By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social

According to the Ancient Greek philosopher Proclus (c.419–486) in his Commentary on Euclid, geometry (from geo metros, “measurement of land”) first arose in surveying practices among the ancient Egyptians, to assist them in redefining property boundaries following the annual the flooding of the Nile Valley. In modern terms, we would refer to the representation of a tract of land by figures drawn on a flat surface as a mathematical model. It captures precisely the aspects of the land surface that are important to perform calculations to re-establish the obliterated boundary markers.

That particular model proved to be so versatile that it developed into an entire branch of mathematics; indeed geometry is the defining, first example of pure mathematics—where a model is studied in its own right, independent of its origins or applications. (Astronomy was another, obvious early application of geometry.)

In the case of geometry, the distinction between the model and any of its applications is apparent to all. We can see how the model captures (features of) the reality it represents, but that in no way leads us to think of the model as being the reality. The model is clearly not the reality.

For many mathematical models, this is not the case; indeed, in the sciences the model often ends up being viewed, presented, and even taught, as the reality. A well-known example is the familiar solar-system-model of the atom. Known officially as the “Niels Bohr atom”, or more fully, the “Niels Bohr model of the atom”, that second variant is the more accurate description.

During the late 19th Century, the discovery of the electron and radioactivity led physicists to propose various models of the atom’s structure. In 1913, the Danish scientist Niels Bohr (1885-1962) proposed a theory for the hydrogen atom based on the quantum theory prediction that some physical quantities take only discrete values. In Bohr’s model for the atom, electrons move around a nucleus, but only in prescribed orbits, and if electrons jump to a lower-energy orbit, the difference manifests as radiation. Bohr was awarded a Nobel Prize for his work in 1922.

Within a few years, flaws were found in Bohr’s model, and others quickly superseded it. But by then, the familiarity of the “Solar System” image and the simplicity of the model had implanted it firmly in the minds of scientists the world over. And not just scientists; society at large latched on to it.

There was a lot to be said in its favor. Even when physicists were well aware of its flaws (i.e., the ways it does not represent reality), they could not (and did not) deny its tremendous power for thinking about atomic-level physical phenomena. To this day, the logo of the International Atomic Energy Authority includes an image of the Bohr Atom.

The fact is—and laypersons rarely have any knowledge of this—scientists generally do not make progress by wrapping their heads completely around natural phenomena. Even everyday concepts such as mass and gravity seem well beyond the capacity of the human mind to fully understand and explain. Instead, what scientists do is develop models—generally mathematical models—that enable them to make progress with them in terms of increasing our understanding of the world and using that understanding to design and create things we find of use.

Those models can rapidly become highly abstract and complex. The quantum theory that took over from the Bohr Atom is a good example, requiring sophisticated mathematical knowledge and skill to pursue. The relativity theory that overcame Newton’s mechanical theory of the universe is another. But whenever scientists dive into the deep, complex ocean of a symbol-rich mathematical theory, they invariably have a simple, visual model at the back of their mind, on which they anchor their thinking.

This, incidentally, explains why the small group that initially proposes a new theory invariably seem to remain way ahead of others who become involved later, with the newcomers being in awe of how the pioneers seem so much more able to cope with all the mathematical complexity. What happens is that the pioneers start with a simple—indeed usually simplistic—visual model, then as they work with it, they discover its flaws, and repeatedly adjust their thinking (i.e., the mathematics built on it) to take account of those flaws whenever they arise. Over months or years, a series of small adjustments to their thinking can result in their having an expertise that a newcomer, who has not had the benefit of that period of development, does not. Rather, the newcomer has to start with the mathematics that the pioneers present in the paper(s) they write after they have sufficient details worked out. Since the newcomers don’t have the experiential knowledge that led from the initial simple image to the mathematical theory the pioneers present, they cannot make effective use of that simple image. To be sure, over time, with a lot of effort, they may be able to reconstruct some of that connecting tissue, but by and large, they can never achieve the fluid way-of-thinking the pioneers display. (I only became aware of this distinction when I was lucky enough to be part of a pioneer group in its early stage; see the final remarks in this essay.)

Though the Bohr model is typically the “picture” of the atom that high school and beginning university students are presented with, it’s generally (at least, I hope it is general) accompanied by an observation that the “real” atom is not like that, together with some examples of phenomena for which such a simplistic model fails (e.g. wave-particle duality).

But that caveat is very definitely not the case for my second example: electricity. The chances are close to 100% that you were taught that electricity flows through wires. And almost certainly, you still think that to be the case. If so, and if you remember some basic high school physics, you might also recall that electrical flow consists of electrons moving through the wire at high speed. Perhaps you even remember the value of 186,000 miles per second for that speed.

Based on that “everyday, basic knowledge of physics”, you can, of course, look at an electrical circuit diagram like the one shown in Figure 3 and interpret it as you would a diagram for the plumbing in your home, where water flows through pipes (with voltage corresponding to water pressure and amperage to amount of water).

And based on that understanding, you could surely (successfully) add a new circuit to your home supply. It’s a useful way to think of it.

Even more, if you were an electrician or an electrical engineer, you could ply your entire  trade based on that interpretation.

But here’s the thing. The “electricity consists of particles flowing through wires at high speed” image works fine when you want to design, build, diagnose, and repair almost all electrical devices (including computers). Indeed, there is no better image to have in mind. But it’s totally false! Electrical energy simply does not flow through wires.

[Caveat: electrons do flow through a wire in a circuit; but at a snail’s pace of about 1 centimeter a minute; it’s called “electron drift”. They do also have a very fast motion, but it’s a back-and forth oscillation across a tiny distance of about half-a-human-hair. Those motions do in fact play an important role in creating electrical current, but it’s not how electrical energy is transmitted along the wire (along the wire, note, not in). Read on.]

The fact is, the “electricity flows through wires” story is not a factual description; it’s a (mathematical) model. A very useful and reliable model to be sure; but a false one.

For most of us, including the electrician who you hire to fix your home circuits, that falsity doesn’t matter. Among the people who know that it’s false—who have to know it’s false—are the engineers who work on the nationwide power grid that delivers electricity to our homes and factories.

Typically, power engineers will describe themselves as delivering not “electricity” but “power” (or sometimes “energy”), presumably because that terminology avoids giving the false impression they are sending stuff through wires. Rather, they would say, what they are doing is making power available, by creating/assembling it in one location and transferring it to another.

Those words creation and assembly require stories all of their own, but that’s for another day. But here’s the scoop. The electrical energy they sell you does not flow through the wires. It is transmitted through the air that surrounds the wires.

[This helps explain how the high voltage current in (actually around, not in) a high-voltage transmission wire can give rise to the 120v current that comes into your home through (i.e., around) the completely separate cable that runs from the transformer on the pole to the supply box inside your home.]

So what is really going on? Well, first of all, no one really understands electricity. But if you get into quantum electrodynamics you’ll encounter a bunch of equations (a mathematical model) which to date is the closest anyone has come to answering the question, “What is electricity?” Though many power-system engineers may not have mastery of all the mathematics, they know enough about what that mathematics says is going on for them to be able to design, build, and operate the world’s power grids. See for example, this article from the international power organization EnergyOne.

In other words, the People-Who-Know have access to a better model than the rest of us, one that is adequate for their professional needs. But it’s still just a mathematical model. Like most mathematical models, what it does is provide a sound utilitarian framework for what, ultimately, we might as well describe as “magic”. (But notice that most incantations of the word “magic” cannot be meaningfully modeled by mathematics. Many people may be gullible; mathematics is not. Electrical power is part of the universe’s magic.)

Here, briefly, is that better model.

When you connect a source of electrical energy (such as a battery or a generator) to a circuit made up of wires and other conducting materials, it sets up an electrical energy field around that circuit. Like gravity, that field is not visible, but it has physical effects on physical entities (including ourselves, where we can feel, detect, observe, or infer its presence). See Figure 4.

Critical to this occurring, the material the wires are made of has to be what we call an electrical conductor. What that means is its atoms have (one or more) electrons that are not tightly bound to the nucleus, leaving them free to move from atom to atom. Copper is one such conductor. If there is an energy difference along that conductor (coming from, say, a battery or a generator), tiny motions of those free electrons back-and-forth cause a ripple that moves along the wire, like a spectator wave at a football stadium. Those waves creates what we call an electrical field.

But that’s just part of what’s going on. Those oscillations also create a magnetic field perpendicular to the electrical field. This is all described by what are called Maxwell’s Equations; I said this was a mathematical model! The pictures where the mathematics takes us are in Figures 4 and 5 (which depict D.C. current and A.C. current, respectively).

The electromagnetic field that arises this way transmits energy along and outside the wire. That transmission of energy is what we call electricity. But note that the energy flow is (carried by) the field, not the rapidly oscillating electrons in the wire.

Notice that the above mathematically-grounded descriptions tell us nothing about what electrical energy is. The equations simply describe its transmission.

I should note that the details differ between D.C. current, as supplied by a battery, and A.C. current as generated by the power companies. Both are covered by this excellent introductory video by an online science educator who goes by the name The Science A: Circuit Energy doesn’t FLOW the way you think!

Be aware that almost any video that tries to convey electric current in an accessible fashion can be (and usually is) critiqued by experts. The phenomenon is, as I’ve noted, complex, and ultimately not fully understood. But the above video is a huge improvement over the familiar “electrons moving rapidly along the inside of a wire” picture.

Another video worth watching is Veritaseum’s The Big Misconception About Electricity. This one is useful at least as much because of what it gets “wrong” as for what it gets right. As you will discover if you subsequently watch his follow-up video How electricity actually works, made a year later, based on feedback he had received to his first one.

Finally, you might find it useful to check out this video regarding the behavior of individual electrons in a wire: What Are Electrons REALLY Doing In A Wire? Quantum Physics and High School Myths

FINAL THOUGHTS

Those of us who teach pure mathematics (and though retired from full-time university life, I still do from time to time) frequently tell our students that mathematics is based on the notion of absolute truth (from accepted axioms), and moreover that it the only discipline that can achieve that goal. Internally, that is true. But when it comes to developing and applying a mathematical model, we rarely achieve truth. On the contrary, though we sometimes achieve a partial truth (as occurs with geometry), quite often what we get is something we know to be flat-out false—but is nonetheless useful. Indeed, as with electricity, for many purposes, such models can be used effectively and safely as if they were the truth.