# K in Coulomb's law

## Main Question or Discussion Point

"K" in Coulomb's law

Hi I want to ask how we can get k in Coulomb's law.

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Simon Bridge
Homework Helper

Welcome to PF;
k is an empirical constant like the acceleration of gravity - you can look it up.

$$k=\frac{1}{4\pi \epsilon_0}$$

The Coulomb constant is in fact the coupling constant of electric interaction.

Its numerical value depends on the charge unit. When you redefine the charge unit (coulomb) this constant will change. If you take the electron (elemental) charge as the base unit, you will get the fine structure constant. If you get the Planck charge as the unit, you will get 1.

Simon Bridge
Homework Helper

Its numerical value depends on the charge unit.
Pretty much like anything you measure ten? Numercal value depends on the units you use - how you measure it.

Pretty much like anything you measure ten? Numercal value depends on the units you use - how you measure it.
Yes, but I meant: the Coulomb constant is a definition of our charge unit, in a sense. It does not say anything about reality. It just says about us - namely about our unit system. This is just a force that two particles of one unit charge extert one on another.

Simon Bridge
Homework Helper

So what you are saying is that since the actual value of something depends on our unit system, then the actual value says something about our unit system....?

Perhaps you mean that scale factors are not part of "reality"? Or just this particular one?

I have seen engineering texts decrying the SI system requirement for the permittivity of free space - saying there is no need for such a thing in the unit systems they are used to.

Perhaps you mean that scale factors are not part of "reality"? Or just this particular one?
Suppose that you have 2 equivalent descriptions of reality and a transformation between them. The "reality" is the thing that is unchanged under this transformation. Everything else is not "reality", it is just a trait of that particular description.

One example of pure mathematical trait is a coordinate system. The second one is a unit system. You can pick several different theories in several different coordinate systems with several different units. All they mean the same. If you have some property that holds only in one of these formulations and not in others, then this is not a physical law. It's just a mathematical artifact of the theory, just as unimportant as the color of the ink in your pen you have chosen to write the theory with.

In this particular case:

$$F = K q_1 q_2 / r^2$$ - this is a physical law. It holds in all unit systems, with different numerical values of the parameters.
$$K = 8.987551787368176$$ - this is not a physical law. This is just one particular formulation. I can provide a different formulation where this does not hold.

Thank you.

Simon Bridge