Physical constant

In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. It can be contrasted with a mathematical constant, which is also a fixed value but does not directly involve any physical measurement.

There are many physical constants in science, some of the most widely recognized being the rationalized Planck's constant ħ, the gravitational constant G, the speed of light in the vacuum c, the electric constant ε₀, and the elementary charge e. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed limit of the universe and is expressed dimensionally as length divided by time; while the fine-structure constant α, which characterizes the strength of the electromagnetic interaction, is dimensionless.

Dimensionful and dimensionless physical constants

The numerical values of dimensionful physical constants depend on the unit system used, such as SI or cgs. As such, numbers such as the speed of light, c, expressed in units of meters per second (299,792,458) are not values that a theory of physics can be expected to predict.

Ratios of like-dimensioned physical constants do not depend on unit systems in this way (the units cancel), so they are pure (dimensionless) numbers whose values a future theory of physics could conceivably hope to predict. Additionally, all equations describing laws of physics can be expressed without dimensionful physical constants via a process known as nondimensionalization, but the dimensionless constants will remain. Thus, theoretical physicists tend to regard these dimensionless quantities as fundamental physical constants.

However, the phrase fundamental physical constant is also used in other ways. For example, the National Institute of Standards and Technology [1] uses it to refer to any universal physical quantity believed to be constant, such as the speed of light, c, and the gravitational constant G.

The fine-structure constant α is probably the best known dimensionless fundamental physical constant. Nobody knows why it takes on the value that it does (currently measured at about 1/137.035999). Many attempts have been made to derive this value from theory, but so far none have succeeded. The same holds for the dimensionless ratios of masses of fundamental particles (the most apparent is m_p/m_e, approximately 1836.152673). But, with the development of quantum chemistry in the 20th century, a vast number of previously inexplicable dimensionless physical constants were successfully computed from theory. As such, some theoretical physicists still hope for continued progress in explaining the values of dimensionless physical constants.

It is known that the universe would be very different if these constants took values significantly different from those we observe. For example, a few percent change in the value of the fine structure constant would be enough to eliminate stars like our Sun. This has prompted attempts at anthropic explanations of the dimensionless physical constants.

How constant are the physical constants?

Beginning with Paul Dirac in 1937, some scientists have speculated that physical constants may actually decrease in proportion to the age of the universe. Scientific experiments have not yet pinpointed any definite evidence that this is the case, although they have placed upper bounds on the maximum possible relative change per year at very small amounts (roughly 10^-5 per year for the fine structure constant α and 10^-11 for the gravitational constant G).

It is currently disputed [2] [3] that any changes in dimensionful physical constants such as G, c, ħ, or ε₀ are operationally meaningful; however, a sufficient change in a dimensionless constant such as α is generally agreed to be something that would definitely be noticed. If a measurement indicated that a dimensionful physical constant had changed, this would be the result or interpretation of a more fundamental dimensionless constant changing, which is the salient metric. From John D. Barrow 2002:

"[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged."

Anthropic principle

Some physicists have explored the notion that if the (dimensionless) fundamental physical constants had sufficiently different values, our universe would be so radically different that intelligent life would probably not have emerged, and that our universe therefore seems to be fine-tuned for intelligent life. The weak anthropic principle simply states that it is only because these fundamental constants acquired their respective values that there was sufficient order and richness in elemental diversity for life to have formed, which subsequently evolved the necessary intelligence toward observing that these constants have taken on the values they have.

Table of universal constants

Quantity	Symbol	Value	Relative Standard Uncertainty
characteristic impedance of vacuum	$Z_{0}=\mu _{0}c\,$	376.730 313 461... Ω	defined
electric constant (permittivity of free space)	$\epsilon _{0}=1/(\mu _{0}c^{2})\,$	8.854 187 817... × 10-12F·m-1	defined
magnetic constant (permeability of free space)	$\mu _{0}\,$	4π × 10-7 N·A-2 = 1.2566 370 614... × 10-6 N·A-2	defined
Newtonian constant of gravitation	$G\,$	6.6742(10) × 10-11m3·kg-1·s-2	1.5 × 10-4
Planck's constant	$h\,$	6.626 0693(11) × 10-34 J·s	1.7 × 10-7
Dirac's constant	$\hbar =h/(2\pi )$	1.054 571 68(18) × 10-34 J·s	1.7 × 10-7
speed of light in vacuum	$c\,$	299 792 458 m·s-1	defined

Table of electromagnetic constants

Quantity	Symbol	Value1 (SI units)	Relative Standard Uncertainty
Bohr magneton	$\mu _{B}=e\hbar /2m_{e}$	927.400 949(80) × 10-26 J·T-1	8.6 × 10-8
conductance quantum	$G_{0}=2e^{2}/h\,$	7.748 091 733(26) × 10-5 S	3.3 × 10-9
Coulomb's constant	$\kappa =1/4\pi \epsilon _{0}\,$	8.987 742 438 × 109 N·m2C-2	defined
elementary charge	$e\,$	1.602 176 53(14) × 10-19 C	8.5 × 10-8
Josephson constant	$K_{J}=2e/h\,$	483 597.879(41) × 109 Hz· V-1	8.5 × 10-8
magnetic flux quantum	$\phi _{0}=h/2e\,$	2.067 833 72(18) × 10-15 Wb	8.5 × 10-8
nuclear magneton	$\mu _{N}=e\hbar /2m_{p}$	5.050 783 43(43) × 10-27 J·T-1	8.6 × 10-8
resistance quantum	$R_{0}=h/2e^{2}\,$	12 906.403 725(43) Ω	3.3 × 10-9
von Klitzing constant	$R_{K}=h/e^{2}\,$	25 812.807 449(86) Ω	3.3 × 10-9

Table of atomic and nuclear constants

Quantity		Symbol	Value1 (SI units)	Relative Standard Uncertainty
Bohr radius		$a_{0}=\alpha /4\pi R_{\infty }\,$	0.529 177 2108(18) × 10-10 m	3.3 × 10-9
Fermi coupling constant		$G_{F}/(\hbar c)^{3}$	1.166 39(1) × 10-5 GeV-2	8.6 × 10-6
fine-structure constant		$\alpha =\mu _{0}e^{2}c/(2h)=e^{2}/(4\pi \epsilon _{0}\hbar c)\,$	7.297 352 568(24) × 10-3	3.3 × 10-9
Hartree energy		$E_{h}=2R_{\infty }hc\,$	4.359 744 17(75) × 10-18 J	1.7 × 10-7
quantum of circulation		$h/2m_{e}\,$	3.636 947 550(24) × 10-4 m2 s-1	6.7 × 10-9
Rydberg constant		$R_{\infty }=\alpha ^{2}m_{e}c/2h\,$	10 973 731.568 525(73) m-1	6.6 × 10-12
Thomson cross section		$(8\pi /3)r_{e}^{2}$	0.665 245 873(13) × 10-28 m2	2.0 × 10-8
weak mixing angle		$\sin ^{2}\theta _{W}=1-(m_{W}/m_{Z})^{2}\,$	0.222 15(76)	3.4 × 10-3

Table of physico-chemical constants

Quantity		Symbol	Value1 (SI units)	Relative Standard Uncertainty
atomic mass unit (unified atomic mass unit)		$m_{u}=1\ u\,$	1.660 538 86(28) × 10-27 kg	1.7 × 10-7
Avogadro's number		$N_{A},L\,$	6.022 1415(10) × 1023	1.7 × 10-7
Boltzmann constant		$k=R/N_{A}\,$	1.380 6505(24) × 10-23 J·K-1	1.8 × 10-6
Faraday constant		$F=N_{A}e\,$	96 485.3383(83)C·mol-1	8.6 × 10-8
first radiation constant		$c_{1}=2\pi hc^{2}\,$	3.741 771 38(64) × 10-16 W·m2	1.7 × 10-7
	for spectral radiance	$c_{1L}\,$	1.191 042 82(20) × 10-16 W · m2 sr-1	1.7 × 10-7
Loschmidt constant	at $T$=273.15 K and $p$=101.325 kPa	$n_{0}=N_{A}/V_{m}\,$	2.686 7773(47) × 1025 m-3	1.8 × 10-6
gas constant		$R\,$	8.314 472(15) J·K-1·mol-1	1.7 × 10-6
molar Planck constant		$N_{A}h\,$	3.990 312 716(27) × 10-10 J · s · mol-1	6.7 × 10-9
molar volume of an ideal gas	at $T$=273.15 K and $p$=100 kPa	$V_{m}=RT/p\,$	22.710 981(40) × 10-3 m3 ·mol-1	1.7 × 10-6
	at $T$=273.15 K and $p$=101.325 kPa		22.413 996(39) × 10-3 m3 ·mol-1	1.7 × 10-6
Sackur-Tetrode constant	at $T$=1 K and $p$=100 kPa	$S_{0}/R={\frac {5}{2}}$ $+\ln \left[(2\pi m_{u}kT/h^{2})^{3/2}kT/p\right]$	-1.151 7047(44)	3.8 × 10-6
	at $T$=1 K and $p$=101.325 kPa		-1.164 8677(44)	3.8 × 10-6
second radiation constant		$c_{2}=hc/k\,$	1.438 7752(25) × 10-2 m·K	1.7 × 10-6
Stefan-Boltzmann constant		$\sigma =(\pi ^{2}/60)k^{4}/\hbar ^{3}c^{2}$	5.670 400(40) × 10-8 W·m-2·K-4	7.0 × 10-6
Wien displacement law constant		$b=(hc/k)/\,$ 4.965 114 231...	2.897 7685(51) × 10-3 m · K	1.7 × 10-6

Table of adopted values

Quantity		Symbol	Value (SI units)	Relative Standard Uncertainty
conventional value of Josephson constant2		$K_{J-90}\,$	483 597.9 × 109 Hz · V-1	defined
conventional value of von Klitzing constant3		$R_{K-90}\,$	25 812.807 Ω	defined
molar mass	constant	$M_{u}=M(\,^{12}{\mbox{C}})/12$	1 × 10-3 kg · mol-1	defined
	of carbon-12	$M(\,^{12}{\mbox{C}})=N_{A}m(\,^{12}{\mbox{C}})$	12 × 10-3 kg · mol−1	defined
standard acceleration of gravity (gee, free fall on Earth)		$g_{n}\,\!$	9.806 65 m·s-2	defined
standard atmosphere		${\mbox{atm}}\,$	101 325 Pa	defined

Notes

¹The values are given in the so-called concise form; the number in brackets is the standard uncertainty, which is the value multiplied by the relative standard uncertainty.
²This is the value adopted internationally for realizing representations of the volt using the Josephson effect.
³This is the value adopted internationally for realizing representations of the ohm using the quantum Hall effect.

See also

• Fundamental physical constant
• Astronomical constant
• Scientific constants named after people
• Fine-tuned universe
• Physical law
• CODATA
• Natural units
• Atomic units
• Planck units

References

ISBN links support NWE through referral fees

• CODATA Recommendations - 2002 CODATA Internationally recommended values of the Fundamental Physical Constants
• John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.

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