Elementary charge

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

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

In the SI system of units, the value of the elementary charge is exactly defined as ${\displaystyle e}$ = 1.602176634×10⁻¹⁹

are defined by 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]

in 1865.

As a unit

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
${\displaystyle {\sqrt {{\text{MeV}}\cdot {\text{fm}}}}}$ ( femtometers )	${\displaystyle {\sqrt {1.4399764}}}$
statC	≘ 4.80320425(10)×10−10

In some

(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

reduced Planck constant

.

Quantization

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.
• 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]

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

(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

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.^[15] 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.^[14]

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.^[14][16]

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

. 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

Any

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

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

is It can be measured directly using the quantum Hall effect.

From these two constants, the elementary charge can be deduced:

CODATA method

The relation used by

to determine elementary charge was:

where h is the

. 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[19]
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.[20]
antiproton	${\displaystyle q_{\bar {\text{p}}}=-e}$	${\displaystyle \left|{q_{\bar {\text{p}}}+q_{\text{p}}}\right|<10^{-9}e}$	Hori et al.[21] as cited in antiproton/proton charge difference listing of the Particle Data Group[22] 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

References

1. ^ ^{a b} "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
2. ISBN 978-92-822-2272-0, archived
   from the original on 18 October 2021
3. S2CID 242934226
   .
4. doi:10.1126/science.32.822.436
   .
5. doi:10.1063/1.2915126
   .
6. S2CID 121189755
   .
7. ^ G. J. Stoney (1894). "Of the "Electron," or Atom of Electricity".
   doi:10.1080/14786449408620653
   .
8. ^ Q is for Quantum, by John R. Gribbin, Mary Gribbin, Jonathan Gribbin, page 296, Web link
9. 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. S2CID 118418446
    .
12. 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. ^
    doi:10.1103/RevModPhys.80.633. Archived from the original
    (PDF) on 2017-10-01. Direct link to value.
15. doi:10.1021/ed031p226
    .
16. doi:10.1063/1.556049. Archived from the original
    (PDF) on 2017-10-01.
17. S2CID 119339791
    .
18. S2CID 4310360
    .
19. ^ 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.
    S2CID 118579475
    .
20. ^ Ahmadi, M.; et al. (2016). "An improved limit on the charge of antihydrogen from stochastic acceleration" (PDF). Nature. 529 (7586): 373–376.
    S2CID 205247209
    . Retrieved May 1, 2022.
21. ^ Hori, M.; et al. (2011). "Two-photon laser spectroscopy of antiprotonic helium and the antiproton-to-electron mass ratio". Nature. 475 (7357): 484–488.
    S2CID 4376768
    .
22. ^ Olive, K. A.; et al. (2014). "Review of particle physics" (PDF). Chinese Physics C. 38 (9): 090001.
    S2CID 118395784
    .

Further reading

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