Elementary charge explained

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

In SI units, the coulomb is defined such that the value of the elementary charge is exactly ^[1] 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.

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%.^{[2] [3]} 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^[4] and (through the Faraday constant) at order-of-magnitude accuracy by Johann Loschmidt's measurement of the Avogadro constant in 1865.

As a unit

See also: 2019 revision of the SI.

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.^[5] 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 conceptual distinction 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

with the result that $$e\; =\; \backslash sqrt\backslash sqrt\; \backslash approx\; 0.30282212088\; \backslash sqrt,$$where is the fine-structure constant, is the speed of light, is the electric constant, and 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  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 types of exception 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 . 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 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  e. Therefore, the "quantum of charge" is  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.^[6] 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.^{[7] [8]} It is currently not known why isolatable particles are restricted to integer charges; much of the string theory landscape appears to admit fractional charges.^{[9] [10]}

Experimental measurements of the elementary charge

The elementary charge as expressed in SI units 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

The first determinations of the elementary charge were based on Faraday's laws of electrolysis.^[11] 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$$e\; =\; \backslash frac.$$(In words, the charge of one mole of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.)

In the late 1800's several physicists including Johann Josef Loschmidt and James Clerk Maxwell used estimates of the average diameter of the gas molecules to estimate Avogadro number, .^[12] Loschmidt's work gave a value of .^[13] Today the value of N_A has been adopted^[1] as a defining constant equal to

The value of 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.

In 1874, George Johnstone Stoney used this method of Faraday's second law to give the first estimate of the elementary charge; he was too small by a factor of 20.^[11] In 1901, Max Planck used his theory of blackbody radiation to estimate the Boltzmann constant, k_B, and combined this with a value for the gas constant, R, to estimate N_A via $$R\; =\; N\_\backslash mathrm\; k\_\backslash mathrm$$and then apply Faraday's second law. His value the elementary charge of is close to the modern value, .^[11]

JJ Thomson's electrostatic measurement

In 1898 J J Thomson measured the current in a gas in an electric field exposed to X-rays. The current would be, where is the number of ions measured, is the charge on the ions, and their mean velocity. The mean velocity in the electric field was previously measured by Rutherford. Thomson had discovered that water droplets would form around the ions if the gas was saturated before the X-ray exposure. The rate at which the vapor cloud fell gave an estimate of the size of the droplets and the mass of the droplets hitting the bottom of the experimental chamber per unit time gave total flux of water. Combined this gave an estimate of the number of droplets and thus the number of ions, . Then the current measured could be converted to an estimate of the elementary charge.^[15]

Oil-drop experiment

See 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

See 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$$K\_\backslash text\; =\; \backslash frac,$$where h is the Planck constant. It can be measured directly using the Josephson effect.

The von Klitzing constant is$$R\_\backslash text\; =\; \backslash frac.$$It can be measured directly using the quantum Hall effect.

From these two constants, the elementary charge can be deduced:$$e\; =\; \backslash frac.$$

CODATA method

The relation used by CODATA to determine elementary charge was:$$e^2\; =\; \backslash frac\; =\; 2h\; \backslash alpha\; \backslash varepsilon\_0\; c,$$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	q e- =-e	exact	by definition
proton	qp=e	\left	\right	< 10^e	by finding no measurable sound when an alternating electric field is applied to SF6 gas in a spherical resonator[18]
	q e+ =e	\left	\right	< 10^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]
	q\bar{p }=-e	\left	\right	< 10^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

Further reading

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

Notes and References

1. Book: Newell . David B. . Tiesinga . Eite . 2019 . The International System of Units (SI) . NIST Special Publication 330 . National Institute of Standards and Technology . Gaithersburg, Maryland . 10.6028/nist.sp.330-2019 . 242934226.
2. Millikan, R. A. . The isolation of an ion, a precision measurement of its charge, and the correction of Stokes's law . Science . 32 . 822 . 436-448 . 10.1126/science.32.822.436 . 1910.
3. Fletcher, Harvey . My work with Millikan on the oil-drop experiment . Physics Today . 35 . 6 . 43-47 . 10.1063/1.2915126 . 1982.
4. Martin J. . Klein . Max Planck and the beginnings of the quantum theory . Archive for History of Exact Sciences . 1 October 1961 . 1432-0657 . 1 . 5 . 459–479 . 10.1007/BF00327765 . 121189755 .
5. G. J. Stoney . 1894 . Of the "Electron," or Atom of Electricity . subscription . . 5 . 38 . 418–420 . 10.1080/14786449408620653.
6. Q is for Quantum, by John R. Gribbin, Mary Gribbin, Jonathan Gribbin, page 296, Web link
7. 10.1146/annurev.ns.34.120184.002333. free. Magnetic Monopoles. 1984. Preskill. J.. Annual Review of Nuclear and Particle Science. 34. 1. 461–530. 1984ARNPS..34..461P.
8. News: Three Surprising Facts About the Physics of Magnets . 17 July 2019 . Space.com . 2018 . en.
9. Schellekens . A. N. . Life at the interface of particle physics and string theory . Reviews of Modern Physics . 2 October 2013 . 85 . 4 . 1491–1540 . 10.1103/RevModPhys.85.1491. 1306.5083 . 2013RvMP...85.1491S . 118418446 .
10. Perl . Martin L. . Lee . Eric R. . Loomba . Dinesh . Searches for Fractionally Charged Particles . . November 2009 . 59 . 1 . 47–65 . 10.1146/annurev-nucl-121908-122035. free. 2009ARNPS..59...47P .
11. Book: Pais, Abraham . Inward bound: of matter and forces in the physical world . 2002 . Clarendon Press [u.a.] . 978-0-19-851997-3 . Reprint . Oxford.
12. Book: "Subtle is the Lord-- ": the science and the life of Albert Einstein . 2005 . Oxford University Press . 978-0-19-152402-8 . Pais . Abraham . Oxford New York.
13. J. . Loschmidt . Johann Josef Loschmidt . Zur Grösse der Luftmoleküle . Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien . 52 . 2 . 395–413 . 1865. English translation .
14. Ehl, Rosemary Gene . Ihde, Aaron . Aaron J. Ihde . Faraday's Electrochemical Laws and the Determination of Equivalent Weights . Journal of Chemical Education . 1954 . 31 . May . 226–232 . 10.1021/ed031p226 . 1954JChEd..31..226E .
15. Book: Whittaker, E. T. (Edmund Taylor) . A history of the theories of aether and electricity: from the age of Descartes to the close of the nineteenth century . 1910 . London; New York: Longmans, Green . University of California Libraries . none .
16. Beenakker . Carlo . Schönenberger . Christian . Christian Schönenberger . 2006 . Quantum Shot Noise . Physics Today . 56 . 5 . 37–42 . cond-mat/0605025 . 10.1063/1.1583532 . 119339791.
17. Nature . 389 . 162–164 . 1997 . 10.1038/38241 . Direct observation of a fractional charge . R. . de-Picciotto . M. . Reznikov . M. . Heiblum . V. . Umansky . G. . Bunin . D. . Mahalu . 162 . 1997Natur.389..162D . cond-mat/9707289 . 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 . 10.1103/PhysRevA.83.052101 . 1102.2766 . 118579475.
19. etal . Ahmadi . M. . 2016 . An improved limit on the charge of antihydrogen from stochastic acceleration . Nature . 529 . 7586 . 373–376 . 10.1038/nature16491 . 26791725 . 205247209 . May 1, 2022.
20. etal . Hori . M. . 2011 . Two-photon laser spectroscopy of antiprotonic helium and the antiproton-to-electron mass ratio. . Nature . 475 . 7357 . 484–488 . 10.1038/nature10260 . 21796208 . 1304.4330 . 4376768.
21. etal . Olive . K. A. . 2014 . Review of particle physics . Chinese Physics C . 38 . 9 . 090001 . 10.1088/1674-1137/38/9/090001 . 118395784 .