Elementary charge

The elementary charge, usually denoted by e, is a fundamental physical constant, defined as the electric charge carried by a single proton (+ 1e) or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 e.^[2][a]

Elementary charge
Common symbols	${\displaystyle e}$
SI unit	coulomb
Dimension	${\displaystyle {\mathsf {TI}}}$
Value	1.602176634×10−19 C‍[1]

In the SI system of units, the value of the elementary charge is exactly defined as ${\displaystyle e}$ = 1.602176634×10⁻¹⁹ coulombs, or 160.2176634 zeptocoulombs (zC).^[3] Since the 2019 revision of the SI, the seven SI base units are defined in terms of seven fundamental physical constants, of which the elementary charge is one.

In the centimetre–gram–second system of units (CGS), the corresponding quantity is 4.8032047...×10⁻¹⁰ statcoulombs.^[b]

Robert A. Millikan and Harvey Fletcher's oil drop experiment first directly measured the magnitude of the elementary charge in 1909, differing from the modern accepted value by just 0.6%.^[4][5] Under assumptions of the then-disputed atomic theory, the elementary charge had also been indirectly inferred to ~3% accuracy from blackbody spectra by Max Planck in 1901^[6] and (through the Faraday constant) at order-of-magnitude accuracy by Johann Loschmidt's measurement of the Avogadro number in 1865.

As a unit

See also: 2019 revision of the SI

Elementary charge
Unit system	Atomic units
Unit of	electric charge
Symbol	e
Conversions
1 e in ...	... is equal to ...
coulombs	1.602176634×10−19‍[1]
${\displaystyle {\sqrt {\varepsilon _{0}\hbar c}}}$ (natural units)	0.30282212088
statC	≘ 4.80320425(10)×10−10

In some natural unit systems, such as the system of atomic units, e functions as the unit of electric charge. The use of elementary charge as a unit was promoted by George Johnstone Stoney in 1874 for the first system of natural units, called Stoney units.^[7] Later, he proposed the name electron for this unit. At the time, the particle we now call the electron was not yet discovered and the difference between the particle electron and the unit of charge electron was still blurred. Later, the name electron was assigned to the particle and the unit of charge e lost its name. However, the unit of energy electronvolt (eV) is a remnant of the fact that the elementary charge was once called electron.

In other natural unit systems, the unit of charge is defined as ${\displaystyle {\sqrt {\varepsilon _{0}\hbar c}},}$ with the result that $${\displaystyle e={\sqrt {4\pi \alpha }}{\sqrt {\varepsilon _{0}\hbar c}}\approx 0.30282212088{\sqrt {\varepsilon _{0}\hbar c}},}$$ where α is the fine-structure constant, c is the speed of light, ε₀ is the electric constant, and ħ is the reduced Planck constant.

Quantization

See also: Partial charge

Charge quantization is the principle that the charge of any object is an integer multiple of the elementary charge. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not ⁠1/2⁠ e, or −3.8 e, etc. (There may be exceptions to this statement, depending on how "object" is defined; see below.)

This is the reason for the terminology "elementary charge": it is meant to imply that it is an indivisible unit of charge.

Fractional elementary charge

There are two known sorts of exceptions to the indivisibility of the elementary charge: quarks and quasiparticles.

• Quarks, first posited in the 1960s, have quantized charge, but the charge is quantized into multiples of ⁠1/3⁠ e. However, quarks cannot be isolated; they exist only in groupings, and stable groupings of quarks (such as a proton, which consists of three quarks) all have charges that are integer multiples of e. For this reason, either 1 e or ⁠1/3⁠ e can be justifiably considered to be "the quantum of charge", depending on the context. This charge commensurability, "charge quantization", has partially motivated grand unified theories.
• Quasiparticles are not particles as such, but rather an emergent entity in a complex material system that behaves like a particle. In 1982 Robert Laughlin explained the fractional quantum Hall effect by postulating the existence of fractionally charged quasiparticles. This theory is now widely accepted, but this is not considered to be a violation of the principle of charge quantization, since quasiparticles are not elementary particles.

Quantum of charge

All known elementary particles, including quarks, have charges that are integer multiples of ⁠1/3⁠ e. Therefore, the "quantum of charge" is ⁠1/3⁠ e. In this case, one says that the "elementary charge" is three times as large as the "quantum of charge".

On the other hand, all isolatable particles have charges that are integer multiples of e. (Quarks cannot be isolated: they exist only in collective states like protons that have total charges that are integer multiples of e.) Therefore, the "quantum of charge" is e, with the proviso that quarks are not to be included. In this case, "elementary charge" would be synonymous with the "quantum of charge".

In fact, both terminologies are used.^[8] For this reason, phrases like "the quantum of charge" or "the indivisible unit of charge" can be ambiguous unless further specification is given. On the other hand, the term "elementary charge" is unambiguous: it refers to a quantity of charge equal to that of a proton.

Lack of fractional charges

Paul Dirac argued in 1931 that if magnetic monopoles exist, then electric charge must be quantized; however, it is unknown whether magnetic monopoles actually exist.^[9][10] It is currently unknown why isolatable particles are restricted to integer charges; much of the string theory landscape appears to admit fractional charges.^[11][12]

See also: Anomaly (physics) § Anomaly cancellation

Experimental measurements of the elementary charge

The elementary charge is exactly defined since 20 May 2019 by the International System of Units. Prior to this change, the elementary charge was a measured quantity whose magnitude was determined experimentally. This section summarizes these historical experimental measurements.

In terms of the Avogadro constant and Faraday constant

If the Avogadro constant N_A and the Faraday constant F are independently known, the value of the elementary charge can be deduced using the formula $${\displaystyle e={\frac {F}{N_{\text{A}}}}.}$$ (In other words, the charge of one mole of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.)

This method is not how the most accurate values are measured today. Nevertheless, it is a legitimate and still quite accurate method, and experimental methodologies are described below.

The value of the Avogadro constant N_A was first approximated by Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas.^[13] Today the value of N_A can be measured at very high accuracy by taking an extremely pure crystal (often silicon), measuring how far apart the atoms are spaced using X-ray diffraction or another method, and accurately measuring the density of the crystal. From this information, one can deduce the mass (m) of a single atom; and since the molar mass (M) is known, the number of atoms in a mole can be calculated: N_A = M/m.

The value of F can be measured directly using Faraday's laws of electrolysis. Faraday's laws of electrolysis are quantitative relationships based on the electrochemical researches published by Michael Faraday in 1834.^[14] In an electrolysis experiment, there is a one-to-one correspondence between the electrons passing through the anode-to-cathode wire and the ions that plate onto or off of the anode or cathode. Measuring the mass change of the anode or cathode, and the total charge passing through the wire (which can be measured as the time-integral of electric current), and also taking into account the molar mass of the ions, one can deduce F.^[1]

The limit to the precision of the method is the measurement of F: the best experimental value has a relative uncertainty of 1.6 ppm, about thirty times higher than other modern methods of measuring or calculating the elementary charge.^[15]

Oil-drop experiment

Main article: Oil-drop experiment

A famous method for measuring e is Millikan's oil-drop experiment. A small drop of oil in an electric field would move at a rate that balanced the forces of gravity, viscosity (of traveling through the air), and electric force. The forces due to gravity and viscosity could be calculated based on the size and velocity of the oil drop, so electric force could be deduced. Since electric force, in turn, is the product of the electric charge and the known electric field, the electric charge of the oil drop could be accurately computed. By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e.

The necessity of measuring the size of the oil droplets can be eliminated by using tiny plastic spheres of a uniform size. The force due to viscosity can be eliminated by adjusting the strength of the electric field so that the sphere hovers motionless.

Shot noise

Main article: Shot noise

Any electric current will be associated with noise from a variety of sources, one of which is shot noise. Shot noise exists because a current is not a smooth continual flow; instead, a current is made up of discrete electrons that pass by one at a time. By carefully analyzing the noise of a current, the charge of an electron can be calculated. This method, first proposed by Walter H. Schottky, can determine a value of e of which the accuracy is limited to a few percent.^[16] However, it was used in the first direct observation of Laughlin quasiparticles, implicated in the fractional quantum Hall effect.^[17]

From the Josephson and von Klitzing constants

Another accurate method for measuring the elementary charge is by inferring it from measurements of two effects in quantum mechanics: The Josephson effect, voltage oscillations that arise in certain superconducting structures; and the quantum Hall effect, a quantum effect of electrons at low temperatures, strong magnetic fields, and confinement into two dimensions. The Josephson constant is $${\displaystyle K_{\text{J}}={\frac {2e}{h}},}$$ where h is the Planck constant. It can be measured directly using the Josephson effect.

The von Klitzing constant is $${\displaystyle R_{\text{K}}={\frac {h}{e^{2}}}.}$$ It can be measured directly using the quantum Hall effect.

From these two constants, the elementary charge can be deduced: $${\displaystyle e={\frac {2}{R_{\text{K}}K_{\text{J}}}}.}$$

CODATA method

The relation used by CODATA to determine elementary charge was: $${\displaystyle e^{2}={\frac {2h\alpha }{\mu _{0}c}}=2h\alpha \varepsilon _{0}c,}$$ where h is the Planck constant, α is the fine-structure constant, μ₀ is the magnetic constant, ε₀ is the electric constant, and c is the speed of light. Presently this equation reflects a relation between ε₀ and α, while all others are fixed values. Thus the relative standard uncertainties of both will be same.

Tests of the universality of elementary charge

Particle	Expected charge	Experimental constraint	Notes
electron	${\displaystyle q_{{\text{e}}^{-}}=-e}$	exact	by definition
proton	${\displaystyle q_{\text{p}}=e}$	${\displaystyle \left|{q_{\text{p}}-e}\right|<10^{-21}e}$	by finding no measurable sound when an alternating electric field is applied to SF6 gas in a spherical resonator[18]
positron	${\displaystyle q_{{\text{e}}^{+}}=e}$	${\displaystyle \left|{q_{{\text{e}}^{+}}-e}\right|<10^{-9}e}$	by combining the best measured value of the antiproton charge (below) with the low limit placed on antihydrogen's net charge by the ALPHA Collaboration at CERN.[19]
antiproton	${\displaystyle q_{\bar {\text{p}}}=-e}$	${\displaystyle \left|{q_{\bar {\text{p}}}+q_{\text{p}}}\right|<10^{-9}e}$	Hori et al.[20] as cited in antiproton/proton charge difference listing of the Particle Data Group[21] The Particle Data Group article has a link to the current online version of the particle data.

See also

• Committee on Data of the International Science Council

Notes

1. The symbol e has another useful mathematical meaning due to which its use as label for elementary charge is avoided in theoretical physics. For example, in quantum mechanics one wants to be able to write compactly plane waves ${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }}$ with the use of Euler's number ${\displaystyle e=e^{1}=\exp(1)}$. In the US, Euler's number is often denoted e (italicized), while it is usually denoted e (roman type) in the UK and Continental Europe. Somewhat confusingly, in atomic physics, e sometimes denotes the electron charge, i.e. the negative of the elementary charge. The symbol q_e is also used for the charge of an electron.
2. This is derived from the CODATA 2018 value, since one coulomb corresponds to exactly 2997924580 statcoulombs. The conversion factor is ten times the numerical value of speed of light in metres per second.

References

1. "2022 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18.
2. International Bureau of Weights and Measures (20 May 2019), The International System of Units (SI) (PDF) (9th ed.), ISBN 978-92-822-2272-0, archived from the original on 18 October 2021
3. Newell, David B.; Tiesinga, Eite (2019). The International System of Units (SI). NIST Special Publication 330. Gaithersburg, Maryland: National Institute of Standards and Technology. doi:10.6028/nist.sp.330-2019. S2CID 242934226.
4. Millikan, R. A. (1910). "The isolation of an ion, a precision measurement of its charge, and the correction of Stokes's law". Science. 32 (822): 436–448. doi:10.1126/science.32.822.436.
5. Fletcher, Harvey (1982). "My work with Millikan on the oil-drop experiment". Physics Today. 35 (6): 43–47. doi:10.1063/1.2915126.
6. Klein, Martin J. (1 October 1961). "Max Planck and the beginnings of the quantum theory". Archive for History of Exact Sciences. 1 (5): 459–479. doi:10.1007/BF00327765. ISSN 1432-0657. S2CID 121189755.
7. G. J. Stoney (1894). "Of the "Electron," or Atom of Electricity". Philosophical Magazine. 5. 38: 418–420. doi:10.1080/14786449408620653.
8. Q is for Quantum, by John R. Gribbin, Mary Gribbin, Jonathan Gribbin, page 296, Web link
9. Preskill, J. (1984). "Magnetic Monopoles". Annual Review of Nuclear and Particle Science. 34 (1): 461–530. Bibcode:1984ARNPS..34..461P. doi:10.1146/annurev.ns.34.120184.002333.
10. "Three Surprising Facts About the Physics of Magnets". Space.com. 2018. Retrieved 17 July 2019.
11. Schellekens, A. N. (2 October 2013). "Life at the interface of particle physics and string theory". Reviews of Modern Physics. 85 (4): 1491–1540. arXiv:1306.5083. Bibcode:2013RvMP...85.1491S. doi:10.1103/RevModPhys.85.1491. S2CID 118418446.
12. Perl, Martin L.; Lee, Eric R.; Loomba, Dinesh (November 2009). "Searches for Fractionally Charged Particles". Annual Review of Nuclear and Particle Science. 59 (1): 47–65. Bibcode:2009ARNPS..59...47P. doi:10.1146/annurev-nucl-121908-122035.
13. Loschmidt, J. (1865). "Zur Grösse der Luftmoleküle". Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien. 52 (2): 395–413. English translation Archived February 7, 2006, at the Wayback Machine.
14. Ehl, Rosemary Gene; Ihde, Aaron (1954). "Faraday's Electrochemical Laws and the Determination of Equivalent Weights". Journal of Chemical Education. 31 (May): 226–232. Bibcode:1954JChEd..31..226E. doi:10.1021/ed031p226.
15. Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998" (PDF). Journal of Physical and Chemical Reference Data. 28 (6): 1713–1852. Bibcode:1999JPCRD..28.1713M. doi:10.1063/1.556049. Archived from the original (PDF) on 2017-10-01.
16. Beenakker, Carlo; Schönenberger, Christian (2006). "Quantum Shot Noise". Physics Today. 56 (5): 37–42. arXiv:cond-mat/0605025. doi:10.1063/1.1583532. S2CID 119339791.
17. de-Picciotto, R.; Reznikov, M.; Heiblum, M.; Umansky, V.; Bunin, G.; Mahalu, D. (1997). "Direct observation of a fractional charge". Nature. 389 (162–164): 162. arXiv:cond-mat/9707289. Bibcode:1997Natur.389..162D. doi:10.1038/38241. S2CID 4310360.
18. Bressi, G.; Carugno, G.; Della Valle, F.; Galeazzi, G.; Sartori, G. (2011). "Testing the neutrality of matter by acoustic means in a spherical resonator". Physical Review A. 83 (5): 052101. arXiv:1102.2766. doi:10.1103/PhysRevA.83.052101. S2CID 118579475.
19. Ahmadi, M.; et al. (2016). "An improved limit on the charge of antihydrogen from stochastic acceleration" (PDF). Nature. 529 (7586): 373–376. doi:10.1038/nature16491. PMID 26791725. S2CID 205247209. Retrieved May 1, 2022.
20. Hori, M.; et al. (2011). "Two-photon laser spectroscopy of antiprotonic helium and the antiproton-to-electron mass ratio". Nature. 475 (7357): 484–488. arXiv:1304.4330. doi:10.1038/nature10260. PMID 21796208. S2CID 4376768.
21. Olive, K. A.; et al. (2014). "Review of particle physics" (PDF). Chinese Physics C. 38 (9): 090001. doi:10.1088/1674-1137/38/9/090001. S2CID 118395784.

Further reading

• Fundamentals of Physics, 7th Ed., Halliday, Robert Resnick, and Jearl Walker. Wiley, 2005