an electromagnetic wave, are discrete, and there are a finite amount of them. So over enormous distances, the photons get diluted, until you get to the point that any detector sensitive enough to see anything at all is just seeing the individual photons as they come in.

This is not only known, it's an essential consideration when doing long-distance optical astronomy, where the detectors don't measure the electric field, but rather see "clicks" for individual photons coming in. This results in "shot noise" due to the discreteness of the photons, just like how shot noise arises in electronics due to the discreteness of electrons. You can even enhance the sensitivity of telescopes by correlating their click times, using the Hanbury-Brown--Twiss effect.

Now, I guess your specific question is what happens to the electric field at such distances. It is not correct to say that the electric field is zero in most places and nonzero at others. Instead, classical electromagnetism breaks down in this limit: we have to remember that electromagnetic fields are fundamentally quantum objects, like everything else. Instead of having a definite field value, we have a superposition of field values at each point. The typical field values fall off as $1/r$ just as advertised (otherwise astronomy would not work), but there's a spread. Measuring the field at a point just collapses this superposition.

The point is that we cannot say that field lines either do or do not go on forever -- that is a overly casual application of words. What's really going on is that the entire idea of a field line eventually stops making physical sense.

This isn't just some weird theorist's fantasy, either. The fact that the electromagnetic field is quantum plays an important role in particle physics (under the name of "quantum electrodynamics"), in astronomy (as mentioned above), in atomic physics (as "quantum optics"), and even in electrical engineering (as "cavity QED" and "circuit QED"). We don't talk about this in introductory E&M courses because the mathematical complications are great, but it is relevant when you make precision devices and measurements.

Answer 2

Yes, you are wrong.

The gauge boson (force-carrying particle) for electromagnetism is the photon. It is massless, therefore its range is infinite. Photons travel at the speed of light (c), which is therefore the speed with which electromagnetic disturbances propagate through a vacuum. No material medium is required to "carry" photons; the vacuum constants for permeability and permittivity yield the numerical value of c.