Richard Feynman said nearly everything is really interesting if you go into it deeply enough. In that spirit I’m going to dig into the units on Coulomb’s constant. This turns out to be an interesting rabbit trail.
Coulomb’s law says that the force between two charged particles is proportional to the product of their charges and inversely proportional to the distance between them. In symbols,
The proportionality constant, the ke term, is known as Coulomb’s constant.
What are the units on Coulomb’s constant? Well, they’re whatever they have to be. The left hand side is a force, so it’s measured in newtons, N. Charges are measured in coulombs and distances in meters, so the right hand side, aside from Coulomb’s constant, has units coulombs squared per meter squared, C² / m². So ke must have units N m² / C².
OK, but what is a coulomb? That’s where things get interesting.
The informal definition that you might see in a textbook is that a coulomb is the amount of charge on a certain number of electrons, and that an ampere is a current of that many electrons flowing per second.
The formal definition, until two years ago, was that a coulomb was defined the amount of charge carried by a current of one ampere per second , and an ampere was defined as
that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10−7 newtons per metre of length.
The great redefinition
There were several things about the definitions of SI units that were less than satisfying. For example, the infinitely long conductors in the definition of ampere are in short supply.
The definitions of fundamental units have changed over time as measurement technology changes. For example, the kilogram was defined as the mass of a particular physical object, the Prototypical International Kilogram. Obviously this is awkward, but it wasn’t technically feasible to do anything better until recently.
The SI base units were redefined effective May 20, 2019.
The elementary charge, the charge on a single electron, is
e = 1.602176634×10−19 coulomb.
This equation used to be an empirical statement, the measured value of the elementary charge in terms of the coulomb. Now the equation is taken to be exact by definition, defining the coulomb.
Now that we know what a coulomb is, let’s go back to Coulomb’s constant. We said that ke must have units N m² / C². We’ve said what coulombs are, but what about newtons and meters? The newton is defined in terms of the kilogram, meter, and second, and the definitions of all these units changed as well.
The speed of light is now
c = 299792458 m⋅s−1
by definition. The second is defined so that the transition frequency of a caesium-133 atom is 9,192,631,770 cycles per second, and the meter is defined in terms of the speed of light and the second.
The Planck constant is now exactly
h = 6.62607015×10−34 kg m² / s
by definition, which defines the kilogram in terms of the meter, the second, and h. Now someone on a distant planet without access to the standard kilogram can determine how much a kilogram is by measuring the speed of light, the frequency of a caesium-133 atom, and the Plank constant.
Coulomb’s constant is equal to
where ɛ0 is vacuum permittivity.
where c is the speed of light and μ0 is vacuum permeability.
It used to be that
μ0 = 4π × 10−7 N/A2
by definition, but now that the speed of light is specified as exact by definition, μ0 is a measured quantity. Still, the measured value is very close to the former definition, accurate to nine significant figures. Now the value of c is exact by definition, and so the product of ɛ0 and μ0 is exact by definition, but ɛ0 and μ0 individually empirically determined.
 The abbreviation for coulomb is C and the abbreviation for ampere is A because units named after people, such as Coulomb and Ampère, are capitalized. But why aren’t the full unit names “coulomb” and “ampere” not capitalized? Because full names of SI units are not capitalized. Except for Celsius. C’est comme ça parce que c’est comme ça.