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Tue, 28 Mar 2006

The speed of electricity
For some reason I have needed to know this several times in the past few years: what is the speed of electricity? And for some reason, good answers are hard to come by.

(Warning: as with all my articles on physics, readers are cautioned that I do not know what I am talking about, but that I can talk a good game and make up plenty of plausible-sounding bullshit that sounds so convincing that I believe it myself. Beware of bullshit.)

If you do a Google search for "speed of electricity", the top hit is Bill Beaty's long discourse on the subject. In this brilliantly obtuse article, Beaty manages to answer just about every question you might have about everything except the speed of electricity, and does so in a way that piles confusion on confusion.

Here's the funny thing about electricity. To have electricity, you need moving electrons in the wire, but the electrons are not themselves the electricity. It's the motion, not the electrons. It's like that joke about the two rabbinical students who are arguing about what makes tea sweet. "It's the sugar," says the first one. "No," disagrees the other, "it's the stirring." With electricity, it really is the stirring.

We can understand this a little better with an analogy. Actually, several analogies, each of which, I think, illuminates the others. They will get progressively closer to the real truth of the matter, but readers are cautioned that these are just analogies, and so may be misleading, particularly if overextended. Also, even the best one is not really very good. I am introducing them primarily to explain why I think M. Beaty's answer is obtuse.

  1. Consider a garden hose a hundred feet long. Suppose the hose is already full of water. You turn on the hose at one end, and water starts coming out the other end. Then you turn off the hose, and the water stops coming out. How long does it take for the water to stop coming out? It probably happens pretty darn fast, almost instantaneously.

    This shows that the "signal" travels from one end of the hose to the other at a high speed—and here's the key idea—at a much higher speed than the speed of the water itself. If the hose is one square inch in cross-section, its total volume is about 5.2 gallons. So if you're getting two gallons per minute out of it, that means that water that enters the hose at the faucet end doesn't come out the nozzle end until 156 seconds later, which is pretty darn slow. But it certainly isn't the case that you have to wait 156 seconds for the water to stop coming out after you turn off the faucet. That's just how long it would take to empty the hose. And similarly, you don't have to wait that long for water to start coming out when you turn the faucet on, unless the hose was empty to begin with.

    The water is like the electrons in the wire, and electricity is like that signal that travels from the faucet to the nozzle when you turn off the water. The electrons might be travelling pretty slowly, but the signal travels a lot faster.

  2. You're waiting in the check-in line at the airport. One of the clerks calls "Can I help who's next?" and the lady at the front of the line steps up to the counter. Then the next guy in line steps up to the front of the line. Then the next person steps up. Eventually, the last person in line steps up. You can imagine that there's a "hole" that opens up at the front of the line, and the hole travels backwards through the line to the back end.

    How fast does the hole travel? Well, it depends. But one thing is sure: the speed at which the hole moves backward is not the same as the speed at which the people move forward. It might take the clerks another hour to process the sixty people in line. That does not mean that when they call "next", it will take an hour for the hole to move all the way to the back. In fact, the rate at which the hole moves is to a large extent independent of how fast the people in the line are moving forward.

    The people in the line are like electrons. The place at which the people are actually moving—the hole—is the electricity itself.

  3. In the ocean, the waves start far out from shore, and then roll in toward the shore. But if you look at a cork bobbing on the waves, you see right away that even though the waves move toward the shore, the water is staying in pretty much the same place. The cork is not moving toward the shore; it's bobbing up and down, and it might well stay in the same place all day, bobbing up and down. It should be pretty clear that the speed with which the water and the cork are moving up and down is only distantly related to the speed with which the waves are coming in to shore. The water is like the electrons, and the wave is like the electricity.

  4. A bomb explodes on a hill, and sometime later Ike on the next hill over hears the bang. This is because the exploding bomb compresses the air nearby, and then the compressed air expands, compressing the air a little way away again, and the compressed air expands and compresses the air a little way farther still, and so there's a wave of compression that spreads out from the bomb until eventually the air on the next hill is compressed and presses on Ike's eardrums. It's important to realize that no individual air molecule has traveled from hill A to hill B. Each air molecule stays in pretty much the same place, moving back and forth a bit, like the water in the water waves or the people in the airport queue. Each person in the airport line stays in pretty much the same place, even though the "hole" moves all the way from the front of the line to the back. Similarly, the air molecules all stay in pretty much the same place even as the compression wave goes from hill A to hill B. When you speak to someone across the room, the sound travels to them at a speed of 680 miles per hour, but they are not bowled over by hurricane-force winds. (Thanks to Aristotle Pagaltzis for suggesting that I point this out.) Here the air molecules are like the electrons in the wire, and the sound is like the electricity.

OK, where did all these analogies get us? I wanted to make clear that in each of these phenomena, there are two kinds of moving things. There is a motion of the concrete particles in the medium itself: water, or air, or people in the queue. And then there is a more subtle phenomenon, which is the motion of the wave of change through the medium. And the speeds of these two things are related in a complex way, if at all.

I believe that when someone asks for the speed of electricity, what they are typically after is something like: When I flip the switch on the wall, how long before the light goes on? Or: the ALU in my computer emits some bits. How long before those bits get to the output bus? Or again: I send a telegraph message from Nova Scotia to Ireland on an undersea cable. How long before the message arrives in Ireland? Or again: computers A and B are on the same branch of an ethernet, 10 meters apart. How long before a packet emitted by A's ethernet hardware gets to B's ethernet hardware?

M. Beaty's answer about the speed of the electrons is totally useless as an answer to this kind of question. It's a really detailed, interesting answer to a question to which hardly anyone was interested in the answer.

Here the analogy with the speed of sound really makes clear what is wrong with M. Beaty's answer. I set off a bomb on one hill. How long before Ike on the other hill a mile away hears the bang? Or, in short, "what is the speed of sound?" M. Beaty doesn't know what the speed of sound is, but he is glad to tell you about the speed at which the individual air molecules are moving back and forth, although this actually has very little to do with the speed of sound. He isn't going to tell you how long before the tsunami comes and sweeps away your village, but he has plenty to say about how fast the cork is bobbing up and down on the water.

That's all fine, but I don't think it's what people are looking for when they want the speed of electricity. So the individual charges in the wire are moving at 2.3 mm/s; who cares? As M. Beaty was at some pains to point out, the moving charges are not themselves the electricity, so why bring it up?

I wanted to end this article with a correct and pertinent answer to the question. For a while, I was afraid I was going to have to give up. At first, I just tried looking it up on the web. Many people said that the electricity travels at the speed of light, c. This seemed rather implausible to me, for various reasons. (That's another essay for another day.) And there was widespread disagreement about how fast it really was. For example:

But then I found this page on the characteristic impedance of coaxial cables and other wires, which seems rather more to the point than most of the pages I have found that purport to discuss the "speed of electricity" directly.

From this page, we learn that the thing I have been referring to as the "speed of electricity" is called, in electrical engineering jargon, the "velocity factor" of the wire. And it is a simple function of the "dielectric constant" not of the wire material itself, but of the insulation between the two current-carrying parts of the wire! (In typical physics fashion, the dielectric "constant" is anything but; it depends on the material of which the insulation is made, the temperature, and who knows what other stuff they aren't telling me. Dielectric constants in the rest of the article are for substances at room temperature.) The function is simply:

$$V = {c\over\sqrt{\varepsilon_r}}$$

where V is the velocity of electricity in the wire, and εr is the dielectric constant of the insulating material, relative to that of vacuum. Amazingly, the shape, material, and configuration of the wire doesn't come into it; for example it doesn't matter if the wire is coaxial or twin parallel wires. (Remember the warning from the top of the page: I don't know what I am talking about.) Dielectric constants range from 1 up to infinity, so velocity ranges from c down to zero, as one would expect. This explains why we find so many inconsistent answers about the speed of electricity: it depends on a specific physical property of the wire. But we can consider some common examples.

Wikipedia says that the dielectric constant of rubber is about 7 (and this website specifies 6.7 for neoprene) so we would expect the speed of electricity in rubber-insulated wire to be about 0.38c. This is not quite accurate, because the wires are also insulated by air and by the rest of the universe. But it might be close to that. (Remember that warning!)

The dielectric constant of air is very small—Wikipedia says 1.0005, and the other site gives 1.0548 for air at 100 atmospheres pressure—so if the wires are insulated only by air, the speed of electricity in the wires should be very close to the speed of light.

We can also work the calculation the other way: this web page says that signal propagation in an ethernet cable is about 0.66c, so we infer that the dielectric constant for the insulator is around 1/0.662 = 2.3. We look up this number in a a table of dielectric constants and guess from that that the insulator might be polyethylene or something like it. (This inference would be correct.)

What's the lower limit on signal propagation in wires? I found a reference to a material with a dielectric constant of 2880. Such a material, used as an insulator between two wires, would result in a velocity of about 2% of c, which is still 5600 km/s. this page mentions cement pastes with "effective dielectric constants" up around 90,000, yielding an effective velocity of 1/300 c, or 1000 km/s.

Finally, I should add that the formula above only applies for direct currents. For varying currents, such as are typical in AC power lines, the dielectric constant apparently varies with time (some constant!) and the analysis is more complicated.

[ Addendum 20180904: Paul Martin suggests that I link to this useful page about dielectric constants. It includes an extensive table of the εr for various polymers. Mostly they are between 2 and 3.   ]

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