# Who determined Coulomb's Constant?

Coulomb's law says that the electrostatic force between two electric charges is given by the formula:

Ke*q1*q2/r^2

Who was the first person (and when) to determine Ke?

How did they figure it out?

Who named it Coulomb's Constant?

Coulomb's constant was discovered and named after Charles-Augustin de Coulomb. He determined the strength of the electric force by measuring the force between charged objects using a torsion balance.

vanhees71
Gold Member
2021 Award
There is nothing to determine, and I've never heard that this constant is named after Coulomb.

This constant is simply a conversion factor from the units of charge to the mechanical units. In the SI it's $K=\frac{1}{4 \pi \epsilon_0}$. In Gaussian units you have $K=1$ and in the rationalized Heaviside-Lorentz units, which in my opinion are the most naturla ones for theoretical E+M, $K=\frac{1}{4 \pi}$.

Actually, Coulomb found by experiment that the force is proportional to charge and distance, but he did not know there was a constant Ke.

Does anyone know who (and when) came up with the idea for Ke?

vanhees71
Gold Member
2021 Award
Have a look at the WikiPedia, where the issue is quite nicely explained:

http://en.wikipedia.org/wiki/Coulomb's_law

There, indeed they also call the constant, Coulomb's constant. I do not know, which convention of units Coulomb historically used, but I guess it's the static Coulomb, which has become part of the Gaussian system of measures, i.e., it defines the electric charge unit by Coulomb's Law in the form
$$|\vec{F}_{\text{estat}}|=\frac{q_1 q_2}{r^2}.$$
This is no longer the most natural definition since nowadays, on the most fundamental level, we understand the electromagnetic field as defined by the relativistically covariant Lagrangian,
$$\mathscr{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{c} j_{\mu} A^{\mu}.$$
Here $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ is Faraday's tensor in terms of the electromagnetic four-vector potential. It includes the electric and magnetic field components, and Maxwell's equation in this convention then read
$$\vec{\nabla} \cdot \vec{E}=\rho, \quad \vec{\nabla} \times \vec{B}-\frac{1}{c} \frac{\partial \vec{E}}{\partial t}=\frac{1}{c} \vec{j}.$$
The constraints that lead to the derivability of the electromagnetic field components from a vector potential, which read in the three-dimensional form
$$\vec{E}=-\frac{\partial \vec{A}}{c \partial t}-\vec{\nabla A^0}, \quad \vec{B}=\vec{\nabla} \times \vec{A},$$
are given by the inhomogeneous Maxwell equations,
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \frac{\partial \vec{B}}{\partial t}=0, \quad \vec{\nabla} \cdot \vec{B}=0.$$
In this interpretation the Coulomb Law reads
$$|\vec{F}_{\text{estat}}|=\frac{q_1 q_2}{4 \pi r^2}.$$
This is close to the Gaussian units but here the factor $4 \pi$ appears in the solution of Maxwell's eqs. and not in the eqs. themselves. This is the Heaviside-Lorentz system of units and used in theoretical high-energy physics.

The SI is invented for everyday use in electrical engineering and experimental physics. For theoretical physics in the relativistically covariant formulation it's a bit inconvenient, but of course the physics doesn't change with the change of units.

I think the Coulomb constant now is technically defined. Is determined by the permittivity of free space, which is defined via the permeability of free space (which is a defined and set constant) and the speed of light in a vacuum (which is also defined and set, due to the definition of a meter being the amount of distance light travels in a vacuum in a certain amount of time.) So the Coulomb constant, I think, is exact. It is irrational, though.

So, you are saying that Coulomb put an amount of charge on the two balls in the torsion balance, and then calculated the force due to the twist in the thread when the balls repelled each other. That force he calculated was equal to the charge divided by the distance squared. So Coulomb did not need any constant because the units he used worked out just right.

I am writing a historical book- so does anyone know when (or who) someone came up with the convention for Coulomb's constant?

Great reply by vanhees71. Thanks for taking the time.

No, sorry, I don't know how the convention arose.

Coulomb's Constant is defined by the speed of light c, in that way:
C=$\sqrt{\mu_{0}\epsilon_{0}}$ where $\mu_{0}$ is permeability of free space, defined as $\ 4\pi\ \times\ 10^{-7}$ and thus $\epsilon_{0}$ is well defined, κ$_{e}$ is$\frac{1}{4\pi \epsilon_{0}}$. So if we know c from experiments we can know k with any amount of accuracy.

Is that because Coulomb's Law is the perfect example of the relation between Magnetic Field and Electric field?

I would say that it's because the light is a electromagnetic wave, obe that oscillate both in the electric field and in the magnetic field. But thwe first estimation of both epsilon and miu dates before the description of electromagnetic waves.

Do you know why there are there are 3.0 x 10E+09 statC in 1 Coulomb.
And the speed of light is 3.0 x 10E+08 m/s ?

It can't be a coincidence can it?

It's just a selection of unit's, you can't explain relationships between units like c and m/s. It's not a coincidence just a measurement. the meter is a unit previously connected with earth circumfence, so obviously it isn't related with a universal property like the speed of light. for more look at Planck units which uses only universal properties.