It’s fun to think about the fundamental physical constants. These are special values used in our models of the physical universe. They include things like the speed of light, the gravitational constant, and Planck’s constant, and they’re “fundamental” in the sense that we can’t derive them theoretically, we can only measure them.

We use these in solving physics problems all the time, so it’s easy to take them for granted. But why are there such numbers in nature, and why do they just happen to have those specific values? Because, listen, if they were only slightly different, the universe might be incapable of supporting life. Did some cosmic clockmaker set these parameters? Isaac Newton thought so.

One of the most basic of these numbers is the electric constant, k. It’s a value that lets us calculate the forces between electric charges. That’s a big deal when you consider that all matter is made of just three things—electrons, neutrons, and protons, two of which have an electric charge. The interaction between electrons is what forms molecules to create you and everything around you. Otherwise it would all be just some undifferentiated soup.

But how do we know the value of the electric constant? Also, what does it have to do with other fundamental constants? And for that matter, is it really fundamental? Let’s investigate.

Coulomb’s Law and Constant

When we say something has an electric charge, we mean it has a different number of protons and electrons. If your clothes dryer removes some electrons from your socks, they become positively charged. If they gain electrons, they’ll be negatively charged. (Note: You can’t take away protons, since they’re in the nucleus of the atom. It would involve a nuclear reaction, which nobody wants.)

If you have two objects with opposite charges, they attract. If they have the same charge, they repel. Here’s a demo you can do yourself: Take a piece of clear tape and place it on a smooth table. Then put a second piece on top of that one, and pull them off together. Now, if you separate them, one will be positive and one will be negative; hold them in proximity and they will bend toward each other.

If you repeat the process, you’ll have two positive and two negative tapes. Hold two with similar charges near each other, and you’ll see that they repel, like in the picture below:

The smaller the distance between the tapes, the greater the repelling force. If you increase the charge on either (or both) tapes, the force will also get stronger. In 1785, Charles-Augustin de Coulomb modeled this electrostatic force, so we call it Coulomb’s law. This is a famous equation that every chemistry and physics student learns. It looks like this:

Here two objects are separated by a distance r. The values of their charges are q₁ and q₂ in units of, well, coulombs. In order to get the force in newtons, the standard unit for measuring a force (F), we need a constant of proportionality—that’s k—the electric constant, also known as Coulomb’s constant. In these units it has a value of k = 8.987 x 10⁹ nm²/C² (newton-meters squared per coulomb squared … don’t worry about it).

That’s a big number, and it shows how strong electric interactions are—much stronger, in fact, than gravitational interactions. Surprised? It’s not something you notice, because all objects contain both positive and negative charges at the molecular level, so they have a roughly equal number of attractive and repulsive forces that mostly cancel. The gravitational interaction that keeps you pinned to the Earth is more obvious because it involves only an attractive force (you can’t have negative mass), and because we are puny specks on a giant rock.

How He Got It

To come up with this, Coulomb made an instrument called a torsion balance. It had a thin horizontal rod hanging from a fiber so it could rotate freely—all contained in a glass cylinder to shield it from errant breezes. Then he had two small metal balls, one stationary and one on the end of the rod (plus a counterweight on the other end so it balanced).

He then gave the two balls similar charges so they repelled, and he measured the deflection of the rod. Then, to vary the charges, he took one ball and touched it to an identical but uncharged ball, cutting its charge in half, and he remeasured. The rod moved away half as far.

This showed that the electric force (F) was proportional to the product of the charges (q₁q₂). Then, by varying the distance between the balls he found that F was inversely proportional to the distance squared (r²). That meant, e.g., that an attractive force between two charges grows very fast as they get closer to each other (i.e., as r gets smaller).

But how did he find the magical number k? You won’t like this answer, but Coulomb didn’t know the value of Coulomb’s constant—which meant he couldn’t quantify the electric force (F). All he could say was that it was all proportional. His problem was that there was no way at the time to measure electrical charges. There were no coulombs in Coulomb’s day.

But by running similar experiments over time, later scientists gradually zeroed in on the value of the electric constant, which we now know is k = 8.987 x 10⁹ nm²/C².

The Permittivity of Free Space

We could stop there, but hey, science never stops. You know that. It turns out there’s another constant that is related to Coulomb’s constant. We call it the “permittivity of free space” (ε₀), which sure sounds like fun. It tells us how hard it is to create an electric field in a vacuum. A smaller ε₀ , or lower permittivity, would mean you’d get a greater electric field from the same charge. Yes, that seems backwards, but it’s just how they defined it. It’s too late to change.

Is there a permittivity value for nonempty space too? Yup. We call that the dielectric permittivity (ε), and it depends on the type of material. For example, it’s harder to generate an electric field in water than it is in glass, so water has a higher ε.

With this permittivity constant, we can rewrite Coulomb’s law as the following:

All I’ve done is replace k with ¼πε₀, where ε₀ = 8.854 x 10^–12 C²/(nm²). That might seem like a pointless digression. But this enables us to do something pretty wonderful: We can create relationships with other fundamental constants. In particular, there’s a very cool relationship between permittivity (ε₀ )and the speed of light (c).

Here, the Greek letter mu, μ₀, is the magnetic constant, aka the permeability of free space. I’m going to do a whole other piece next week on this one, so stay tuned. For now, suffice it to say that both constants are in there because light is an electromagnetic wave.

This relationship holds for non-empty space too, where light has to travel through a medium like, say, water. But both constants would be much higher, which means light would travel much slower in water.

Remember when I said up top that physics constants were “fundamental”—they couldn’t be derived, only measured empirically? Well, as you can see, that wasn’t entirely true. The equation above places a constraint on these three particular constants, so we only need to measure two of them, and then we can compute the third. If we know the speed of light and the permeability, we can derive the permittivity and by extension the electric constant, k.

I know that sounds crazy, but at some point you have to realize that all of our units and constants are arbitrary. We have to pick some place to start finding values and then build our units like a house of cards. If you change one of them, the whole thing comes crashing down.