Interlude: Crazy Units
I.3 Electromagnetic Units
Art: Okay, Lenny, I get what you are saying. But what about electromagnetic units? They seem to be especially annoying. What’s that thing 0 in all the equations, the thing the textbooks call the dielectric constant of the vacuum?
2 Why does the vacuum have a dielectric constant anyway, and why is it equal to 8.85 × 10−12? That seems really weird.
Art is right; electromagnetic units are a nuisance all of their own. And he is right that it doesn’t make sense to think of the vacuum as a dielectric—not in classical physics, anyway. The language is a holdover from the old ether theory.
The real question is: Why was it necessary to introduce a new unit for electric charge—the so-called coulomb? The history is interesting and actually based on some physical facts, but probably not the ones you imagine. I’ll start by telling you how I would have set things up, and why it would have failed.
What I would have done is to start by trying to accurately measure the force between two electric charges, let’s say by rubbing two pith balls with cat’s fur until they were charged. Presumably I would have found that the force was governed by Coulomb’s law,
Then I would have declared that a unit charge—one for which q = 1—is an amount of charge such that two of them, separated by 1 meter, have a force
between them of 1 newton. (The newton is a unit of force needed to accelerate a 1-kilogram mass by 1 meter per second per second.) In that way there would be no need for a new independent unit of charge, and Coulomb’s law would be simple, just like I wrote earlier.
Maybe if I had been particularly clever and had a bit of foresight, I might have put a factor of 4π in the denominator of the Coulomb law:
But that’s a detail.
Now why would I have failed, or at least not had good accuracy? The reason is that it is difficult to work with charges; they are hard to control.
Putting a decent amount of charge on a pith ball is hard because the electrons repel and tend to jump off the ball. So historically a different strategy was used.
Figure I.2: Parallel Wires. No current flows because the switch is open.
Figure I.3: Parallel Wires with Currents Flowing in Opposite Directions. The switch
is closed.
By contrast with charge, working with electric current in wires is easy.
Current is charge in motion, but because the negative charge of the moving electrons in a wire is held in place by the positive charges of the nuclei, they are easy to control. So instead of measuring the force between two static charges, we instead measure the force between two current-carrying wires.
Figures I.2 and I.3 illustrate how such an apparatus might work. We start with a circuit containing a battery, a switch, and two long parallel wires stretched tight and separated by a known distance. For simplicity the distance could be a meter, although in practice we may want it to be a good deal smaller.
Now we close the switch and let the current flow. The wires repel each other for reasons that will be explained later in this book. What we see is that the wires belly out in response to the force. In fact, we can use the amount of bellying to measure the force (per unit length). This allows us to define a unit of electric current called an ampere or amp.
One amp is the current needed to cause parallel wires separated by 1 meter, to repel with a force of 1 newton per meter of length.
Notice that in this way, we define a unit of current, not a unit of electric charge.
Current measures the amount of charge passing any point on the circuit per unit time. For example, it is the amount of charge that passes through the battery in 1 second.
Art: But wait, Lenny. Doesn’t that also allow us to define a unit of charge? Can’t we say that our unit of charge—call it 1 coulomb of charge—
is the amount of charge that passes through the battery in one second, given that the current is one amp?
Lenny: Very good! That’s exactly right. Let me say it again: The coulomb is by definition the amount of charge passing through the circuit in one second, when the current is one amp; that is, when the force on the wires is one newton per meter of length (assuming the wires are separated by one meter).
The disadvantage is that the definition of a coulomb is indirect. The advantage is that the experiment is so easy that even I did it in the lab. The problem, however, is that the unit of charge defined this way is not the same unit that would result from measuring the force between static charges.
How do the units compare? To answer that, we might try to collect two buckets of charge, each a coulomb, and measure the force between them. This would be dangerous even if it were possible; a coulomb is a really huge amount of charge. The bucket would explode and the charge would just fly apart. So the question becomes, why does it take such a huge amount of charge flowing in the wires to produce a modest force of 1 newton?
Art: Why is there a force between the wires anyway? Even though the wires have moving electrons, the net charge on the wire is zero. I don’t see why there is any force.
Lenny: Yes, you are right that the net charge is zero. The force is not electrostatic. It’s actually due to the magnetic field caused by the motion of the charges. The positive nuclei are at rest and don’t cause a magnetic field,
but the moving electrons do.
Art: Okay, but you still haven’t told me why it takes such a whopping big amount of moving charge to create a mere newton of force between the wires.
Am I missing something?
Lenny: Only one thing. The charges move very slowly.
The electrons in a typical current-carrying wire do indeed move very slowly.
They bounce around very quickly, but like a drunken sailor, they mostly get nowhere; on the average it takes an electron about an hour to move 1 meter along a wire. That seems slow, but compared to what? The answer is that they move very slowly compared to the only natural physical unit of velocity, the speed of light. In the end that’s why it takes a huge amount of charge, moving through the wires of a circuit, to produce a significant force.
Now that we know that the standard unit of charge, the coulomb, is a tremendously large amount of charge, let’s go back to Coulomb’s law. The force between two coulomb-size charges is enormous. To account for this we have to put a huge constant into the force law. Instead of
we write
where 0 is the small number 8.85 × 10−12.
Art: So ultimately the weird dielectric constant of the vacuum has nothing to do with dielectrics. It has more to do with the slow molasses-like motion of electrons in metallic wires. Why don’t we just get rid of 0 and set it equal to 1?
Lenny: Good idea, Art. Let’s do that from now on. But don’t forget that we will be working with a unit of charge that is about one three-hundred-thousandth of a Coulomb. Forgetting about the conversion factor could lead to a nasty explosion.
1 If you’ve read our previous book on quantum mechanics, you’ve heard this sermon before. It’s interesting that the same issues of scale that affect our choice of units also limit our ability to directly perceive quantum effects with our senses.
2 Also called vacuum permittivity, or permittivity of free space.