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Physical constant (fundamental physical constants and their role in science)

A physical constant is a measured quantity believed to be universal and invariable. This article explains types, examples, how they are measured, their role in units and theories, and debates about variability.

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A physical constant is a quantity in the natural sciences that is taken to be universal and unchanging in time and place. Unlike a mathematical constant, which is a pure number derived from abstract reasoning, a physical constant is associated with a physical quantity and is determined by experiment. Physical constants appear as fixed parameters in the laws of physics and provide the numerical ties between theory and observation. They are central to the practice of science, from characterizing microscopic interactions to describing cosmology.

Common examples

  • Speed of light — the invariant speed that appears in relativity and electromagnetic theory: speed of light.
  • Planck's constant — a scale factor setting the size of quantum effects.
  • Gravitational constant — governing the strength of Newtonian gravity: gravitational constant.
  • Avogadro's number — linking amount of substance to particle count in chemistry.
  • Fine-structure constant — a dimensionless measure of electromagnetic interaction strength.

Some constants carry units (for example length or mass per time), while others are dimensionless pure numbers. Dimensionless constants are often considered more fundamental in the sense that their values cannot be altered by changing the system of units; the fine-structure constant is a frequently discussed example.

Characteristics, measurement and standards

Determining physical constants is an experimental enterprise: metrologists perform precise measurements and statistical analyses to obtain the best values. International bodies collect and recommend values that laboratories worldwide use as references. Modern unit systems have been increasingly linked to invariant properties of nature; recent reforms have redefined base units so that several SI units are specified by fixed numerical values of certain constants. The process reduces reliance on physical artefacts and improves reproducibility across experiments.

History and scientific importance

Many constants entered physics as empirical parameters that later acquired deeper theoretical roles. For example, the speed of light was measured in classical experiments long before it became a cornerstone of relativity, and Planck's constant appeared from fitting blackbody radiation data before the development of quantum mechanics. Because constants appear in fundamental equations, they shape the behaviour of matter and radiation across scales and thus underpin technology, from electronics to space navigation.

Debates and notable issues

Scientists sometimes ask whether purportedly constant quantities might vary slowly across space or time. Observations and laboratory tests place strong constraints on any such variation, but the question motivates sensitive astronomical and atomic-clock experiments that probe epochs in the history of the universe. Another recurring topic is the apparent fine-tuning of some constants: small changes in certain values would make complex chemistry or stable stars impossible, a point often invoked in discussions about the emergence of life and in philosophical arguments such as the anthropic principle.

Practical uses and distinctions

  1. Reference values: constants provide the numerical backbone for calculations in physics, chemistry and engineering.
  2. Unit definition: tying units to constants improves precision and universality of measurements.
  3. Theory testing: high-precision measurements of constants can reveal discrepancies that hint at new physics.

In summary, physical constants serve as measured, reproducible anchors connecting theory to experiment. They are indispensable to quantitative science, and ongoing research continues to refine their values, probe their constancy, and explore the implications that their numerical values have for natural phenomena and fundamental theory.

Table of some constants

The digits in parentheses after a numerical value denote the uncertainty in the last digits of the value. (Example: The so-called shorthand notation 6.674 30(15) is equivalent to 6.674 30 ±0.000 15). The uncertainty is given as the estimated standard deviation of the given numerical value from the actual value. The numerical values are based on CODATA 2018.

Designation of the constant

Symbol(s)

Value (SI)

Fundamental

Note

Electromagnetism

Speed of light in vacuum

c

299792458 {\displaystyle \textstyle \mathrm {\frac {m}{s}} }

Yes

Nk. F.

Elementary charge

e

1.602176634e-19 {\displaystyle \textstyle \mathrm {C} }

Yes

Nk. F.

Magnetic field constant

\mu _{0}

1.25663706212(19)e-6 {\displaystyle \textstyle \mathrm {\frac {H}{m}} }

Yes

fK. a. M.

Electric field constant

{\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}\,c^{2}}}}

8.8541878128(13)e-12 {\displaystyle \textstyle \mathrm {\frac {A\,s}{V\,m}} }

Yes

fK. a. M.

Coulomb constant

k_{C}={\frac {1}{4\pi \varepsilon _{0}}}

8.9875517922(14)e9 {\displaystyle \textstyle \mathrm {\frac {m}{F}} }

Yes

fK. a. M.

Wave impedance of the vacuum

{\displaystyle Z_{w0}=\mu _{0}\,c}

3.76730313667(57)e2 Ω {\displaystyle \textstyle \mathrm {\Omega } }

Yes

fK. a. M.

Gravity and cosmology

Gravitational constant

G

6.67430(15)e-11 {\displaystyle \textstyle \mathrm {\frac {m^{3}}{kg\,s^{2}}} }

Yes

Nk. M.

Planck mass

{\displaystyle m_{\text{Planck}}={\sqrt {\frac {\hbar \,c}{G}}}}

2.176434(24)e-8 {\displaystyle \textstyle \mathrm {kg} }

Yes

Planck length

{\displaystyle l_{\text{Planck}}={\frac {\hbar }{m_{\text{Planck}}\,c}}}

1.616255(18)e-35 {\displaystyle \textstyle \mathrm {m} }

Yes

Planck time

{\displaystyle t_{\text{Planck}}={\frac {l_{\text{Planck}}}{c}}}

5.391247(60)e-44 {\displaystyle \textstyle \mathrm {s} }

Yes

Gravitational coupling constant

{\displaystyle \alpha _{G}={\frac {G\,m_{\text{e}}^{2}}{\hbar \,c}}={\frac {m_{e}^{2}}{m_{\text{Planck}}^{2}}}}

1.751810(39)e-45

No

Nk. a. M.

Thermodynamics

Boltzmann constant

k_{B}

1.380649e-23 {\displaystyle \textstyle {\frac {\mathrm {J} }{\mathrm {K} }}}= 8.617333262...e-5 {\displaystyle \textstyle \mathrm {\frac {eV}{K}} }

Yes

fK. F.

Stefan-Boltzmann constant

{\displaystyle \sigma ={\frac {2\pi ^{5}k_{B}^{4}}{15\,h^{3}c^{2}}}}

5.670374419...e-8 {\displaystyle \textstyle \mathrm {\frac {W}{m^{2}\,K^{4}}} }

Yes

fK. a. F.

Vienna Constant

{\displaystyle b={\frac {hc}{4{,}965\,114\cdot k_{B}}}}

2.897771955...e-3 {\displaystyle \textstyle \mathrm {m\cdot K} }

Yes

fK. a. F.

Avogadro constant

N_A

6.02214076e23 {\displaystyle \textstyle {\frac {1}{\mathrm {mol} }}}

No

fK. F.

Faraday constant

{\displaystyle F=e\,N_{A}}

96485.3321233100184... {\displaystyle \textstyle \mathrm {\frac {C}{mol}} }

No

fK. a. F.

Gas constant

{\displaystyle R=N_{A}\,k_{B}}

8.31446261815324 {\displaystyle \textstyle \mathrm {\frac {J}{K\cdot mol}} }

No

fK. a. F.

Loschmidt constant at T0=273.15 K and p0=101.325 kPa

N_{L}or {\displaystyle n_{0}=N_{A}\cdot {\frac {p_{0}}{RT_{0}}}}

2.686780111...e25 {\displaystyle \textstyle \mathrm {\frac {1}{m^{3}}} }

No

fK. a. F.

Molar volume of an ideal gas

{\displaystyle V_{m_{0}}={\frac {R\,T_{0}}{p_{0}}}={\frac {N_{A}}{n_{0}}}}

0.02241396954... {\displaystyle \textstyle \mathrm {\frac {m^{3}}{mol}} }

No

fK. a. F.

Atomic Physics

Rydberg constant

{\displaystyle R_{\infty }={\frac {e^{4}\,m_{e}}{8\varepsilon _{0}^{2}h^{3}c}}={\frac {\alpha ^{2}}{2}}\,{\frac {m_{\mathrm {e} }c}{h}}}

1.0973731568160(21)e7 {\displaystyle \textstyle {\frac {1}{\mathrm {m} }}}

No

F. a. M.

Rydberg Energy

{\displaystyle R_{\infty }hc={\frac {E_{h}}{2}}={\frac {\alpha ^{2}}{2}}m_{\mathrm {e} }c^{2}}

13.605693122994(26) {\displaystyle \textstyle \mathrm {eV} }= 2.1798723611035(42)e-18 {\displaystyle \textstyle \mathrm {J} }

No

Rydberg frequency

R_{\infty }\,c

3.2898419602508(64)e15 {\displaystyle \textstyle \mathrm {Hz} }

No

Hartree Energy

{\displaystyle E_{h}={\frac {e^{4}\,m_{\mathrm {e} }}{4\,\varepsilon _{0}^{2}\,h^{2}}}=m_{\mathrm {e} }c^{2}\alpha ^{2}}

4.3597447222071(85)e-18 {\displaystyle \textstyle \mathrm {J} }

No

Quantum and particle physics

Planck's constant

h

6.62607015e-34 {\displaystyle \textstyle \mathrm {J\,s} }= 4.135667696...e-15 {\displaystyle \textstyle \mathrm {eV\,s} }

Yes

Nk. F.

Planck's reduced quantum of action

\hbar ={\frac {h}{2\pi }}

1.054571817...e-34 {\displaystyle \textstyle \mathrm {J\,s} }

Yes

Nk. a. F.

Spectral radiation constant

{\displaystyle c_{1L}={\frac {2hc^{2}}{sr}}}

1.191042972...e-16 {\displaystyle \textstyle \mathrm {\frac {W\,m^{2}}{sr}} }

Nk. a. F.

First radiation constant

{\displaystyle c_{1}=2\pi \,hc^{2}}

3.741771852...e-16 {\displaystyle \textstyle \mathrm {W\,m^{2}} }

Yes

Nk. a. F.

Second radiation constant

{\displaystyle c_{2}={\frac {hc}{k_{B}}}}

1.438776877...e-2 {\displaystyle \textstyle \mathrm {m\cdot K} }

Yes

fK. a. F.

Fine structure constant

{\displaystyle \alpha ={\frac {\mu _{0}\,e^{2}c}{2h}}}

7.2973525693(11)e-3 = (137.035999084(21))−1

No

F. a. M.

Nuclear Magneton

{\displaystyle \mu _{N}={\frac {e\,\hbar }{2\,m_{p}}}}

5.0507837461(15)e-27 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

Nk. a. M.

magnetic flux quantum

{\displaystyle \Phi _{0}={\frac {h}{2e}}}

2.067833848...e-15 {\displaystyle \textstyle \mathrm {Wb} }

No

Nk. a. F.

Josephson constant

{\displaystyle K_{J}={\frac {1}{\Phi _{0}}}={\frac {2e}{h}}}

4.835978484...e14 {\displaystyle \textstyle \mathrm {\frac {Hz}{V}} }

No

Nk. a. F.

Von Klitzing constant

{\displaystyle R_{K}={\frac {h}{e^{2}}}}

25812.80745... Ω {\displaystyle \textstyle \mathrm {\Omega } }

No

Nk. a. F.

Conductance quantum

{\displaystyle G_{0}={\frac {2e^{2}}{h}}}

7.748091729...e-5 {\displaystyle \textstyle \mathrm {\frac {s\,C^{2}}{m^{2}\,kg}} }

No

Nk. a. F.

Fermi constant

{\displaystyle G_{\rm {F}}^{0}={\frac {G_{\rm {F}}}{(\hbar c)^{3}}}={\frac {\sqrt {2}}{8}}{\frac {g^{2}}{m_{\text{W}}^{2}}}}

4.5437957(23)e14 {\displaystyle \textstyle \mathrm {\frac {1}{J^{2}}} }= 1.1663787(6)e-5 {\displaystyle \textstyle \mathrm {\frac {1}{GeV^{2}}} }

No

???

Electron

Electron mass

m_{{\mathrm e}}

9.1093837015(28)e-31 {\displaystyle \textstyle \mathrm {kg} }= 5.48579909065(16)e-4 {\displaystyle \textstyle \mathrm {u} }

No

Nk. M.

Compton wavelength of the electron

{\displaystyle \lambda _{C}={\frac {h}{m_{\mathrm {e} }c}}}

2.42631023867(73)e-12 {\displaystyle \textstyle \mathrm {m} }

No

Nk. a. M.

Drilling radius

{\displaystyle a_{0}={\frac {4\pi \varepsilon _{0}\hbar ^{2}}{e^{2}\,m_{\mathrm {e} }}}={\frac {1}{\alpha }}{\frac {\lambda _{C}}{2\pi }}={\frac {\hbar }{\alpha m_{\mathrm {e} }c}}}

5.29177210903(80)e-11 {\displaystyle \textstyle \mathrm {m} }

No

fK. a. M.

Classical electron radius

{\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}\,{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}=\alpha ^{2}\,a_{0}}

2.8179403262(13)e-15 {\displaystyle \textstyle \mathrm {m} }

No

Nk. a. M.

Bohr's magneton

{\displaystyle \mu _{B}={\frac {e\,\hbar }{2\,m_{\mathrm {e} }}}}

9.2740100783(28)e-24 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

fK. a. M.

Magnetic moment of the electron

{\displaystyle \mu _{\mathrm {e} }}

-9.2847647043(28)e-24 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

???

Landé factor of the electron

{\displaystyle g_{\mathrm {e} }=-2{\frac {\mu _{\mathrm {e} }}{\mu _{B}}}}

-2.00231930436256(35)

No

???

Gyromagnetic ratio of the electron

{\displaystyle \gamma _{\mathrm {e} }=-2{\frac {\mu _{\mathrm {e} }}{\hbar }}={\frac {g_{\mathrm {e} }\mu _{B}}{\hbar }}}

1.76085963023(53)e11 {\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}

No

???

Specific charge of the electron

{\displaystyle {\frac {e}{m_{\mathrm {e} }}}}

-1.75882001076(53)e11 {\displaystyle \textstyle \mathrm {\frac {C}{kg}} }

No

Nk. a. M.

Neutron

Neutron mass

{\displaystyle m_{\mathrm {n} }}

1.67492749804(95)e-27 {\displaystyle \textstyle \mathrm {kg} }= 1.00866491595(49) {\displaystyle \textstyle \mathrm {u} }

No

Nk. M.

Gyromagnetic ratio of the neutron

{\displaystyle \gamma _{\mathrm {n} }}

1.83247171(43)e8 {\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}

No

???

Magnetic moment of the neutron

{\displaystyle \mu _{\mathrm {n} }}

-9.6623651(23)e-27 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

???

Proton

Proton mass

{\displaystyle m_{\mathrm {p} }}

1.67262192369(51)e-27 {\displaystyle \textstyle \mathrm {kg} }= 1.007276466621(53) {\displaystyle \textstyle \mathrm {u} }

No

Nk. M.

Gyromagnetic ratio of the proton

{\displaystyle \gamma _{\mathrm {p} }}

2.6752218744(11)e8 {\displaystyle \textstyle {\frac {1}{\mathrm {s\,T} }}}

No

???

Magnetic moment of the proton

{\displaystyle \mu _{\mathrm {p} }}

1.41060679736(60)e-26 {\displaystyle \textstyle \mathrm {\frac {J}{T}} }

No

???

Ratio of proton mass to electron mass

{\displaystyle {\frac {m_{\mathrm {p} }}{m_{\mathrm {e} }}}}

1836.15267343(11)

No

???

Abbreviation

Meaning

Nk. F.

Natural constant, determination of the measure

Nk. a. F.

Derived from natural constants only, determination of the measured value

fK. F.

freely defined constant with definition of the dimension number

fK. a. F.

user-defined derived constant with definition of the measured value

fK. a. M.

freely defined, derived constant, measured value

Nk. M.

Natural constant, measured value

Nk. a. M.

Derived from natural constants only, measured value
  1. ↑ a b c d e value is used to define SI units.
  2. ↑ a b c d Until the revision of the SI units in 2019, μ0 had the exact value 4π-10-7 H/m. Thus, ε0, kC, and Zw0 were also exactly fixed.
  3. ↑ from me and mPlanck
  4. ↑ Since 2019, the unit "kelvin" is no longer independent, but defined by the thermodynamic energy; the Boltzmann constant has since become an arbitrary conversion factor to the unit "joule".
  5. ↑ a b c d e f g h i j k l m n Derived value
  6. ↑ Avogadro's constant as well as pressure and temperature at standard conditions are not natural constants but arbitrarily determined.
  7. ↑ a b At standard conditions

Constancy of the constants of nature

Whether the constants of nature are also really constant over astronomical time periods is the subject of current research. For example, measurements of the spectral lines of quasars with the Keck telescope in Hawaii seemed to indicate a slight decrease in the fine structure constant by about one hundredth of a part per thousand over the course of ten billion years. This result was controversial from the beginning; on the one hand, researchers pointed out the uncertain error estimate of the data analysis, and on the other hand, there are data from the Oklo mine in West Africa, where about 2 billion years ago uranium had accumulated to such an extent and had such a high content of the isotope U-235 that a nuclear fission chain reaction took place. According to these data, the fine structure constant had the same numerical value then as it does today. Recent measurements of the spectral lines of quasars with the Very Large Telescope of the European Southern Observatory in Chile contradict the earlier results at the Keck telescope and point to the constancy of the fine structure constant.

In the meantime, precision measurements are possible that can verify any steady fluctuations of the order of magnitude suggested by the observations with the Keck telescope, even in the laboratory over short periods of time. Investigations by Theodor Hänsch and his research group at the Max Planck Institute of Quantum Optics prove the constancy of the fine structure constant with an accuracy of 15 decimal places over a period of four years.

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