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

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    $\begingroup$ From perusal of the literature, I find that electric potential was denoted by V until the 1930s (including publications by Fermi). The earliest paper I could find (my search hasn't been very thorough) that uses $\phi$ is: H. Stommel, "The theory of the electric field induced in deep ocean currents". J. Mar. Res, 7(3), 1948, pp. 386-392. This suggests that one might want to check the literature between 1930 and 1950 for additional occurrences and (possibly) a footnote somewhere that explains the choice of $\phi$. $\endgroup$
    – njuffa
    Mar 27, 2022 at 21:01
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    $\begingroup$ note that voltage begins with V, and $\phi$ may be the Greek letter closest to V $\endgroup$ Mar 27, 2022 at 22:42
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    $\begingroup$ This is not a universal symbol for electric potential. There are too many quantities in science and not enough symbols. Why theta is used for angles? L for angular momentum, etc. Not all have a reason. See this just for fun hsm.stackexchange.com/questions/728/… $\endgroup$
    – AChem
    Mar 27, 2022 at 22:43
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    $\begingroup$ @njuffa, My main concern is that the OP is worrying about a non-universal symbol. There are hundreds of symbols for various quantities. $\endgroup$
    – AChem
    Mar 28, 2022 at 21:44
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    $\begingroup$ @M.Farooq My approach here was: Let's see whether we can find out who first used $\phi$ for electric potential, and whether they gave a reason for doing so. Then attempt to track how this became more popular over time, because best I can tell, use of $\phi$ does seem dominant now. That is an open-ended process: Maybe there is a definite answer, maybe not. Therefore only a comment trail for now. I managed to successfully follow this process for an SI unit on a different question, if I recall correctly. $\endgroup$
    – njuffa
    Mar 28, 2022 at 23:17

2 Answers 2

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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.

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    $\begingroup$ I'm not sure if the variability of some Greek letters in math/physics deserves a new question or not, but it certainly took me a while to understand for example that $\varphi$ was $\phi$. In MathJax $\varphi$ and $\phi$ and in unicode Greek Phi Symbol U+03D5 and Greek Small Letter Phi U+03C6 Do you think there's a new question in there somewhere? $\endgroup$
    – uhoh
    Apr 25, 2022 at 3:42
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    $\begingroup$ @uhoh Funny, I was just thrown off by the symbol $\varrho$ ($\\varrho)$, variant of $\rho$), I couldn't tell what symbol it was until I looked it up and remembered. But I doubt there's any historical-math-and-science question there, maybe a question for orthography folks $\endgroup$ Apr 25, 2022 at 16:47
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As @njuffa rightly says, one should first trace the earliest $\phi$ through citations. I think the answer to that is Helmholtz [1870] — a paper which (quoth W. Kaiser, p. 390) “had an enormous impact on the process of reception of Maxwell’s theory by Continental physicists”. Moreover, the evidence collected then suggests a simple answer to the OP’s question: 𝝓 stands for function.

Helmholtz’s notation and its diffusion (1870–1904):

  • [1870] H. von Helmholtz, Ueber die Bewegungsgleichungen der Elektricität für ruhende leitende Körper. J. Reine Angew. Math. 72 (1870) 57–129. (p. 77: “Bezeichnen wir mit $\varphi$ die Potentialfunction der freien Elektricität”.)

  • [1875] H. A. Lorentz, Over de theorie der terungkaatsing en breking van het licht. Thesis, Leiden, 1875. (p. 1: cites [1870]; p. 29: “De electrische potentiaalfunctie, die wij door $\varphi$ zullen voorstellen”.)

  • [1878] H. A. Lorentz, Over het verband tusschen de voortplantingssnelheid van het licht en de dichtheid en samenstelling der middenstoffen. Verh. Kon. Ned. Akad. Wetensch. 18 (1878), no. 2, 1–112. (p. 6: cites [1870]; p. 4: “Is nu $\varphi_1$ de potentiaalfunctie voor deze ladingen”.)

  • [1880] H. Hertz, Ueber die Induction in rotirenden Kugeln. Thesis, Berlin, 1880. (p. 4: cites [1870]; p. 4: “$\varphi$ die Potentialfunction der freien Elektricität”.)

  • [1884] H. A. Lorentz, Le phénomène découvert par Hall et la rotation électromagnétique du plan de polarisation de la lumière. Arch. Neerl. Sci. Exact. Nat. 19 (1884) 123–152. (p. 139: cites [1870]; p. 138: “En désignant donc par $\varphi$ et $\chi$ les fonctions potentielles électrique et magnétique”.)

  • [1886] J. J. Thomson, Report on Electrical Theories. Rep. Brit. Assoc. Adv. Sci. 55 (1886) 97–155. (pp. 115, 133: cites [1870]; pp. 117, 133: “if $\phi$ denote the electrostatic potential of the free electricity”.)

  • [1890] H. Hertz, Ueber die Grundgleichungen der Electrodynamik für ruhende Körper. Ann. Physik (2) 40 (1890) 577–624. (p. 577: cites [1870]; p. 605: “Die Kräfte besitzen demnach ein Potential $\varphi$”.)

  • [1891] G. Kirchhoff, Vorlesungen über Elektricität und Magnetismus. Teubner, Leipzig, 1891. (p. 219: cites [1870]; p. 218: “sei $\dots\varphi$ das Potential der vorhandenen freie Elektricität”.)

  • [1892a] H. A. Lorentz, La théorie électromagnétique de Maxwell et son application aux corps mouvants. Arch. Neerl. Sci. Exact. Nat. 25 (1892) 363–552. (p. 367: cites [1890]; p. 389: “... le potentiel au point $P$. Cette fonction sera représentée par $\varphi$”; also pp. 456, 465–473.)

  • [1892b] H. Poincaré, Théorie mathématique de la lumière. Carré, Paris, 1892. (p. 21: “$\varphi$ est le potentiel électrostatique”.)

  • [1893a] L. Boltzmann, Vorlesungen über Maxwells Theorie der Elektricität und des Lichtes. II. Theil. Barth, Leipzig, 1893. (pp. 4, 133–140: cites [1870]; p. 42: “Es ist also $\varphi$ das elektrostatische Potential”.)

  • [1893b] J. J. Thomson, Notes on recent researches in electricity and magnetism. Clarendon Press, Oxford, 1893. (p. 251: “let $\phi$ be the electrostatic potential”.)

  • [1896] W. Voigt, Kompendium der theoretischen Physik. Zweiter Band: Elektricität und Magnetismus. Optik. Veit & Comp., Leipzig, 1896. (p. iii: cites [1890]; p. xiii: “$\varphi, \varphi', \varphi''$ Potentialfunktion wahrer, freier, influenzierter Elektricitäten”.)

  • [1897] H. von Helmholtz, Vorlesungen über die Elektromagnetische Theorie des Lichts. Leopold Voss, Leipzig, 1897. (p. 60: “Man bezeichnet im Allgemeinen die Function $\varphi\dots$ als die Potentialfunction der gegebenen Dichtigkeit”.)

  • [1898] C. Neumann, Die elektrischen Kräfte. Zweiter Theil. Teubner, Leipzig, 1898. (p. 181: cites [1870]; p. 180: “$\varphi$ das Potential aller im Conductor enthaltenen freien Elektricität”.)

  • [1899] E. Wiechert, Grundlagen der Elektrodynamik. Teubner, Leipzig, 1899. (p. 67: cites [1870]; p. 67: “Potential $\varphi$ der freien Elektricität”; also pp. 53, 60, 71, 78.)

  • [1900a] E. Cohn, Das Electromagnetische Feld. Hirzel, Leipzig, 1900. (p. 11: “𝜑 heisst das „elektrische Potential“ im Punkte $p$”.)

  • [1900b] H. Weber, Die partiellen Differential-Gleichungen der mathematischen Physik. Vieweg, Braunschweig, 1900. (p. 305: cites [1890, 1893a, 1897]; p. 312: “Die Function $\varphi$ heisst das elektrische Potential oder auch die elektrische Spannung”.)

  • [1901a] C. Neumann, Über die Maxwell-Hertz’sche Theorie. Leipz. Abh. 27 (1901, 1902) 213–348, 755–860; 28 (1903) 77–99. (p. 316: cites [1890]; p. 313: “$\varphi$ das Potential aller freien Elektricität”.)

  • [1901b] E. Wiechert, Elektrodynamische Elementargesetze. Ann. Physik (4) 4 (1901) 667–689. (p. 667: cites [1899]; p. 670: “$\dots$ der ein scalares Potential besitzt. Bezeichnen wir dieses mit $\varPhi\dots$”)

  • [1902] M. Abraham, Dynamik des Electrons. Gött. Nachr. 1902 (1902) 20–41. (p. 33: “Das electrostatische Potential $\varphi'$.)

  • [1903a] M. Abraham, Prinzipien der Dynamik des Elektrons. Ann. Physik. (4) 10 (1903) 105–179. (p. 114: “$\varPhi =$ skalares Potential”.)

  • [1903b] Anonymous, Vorschläge des wissenschaftlichen Ausschusses der Deutschen Physikalischen Gesellschaft für einheitliche Bezeichnungen, Benennungen, Definitionen und Regeln in der Physik. Verh. Deutsche Phys. Ges. 5 (1903) 68–71. (p. 70: “Potential $.........V,\varphi$”.)

  • [1904] A. Sommerfeld, Bezeichnung und Benennung der elektromagnetischen Grössen in der Enzyklopädie der mathematischen Wissenschaften V. Physik. Z. 5 (1904) 467–470.

Standardization (1904–1921)

Sommerfeld’s paper, [1904] above, explains how standardization came about: as editor of the Enzyklopädie, he consulted Lorentz, Cohn, Wien and made harmonizing recommendations “almost matching [1903b]”. Thus we get:

  • [1904a] H. A. Lorentz, V 13. Maxwells elektromagnetische Theorie. Encykl. d. math. Wiss. V 2 (1904) 63–144. (p. 141: cites [1870]; p. 142: “$\dots$ den skalaren Potentialen $\chi$ und $\varphi$”; also pp. 93, 116.)

  • [1904b] H. A. Lorentz, V 14. Weiterbildung der Maxwellschen Theorie. Elektronentheorie. Encykl. d. math. Wiss. V 2 (1904) 145–280. (p. 147: “skalares Potential $\varphi$”.)

  • [1907a] R. Gans, V 15. Elektrostatik und Magnetostatik. Encykl. d. math. Wiss. V 2 (1907) 289–349. (p. 293: “$\varphi$ und $\psi$ heißen elektrisches resp. magnetisches Potential”.)

  • [1907b] F. Pockels, V 16. Beziehungen zwischen elektrostatischen und magnetostatischen Zuständsänderungen einerseits und elastischen und thermischen andererseits. Encykl. d. math. Wiss. V 2 (1907) 350–392. (p. 352: “$\varphi$ das elektrische Potential”.)

  • [1910a] P. Debye, V 17. Stationäre und quasistationäre Felder. Encykl. d. math. Wiss. V 2 (1910) 393–482. (pp. 396, 441: “Potential $\dots\varphi$”.)

  • [1910b] M. Abraham, V 18. Elektromagnetische Wellen. Encykl. d. math. Wiss. V 2 (1910) 483–538. (p. 492: “die beiden Potentiale $\dots$ das skalare $\varphi$ und das vektorielle $\mathfrak a$”.)

  • [1921] W. Pauli, V 19. Relativitätstheorie. Encykl. d. math. Wiss. V 2 (1921) 539–775. (p. 632: “skalares Potential $\varphi$, Vektorpotential $\mathfrak A$ der Lorentzschen Theorie”.)

This was pretty informal: nowadays huge bureaucracies make such decisions, and give not $\varphi$ but U or V (the former apparently chosen after 20 years of debate in 1926, the latter who knows when).

Prehistory (1752–1869)

That $\phi$ stands for function is suggested not only by (e.g.) the first five references above [1870–1884], but also by a much older tradition: as observed by R. Brenneke, exact 1-forms (a.k.a. complete or perfect differentials) $$ Xdx + Ydy + Zdz = d\phi $$ go back to at least Euler [1752a, 1752b, 1753], who sometimes called $\phi$ “the effort”; and before Green and Gauss invented for it the name “potential function” or “potential” [1828, 1840], it was often just “the function” (perhaps qualified, as in fonction des forces, force function, Kräftefunction) and/or denoted $\phi$, $\varphi$, $\varPhi$. Here are some examples (note that the earliest published potential, [1752b], was already denoted $\Phi$):

  • [1752a] L. Euler, Principia motus fluidorum. Novi Comm. Acad. Sci. Petrop. 6 (1761) 24–26, 271–311. (E258 “Read on August 31, 1752”; translation: Physica D 237 (2008) 1840–1854; §§60–67, velocity potential $S$ of an exact 1-form $udx+vdy+wdz$.)

  • [1752b] L. Euler, Harmonie entre les principes généraux de repos et de mouvement de M. de Maupertuis, Hist. Acad. Roy. Berlin 7 (1753) 169–198. (E197 “Presented on November 9, 1752”; §§10–16, force potential or “effort” $\Phi$ of an exact 1-form $Vdv + V'dv' + V''dv''$.)

  • [1753] L. Euler, Principes généraux de l’état d'équilibre des fluides. Hist. Acad. Roy. Berlin 11 (1757) 217–273. (E225 “Presented on October 11, 1753”; §§32–34, force potential or “effort” $s$ of an exact 1-form $Pdx + Qdy + Rdz$.)

  • [1781] J. L. Lagrange, Mémoire sur la Théorie du mouvement des fluides. Nouv. Mém. Acad. Roy. Sci. Berlin 1781 (1783) 151–198. (“Lu le 22 novembre 1781”; §15, a velocity potential denoted $\varphi$.)

  • [1811] S. D. Poisson, Traité de mécanique. Tome Second. Ve Courcier, Paris, 1811. (p. 485, a velocity potential denoted $\varphi$.)

  • [1815] J. L. Lagrange, Mécanique analytique. Tome Second. Ve Courcier, Paris, 1815. (p. 311, a velocity potential denoted $\varphi$.)

  • [1824] S. D. Poisson, Mémoire sur la théorie du magnétisme. Mém. Acad. Sci. Paris 5 (1826) 247–338. (“Lu le 2 février 1824”; p. 302, a magnetic potential denoted $\varphi$.)

  • [1828] G. Green, An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism. T. Wheelhouse, Nottingham, 1828. (p. 56: a potential denoted $\phi'$.)

  • [1836] S. Earnshaw, On fluid motion, so far as it is expressed by the equation of continuity. Trans. Cambridge Philos. Soc. 6 (1838) 203–233. (“Read March 21, 1836”; p. 207, a velocity potential denoted $\phi$.)

  • [1840] C. F. Gauss, Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs-Kräfte. Weidmannsche Buchhandlung, Leipzig, 1840.

  • [1843] G. G. Stokes, On some cases of fluid motion. Trans. Cambridge Philos. Soc. 8 (1849) 105–137. (“Read May 29, 1843”; p. 108, a velocity potential denoted $\phi$.)

  • [1845] F. E. Neumann, Allgemeine Gesetze der inducirten elektrischen Ströme. Abhandl. Akad. Berlin 1845 (1847) 1–88. (p. 44, a magnetic potential denoted $\varphi$.)

  • [1850] W. Thomson, A mathematical theory of magnetism.—Continuation of part I. Philos. Trans. Roy. Soc. London 141 (1851) 269–285. (“Read June 20, 1850”; p. 274, a magnetic potential denoted $\varphi$.)

  • [1852] H. von Helmholtz, Akustik. 1. Theorie. Die Fortschritte der Physik 4 (1852) 101–118. (p. 103, a velocity potential denoted $\varphi$.)

  • [1858] H. von Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55 (1858) 25–55. (p. 28, a velocity potential denoted $\varphi$.)

  • [1860] H. von Helmholtz, Theorie der Luftschwingungen in Röhren mit offenen Enden. J. Reine Angew. Math. 57 (1860) 1–72. (p. 13, a velocity potential denoted $\varPhi$.)

  • [1865] J. C. Maxwell, A Dynamical Theory of the Electromagnetic Field. Philos. Trans. Roy. Soc. London 155 (1865) 459–512. (p. 482, a magnetic potential denoted $\varphi$.)

  • [1868] H. von Helmholtz, Über discontinuirliche Flüssigkeits-Bewegungen. Monatsber. Akad. Berlin 1868 (1869) 215–228. (p. 223, a velocity potential denoted $\phi$.)

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