# Coulomb's Law

## Main Question or Discussion Point

I know people might think this is an unworthy question, but I want to know anyway.

Does anyone know why "r" was chosen as distance in this formula? I know many formulas and most of them follow the same variable usage for measurement of distance ("s"). The letter "r" in my experience is mostly used as radius. Being that they are both measures of length, it seems out of place to use "r" if your talking about point charges in, say, a spherical shape because it may be confusing to someone who doesn't use the formula all the time. If the variables didn't have a generally set and known usage then equations would be unnecessarily hard to remember. Why veer from the norm on equations like this, when I don't see any other "s" that might be used already in this particular equation?

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Yea, I think of "r" as being used because of radius. There are simply too many formula's, too many languages and too much historical and developmental baggage along with our notation. You see "s", "r" and "d". In cylindrical coordinates you often use "rho".

Context is key and it always has to be taken into account.

Yea, I think of "r" as being used because of radius.

Well the r is not radius in this equation. It is distance. But, otherwise I agree. It just seems like it would make things much easier for everyone if the scientific community agreed on having a specific use for each variable. If people don't like change, you inevitably have to change without them. Hence, Americas problem changing to the much easier and widely adopted metric system.

WannabeNewton
One way to look at it is if you simply have a point charge in space or are outside of a spherically symmetric distribution of charge then you can figure out the field at a distance r from the center of the sphere \ point charge by enclosing it in a larger concentric sphere whose radius is r. This is just Gauss's Law. Anyways what does it matter in the end? It is just a label.

Well the r is not radius in this equation. It is distance.
Whats the difference though? Any straight line distance is the radius of a circle.

Generally you recognize the form of a formula rather than the variables. If you see a kqq' divided by something squared you know that something is distance (or radius )

Anyways what does it matter in the end? It is just a label.
I was waiting for the first person to come in here and say this. Here you are..

Simplification and using standards is a fail-safe. I dont think I need to point out all the times in history when things failed because there were multiple standards. (Hubble measurement mismatch; had there been one simple measurement system, the costly failure could have been avoided.)

WannabeNewton
Regardless your argument is moot because it does make sense to speak of r as being a radius in aforementioned cases so I still don't see the point of this.

Sure guy. Im not going to argue with you. If you cant see my point, no problem. In my mind, if you are talking about a distance between two points it is usually "s" and not "r" because that is known as radius of a circle. Very simple concept to understand. Coming up with reasons to not standardize variables or encouraging me to avoid asking such questions instead of actually answering my original question of "why" (I was wondering if there was a logical reason in this particular equation) has gotten me and you nowhere. Thanks for your efforts.

Jano L.
Gold Member
Why veer from the norm on equations like this, when I don't see any other "s" that might be used already in this particular equation?
Many established things in physics are just convenient choices, letter r for distance including. It may be surprising for you that I use r for distance most of the time, while s comes to my mind only in extreme situations such as when I need to denote two different distances. Then I use r,s. The letter r precedes s in the alphabet, so this order of their introduction is quite neat and logical.

The reason for the usage of r for distance, instead of some other letter (such as a), can perhaps be found in the fact that often distance is calculated as a magnitude of certain radius vector.

In case of two point-like particles, we can introduce relative /r/adius vector of particle 2 with respect to the particle 1

$$\mathbf r = \mathbf r_2 - \mathbf r_1$$

Its magnitude is conveniently denoted by
$$r = |\mathbf r|$$

So, the r in Coulomb's law is naturally thought of as the magnitude of the radius vector $\mathbf r$.

I doubt there is hope for such norm in theoretical physics as you propose. There are not enough symbols, and if there were, it is very likely it would still not be very useful. The situation is similar to one in written language: one could invent different letter to each different word(concept), but the result would be extremely complex and awkward.

In practice, both in language and in formulary, we use repeatedly few (40?) symbols for millions of different things. As already pointed out, the key to the understanding of this situation is indeed the fact that it is the context in which the symbol is used which gives us grasp of its meaning, not its graphical (or other) representation. The symbol r or s, standing on its own without assumed context, can mean anything and thus would mean nothing.

Jano L.

Thank you very much for your insight. That is very useful information. I am in Physics 1 and in the past physics classes I have had we used s for distance variables in almost all cases I can remember. These were pulled straight from the physics books. So, indeed it is a surprise to me that you use r instead. I understand the radius vector concept you explained and that makes the use of r much more understandable.

I guess it is because r is the symbol used for the position vector.

Jano L.
Gold Member
I am in Physics 1 and in the past physics classes I have had we used s for distance variables in almost all cases I can remember. These were pulled straight from the physics books. So, indeed it is a surprise to me that you use r instead. I understand the radius vector concept you explained and that makes the use of r much more understandable.
That's great. Now I recall we used s as well in our classes on mathematics at high school, mostly for representing the length of the trajectory (path) that some body has traveled (train, car). Perhaps this came from the Latin word semita(path) or spatium(interval). This trajectory does not need to be straight, it can be helix, and the s was still used to denote its length (we called the quantity s "path" - I do not know if this is also customary in English speaking countries). The symbol r, however, is usually used only for straight distance of two points. So it seems there is a good rationale behind the use of r in Coulomb's law - the symbol s is customary for lengths of curved lines, while the symbol r is used to denote distance between two points (as measured along straight line).