Why is electric potential denoted by $\phi$?

Question

I haven't found any explanation for it, and I'm curious.

Answer 1

This is not an answer, but is I think closer to an answer than some of the comments. The symbol $V$ was used by Laplace to denote the gravitational potential in Mécanique Céleste (1798, see e.g. book III chapter I $\S 4$, found in tome/volume 2). Laplace does not give a reason for using the symbol.

Si l'on désigne par $V$, la somme de toutes les molécules du sphéroïde, divisées par leurs distances respectives au point attiré, et que l'on nomme $x,y,z$, les coordonnées de la molécule $dM$ du sphéroïde, et $a,b,c$, celles du point attiré; ...

If we denote by $V$, the sum of all the molecules of the spheroid, divided by their respective distances to the attracted point, and we call $x,y,z$, the coordinates of the molecule $dM$ of the spheroid, and $a,b,c$ those of the attracted point; ...

The study of potentials goes back much further than Mécanique Céleste, but it was Laplace's work that influenced Poisson in his Mémoires (1813) on the distribution of electricity at the surface of spheroidal conductors. Poisson also used $V$, to denote the same concept of a function whose gradient (what he called the sum of the différences partielles) gives the electric force that would be experienced per unit charge at that point.

By the time Maxwell wrote his Treatise in 1873, $V$ was still the most common symbol for the electrostatic potential. But in some theorems, and the more general case of electrodynamics, he uses $\phi$, $\Phi$, and $\Psi$. For example, in $\S95b$,

We have hitherto used the symbol $V$ for the potential, and we shall continue to do so whenever we are dealing with electrostatics only. In this chapter [chapter IV], however, and in those parts of the second volume in which the electric potential occurs in electro-magnetic investigations, we shall use $\Psi$ as a special symbol for the electric potential.

By the time of Webster's The Theory of Electricity and Magnetism (1897), the symbol $\phi$ is mentioned early on, see e.g. p. 59,

Accordingly the three equations of condition equivalent to curl $R=0$ are simply the conditions that $X,Y,Z$ may be represented as the derivatives of a point-function. ...The scalar function $\phi$ (or its negative) will sometimes be termed the potential of the vector $R$.

But in most of the book Webster uses $V$ to denote electric potential. In the section on magneto-statics, the symbol $\Omega$ is used for a magnetic potential.

In Thomson's Elements of the Mathematical Theory of Electricity and Magnetism (first published 1895, version I have available is from 1909) he uses $V$ to denote potential, and $\Omega$ for magnetic potential. Jeans, The Mathemetical Theory of Electricity and Magnetism (1908, 1925), uses $V$ and $\Omega$ as well.

The first textbook that I'm familiar with that uses $\phi$ to denote potential is Abraham and Foppl, Theorie der Elektrizität [Theory of Electricity], first published in 1904, and the versions I found are from 1905 and 1918. This is a German text, which might give us a hint that the notation $\Phi$ and $\phi$ (both used at different times by Abraham) were more popular in German texts than in English. The translations of this text (8th and 14th editions, under the names Abraham and Becker) was very popular as well in its time, but I only have a copy of the 2nd edition from 1950. It uses $\phi$ to denote potential, keeping with the German convention.

From what I can tell, Mason and Weaver's The Electromagnetic Field (1929) uses $\Phi$ to denote the electrostatic potential, which might be one of the earlier popular English textbooks on classical electromagnetism to use this convention.

So it seems like in terms of elementary textbooks, while the notation $V$ was predominant early on, the use of $\phi$, $\Phi$, and $\Psi$ goes back at least to Maxwell, possibly earlier, but it didn't catch on until the 1930s in the English-speaking world. It may have been adopted due to the popularity of German texts, but that's just my guess. This is all supposing that textbooks are a reasonable source of information on common conventions, but obviously published papers and more specialized texts could have been using different symbols at an earlier time, it's just harder to sort through to find occurrences.

As for why these Greek letters were chosen, if I had to propose a reason, it was just because they were (and still are) commonly used to denote auxiliary functions during proofs and intermediate steps, or otherwise because those symbols were not being used to denote other quantities.