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One of Maxwell's other equations links the E-field to charge density. You can't talk about the time derivative of the E-field without also talking about the time derivative of charge density.

It's true that interactions between charged particles invoke photons, but you can have charged particles (and their associated E-fields) travelling at constant velocity in a vaccum and still define an associated current density without considering photons. I'm not sure your interpretation produces a useful intuition for this situation.

One of the underlying ideas of the paper is that, in conventional metals where conduction band electrons are much more wave than particle, you can still measure when they 'enter' and 'exit' the material through perturbations in the field, that is bursts of photons that occur when electron waves interact with an obstacle.

> you can have an electromagnetic wave traversing the wire at some proportion of the speed of light, while the electrons are moving at speeds closer to meters per second.

This isn't as unintuitive as it sounds though.

In a water pipe you can have a pressure wave reach from one end to the other very quickly, even as the individual water molecules move very slowly.

If water pressure is voltage, water molecules are electrons, and the pipe is the wire, then it's easy to see how a "voltage" can reach the other end of a "wire" without any of the "electrons" having to move very far at all.

The problem with this analogy is that pressure waves can only travel in a medium. The pressure wave exists because the particles in the medium bump against eachother to propagate energy so the analogy gives people the impression that electrons are bumping against eachother to propagate energy. Photons are distinct entities that can travel without a medium. See michelson-morley experiment. Electricity is nothing like what we experience on our macro scale. Analogies never convey the full story.

> the analogy gives people the impression that electrons are bumping against eachother to propogate energy.

If this is not approximately how electricity works, why does the current stop flowing when you disconnect the wires, or equivalently when you "connect" them via something that doesn't have free electrons (i.e. an insulator)?

Unfortunately the explanation to your question is something that takes years of education to understand and even then, most people just come out understanding some parts. The best I can do on an internet forum is tell you that the analogy is wrong, because experiments told us its wrong. My advice is if you want to understand this stuff, do the math! ;)

The water analogy makes perfectly good predictions for the kinds of things I do with electricity, so in that respect I find it useful.

As any analogy, it is not the same as the thing it is an analogy of, but an analogy can be useful without being complete.

There was a whole mess of STEM YouTubers arguing about this last year, e.g.: https://youtu.be/oI_X2cMHNe0?si=-PzCPZ5qNiS9nYFW

TL;DW: the power is transferred by the electric fields, it's the electric fields which move the electrons, if there's a break in the circuit the electrons necessarily accumulate in the places that minimise the electric fields.

The analogy does not completely explain the actual behavior of electrical current, that's true. But what's more important, and what it does clearly show, is that a fast wave can exist without fast movement of the particles involved.

This is not even approximately true. j is not “defined in terms of E”. It occurs as a source term in one of the inhomogeneous Maxwell equations, and consistency with the other inhomogeneous one (div D=rho) requires that div j = dot rho, which means that j is the flow of charge density. So, I guess if electron movement shouldn’t be referred to as current, we shouldn’t refer to electrons as charges, either…

When solving for the wave equation, the substitution made for J is sigma * E. So defined in terms of the E field and some conductivity constant. When solving the wave equation, there is no mention of charge density, everything is fields. This predicted the existence of electromagnetic waves. In reality, we know there is only electrons and photons. So thus the wave equation is the source of truth and what we call fields are just a consequence of the complex interactions between electrons and photons and this is studied in quantum electrodynamics.

> while the electrons are moving at speeds closer to meters per second

Plus, with AC, the electrons move back and forth, instead of just moving forward!

It is however still movement, which is important because static electric fields don't produce magnetism, whereas moving charges do - which is the principle benefit of AC current.

Well, technically that depends on your reference frame. If the _observer_ starts moving, they will observe a magnetic field where before there was only electric.

It’s not that static electric fields “produce” magnetism. Magnetism is a result of relativity.

(But yes, it is still movement…).

It's true that waves carry the energy. But shot noise is still determined by the amount of charge on an electron.

Does the thing at one end of the wire get heavier and the thing at the other end lighter?

Because of the mass of the electrons moved from one thing to the other.

An electron has .05% the mass of a proton, and only a small imbalance of electrons and protons is necessary to generate extremely strong electric fields by earthly standards.

That's fabulous man, but it doesn't actually answer my question.

If the answer is yes then we have moving electrons.

If the answer is no then maybe we don't.

Bruh its simple physics, does one end or the other get lighter, by all measures we care about, not really, the mass of a proton or electron is beyond any consumer hardware measurement. I doubt it would matter beyond extreme scenarios or controlled experiment

Imagine a loop of pipe filled with water, and a pump pushing the water: The water moves around the pipe. The movement doesn't cause any part of the pipe to get heavier.

Imagine a different scenario, where the pipe ends in a big box: This time the box does fill up with water, and gets heavier.

Mapping the analogy from water back to electrons: a loop of pipe is like a loop of wire and a battery; while the pipe ending with a box becomes a capacitor or antenna, and that will leak[0] before you can measure the mass change — but technically yes the the mass of any given wall of a capacitor or of an antenna will be very slightly changed by this sort of thing.

For a sense of scale, to get a total charge of 1 coulomb using electrons, the mass of those electrons will be about 5.7 nanograms, and trying to squeeze that much charge into the last millimetre of some length of a wire 1mm in cross section diameter, involves about 60% of the energy in this explosion: https://www.youtube.com/watch?v=wqKn_3iJOP4

As nothing gets close to being able to hold that kind of energy, even if you're trying to accumulate a lot of excess electrons, those electrons leak well before even coming close to nanograms of excess mass.

[0] The bit on the top with the sparks coming out is a capacitive electrode: https://en.wikipedia.org/wiki/Tesla_coil

Yes, but to an unmeasurably small degree. Electromagnetism is very strong, so you never see the density of electrons change by very much, or the situation will correct itself quite violently.

No.

Imagine a simple circuit, say a light bulb and a battery. Electrons move from the negative terminal, through the bulb, and back to the battery. The net change in number of electrons at any one point is zero. The energy isn't in the electrons themselves, but in the motion of those electrons. Electrons in must equal erlctrons out.

Even a battery doesn't store electrons. It uses the energy carried by those electrons to reverse a chemical reaction. The energy is stored chemically.

If you think about it, the electrons belong to the physical materials in the circuit. You can't really add or remove electrons* as electron count is a fundamental property of those atoms. If you somehow removed electrons from the system, you'd be changing those atoms and the system would no longer be able to pass current at all.

*you can, of course ionize atoms by adding or removing electrons, but that's not exactly what happens in electric circuits

Electrons are not electricity, they just carry it. Kind of. It's really complicated.

So electrons aren't actually moving from one place to another?

Ben Franklin’s definition of current, otherwise known as the standard one, has positive charge carriers.

You're talking about conventional current, and it's nothing more than a polite fiction. We model circuits as current moving from positive to negative, but with the implicit understanding that the real charge moves the opposite direction.

Positive charge carriers do not actually exist[0]. There's only electrons and holes they can go into. We can talk about the movement of holes, but that's a virtual charge carrier at best.

Conventional current is just a convention. It's what we started with (because Franklin was wrong) and it's too much effort to change now. In practice, the distinction almost never matters. Sometimes it does, but not enough that it's worth overhauling the entire field of electronics.

[0] of course positrons and protons exist, but they aren't relevant to electronic circuits

In order to get a sustained current, you need to put electrons in one end and take them out from the other in equal numbers. Thus mass doesn't change.

However, if you're talking about static electricty, you can actually create a mass imbalance by taking a few electrons away from one side or putting some electrons on the other. It's a very, very, very small change in mass.

A difference in voltage between two points in a circuit is a difference in concentration of electrons at those two points, and since electrons have mass that's not quite zero, the location with the higher concentration of electrons will have very slightly higher mass density. But this is true for any electric potential (voltage) difference, whether or not there's any current flowing.

See the vsauce video "what does the internet weigh"

The answer is yes but you can't practically observe it. The electrons repel each other so strongly that you can't accumulate enough of them in one place to be able to observe change of mass of that object.

You can move whole charged atoms, that's a form of electricity too, and it can add observable amount of mass, like with electroplating or welding. But these very quickly turn electrically neutral after depositing.