formula.expert

Constants library

61 values, each with its units, its uncertainty, and where it came from.

Universal & Atomic 42

Speed of light in vacuum exact

c=299,792,458 m/sc = 299,792,458\ \text{m/s}

m/sThe invariant speed of light in vacuum, exactly 299 792 458 m/s since 1983 — the definition that now fixes the length of the metre.

Speed of light squared (mass-energy conversion factor) exact

c2=8.987551787368176×1016 J/kgc^{2} = 8.987551787368176 \times 10^{16}\ \text{J/kg}

J/kgThe exchange rate between mass and energy in E = mc²: 8.988 × 10¹⁶ joules locked in every kilogram of rest mass.

Planck constant exact

h=6.62607015×10−34 J⋅sh = 6.62607015 \times 10^{-34}\ \text{J}{\cdot}\text{s}

J·sThe quantum of action, exactly 6.626 070 15 × 10⁻³⁴ J·s — the constant that has defined the kilogram since 2019.

Reduced Planck constant (Dirac constant) exact

ℏ=1.054571817×10−34 J⋅s\hbar = 1.054571817 \times 10^{-34}\ \text{J}{\cdot}\text{s}

J·sPlanck's constant divided by 2π, 1.054 571 817 × 10⁻³⁴ J·s — the natural quantum of angular momentum and spin.

Newtonian constant of gravitation measured

G=6.6743×10−11 m3/(kg⋅s2)G = 6.6743 \times 10^{-11}\ \text{m}^{3}\text{/(kg}{\cdot}\text{s}^{2}\text{)}

m³/(kg·s²)The coupling strength of gravity, 6.674 30 × 10⁻¹¹ m³/(kg·s²) — the worst-measured constant in all of physics.

Elementary charge exact

e=1.602176634×10−19 Ce = 1.602176634 \times 10^{-19}\ \text{C}

CThe charge of a proton, exactly 1.602 176 634 × 10⁻¹⁹ C — the quantum of free charge and the SI definition of the ampere.

Electronvolt (in joules) exact

eV=1.602176634×10−19 J\mathrm{eV} = 1.602176634 \times 10^{-19}\ \text{J}

JThe energy one electron gains crossing one volt: exactly 1.602 176 634 × 10⁻¹⁹ J, the working currency of atomic physics.

Standard acceleration of gravity exact

gn=9.80665 m/s2g_{n} = 9.80665\ \text{m/s}^{2}

m/s²The conventional value of free-fall acceleration, exactly 9.806 65 m/s² — a defined reference, not a measurement of your local g.

Electron mass measured

me=9.1093837139×10−31 kgm_{\mathrm{e}} = 9.1093837139 \times 10^{-31}\ \text{kg}

kgThe rest mass of the electron, 9.109 383 714 × 10⁻³¹ kg — equivalently 5.485 799 091 × 10⁻⁴ u or 0.510 999 MeV/c².

Electron mass energy equivalent measured

mec2=8.187105788×10−14 Jm_{\mathrm{e}} c^{2} = 8.187105788 \times 10^{-14}\ \text{J}

JThe electron's rest energy, 8.187 105 788 × 10⁻¹⁴ J or 510.999 keV — the photon energy of every positron annihilation.

Proton mass measured

mp=1.67262192595×10−27 kgm_{\mathrm{p}} = 1.67262192595 \times 10^{-27}\ \text{kg}

kgThe rest mass of the proton, 1.672 621 926 × 10⁻²⁷ kg — 1.007 276 u, 938.272 MeV/c², and 1836 times the electron.

Proton mass energy equivalent measured

mpc2=1.50327761802×10−10 Jm_{\mathrm{p}} c^{2} = 1.50327761802 \times 10^{-10}\ \text{J}

JThe proton's rest energy, 1.503 277 618 × 10⁻¹⁰ J or 938.272 MeV — the yardstick for accelerator and nuclear energies.

Neutron mass measured

mn=1.67492750056×10−27 kgm_{\mathrm{n}} = 1.67492750056 \times 10^{-27}\ \text{kg}

kgThe rest mass of the neutron, 1.674 927 501 × 10⁻²⁷ kg — 1.008 665 u or 939.565 MeV/c², just heavier than the proton.

Neutron mass energy equivalent measured

mnc2=1.50534976514×10−10 Jm_{\mathrm{n}} c^{2} = 1.50534976514 \times 10^{-10}\ \text{J}

JThe neutron's rest energy, 1.505 349 765 × 10⁻¹⁰ J or 939.565 MeV — exceeding the proton's by the 1.293 MeV that drives beta decay.

Muon mass measured

mμ=1.883531627×10−28 kgm_{\mu} = 1.883531627 \times 10^{-28}\ \text{kg}

kgThe rest mass of the muon, 1.883 531 627 × 10⁻²⁸ kg or 105.658 MeV/c² — 207 electrons in one unstable package.

Tau lepton mass measured

mτ=3.16754×10−27 kgm_{\tau} = 3.16754 \times 10^{-27}\ \text{kg}

kgThe rest mass of the tau lepton, 3.167 54 × 10⁻²⁷ kg or 1776.86 MeV/c² — heavier than a proton, and the shortest-lived lepton.

Proton-electron mass ratio measured

mp/me=1836.152673426m_{\mathrm{p}}/m_{\mathrm{e}} = 1836.152673426

dimensionlessThe proton outweighs the electron by 1836.152 673 4 — a pure number, known to 17 parts per trillion, that shapes all of chemistry.

Neutron-proton mass ratio measured

mn/mp=1.00137841946m_{\mathrm{n}}/m_{\mathrm{p}} = 1.00137841946

dimensionlessThe neutron is heavier than the proton by just 0.1378 % — the 1.293 MeV difference that makes free neutrons decay and stars burn.

Muon-electron mass ratio measured

mμ/me=206.7682827m_{\mu}/m_{\mathrm{e}} = 206.7682827

dimensionlessThe muon is 206.768 times heavier than the electron — the same particle in every respect except mass, and nobody knows why.

Fine-structure constant measured

α=0.0072973525643\alpha = 0.0072973525643

dimensionlessThe dimensionless strength of the electromagnetic interaction, 0.007 297 352 564 — roughly 1/137, and pure number with no units at all.

Inverse fine-structure constant measured

α−1=137.035999177\alpha^{-1} = 137.035999177

dimensionlessThe reciprocal of the fine-structure constant, 137.035 999 177 — famously near 137, and definitively not equal to it.

Rydberg constant measured

R∞=10973731.568157 m−1R_{\infty} = 10973731.568157\ \text{m}^{-1}

m⁻¹The wavenumber scale of atomic spectra, 10 973 731.568 157 m⁻¹ — the most precisely measured constant in all of physics.

Rydberg energy (hcR∞) measured

hcR∞=2.179872361103×10−18 JhcR_{\infty} = 2.179872361103 \times 10^{-18}\ \text{J}

JThe ionisation energy of ground-state hydrogen, 2.179 872 361 × 10⁻¹⁸ J or 13.605 693 eV — the natural unit of atomic energy.

Hartree energy measured

Eh=4.359744722206×10−18 JE_{\mathrm{h}} = 4.359744722206 \times 10^{-18}\ \text{J}

JThe atomic unit of energy, 4.359 744 722 × 10⁻¹⁸ J or 27.211 386 eV — twice the Rydberg and the currency of quantum chemistry.

Bohr radius measured

a0=5.29177210544×10−11 ma_{0} = 5.29177210544 \times 10^{-11}\ \text{m}

mThe most probable electron-proton distance in ground-state hydrogen, 5.291 772 105 × 10⁻¹¹ m — the natural size of an atom.

Compton wavelength of the electron measured

λC=2.42631023538×10−12 m\lambda_{\mathrm{C}} = 2.42631023538 \times 10^{-12}\ \text{m}

mλ_C = h/(m_e c) = 2.426 310 235 × 10⁻¹² m — the wavelength shift of a photon scattered through 90° by a free electron.

Reduced Compton wavelength of the electron measured

λˉC=3.8615926744×10−13 m\bar{\lambda}_{\mathrm{C}} = 3.8615926744 \times 10^{-13}\ \text{m}

mħ/(m_e c) = 3.861 592 674 × 10⁻¹³ m — the Compton wavelength divided by 2π, and the natural length scale of the Dirac equation.

Classical electron radius measured

re=2.8179403205×10−15 mr_{\mathrm{e}} = 2.8179403205 \times 10^{-15}\ \text{m}

mr_e = α²a₀ = 2.817 940 321 × 10⁻¹⁵ m — the radius a classical sphere of charge e would need to have rest energy m_e c².

Thomson cross section measured

σe=6.6524587051×10−29 m2\sigma_{\mathrm{e}} = 6.6524587051 \times 10^{-29}\ \text{m}^{2}

m²The low-energy scattering cross section of a photon on a free electron, 6.652 458 705 × 10⁻²⁹ m² — that is 0.665 barn.

Electron g-factor measured

ge=−2.00231930436092g_{\mathrm{e}} = -2.00231930436092

dimensionlessThe electron's magnetic moment in Bohr magnetons, −2.002 319 304 360 92 — the most precisely tested prediction in all of science.

Quantum of circulation measured

h2me=0.00036369475467 m2/s\frac{h}{2m_{\mathrm{e}}} = 0.00036369475467\ \text{m}^{2}\text{/s}

m²/sh/(2m_e) = 3.636 947 547 × 10⁻⁴ m²/s — the ratio of Planck's constant to mass that atom interferometers measure directly.

Electron charge-to-mass quotient measured

−e/me=−1.75882000838×1011 C/kg-e/m_{\mathrm{e}} = -1.75882000838 \times 10^{11}\ \text{C/kg}

C/kgThe electron's charge divided by its mass, −1.758 820 008 × 10¹¹ C/kg — the quantity J. J. Thomson measured in 1897 to discover the electron.

Proton charge-to-mass quotient measured

e/mp=95788331.43 C/kge/m_{\mathrm{p}} = 95788331.43\ \text{C/kg}

C/kg9.578 833 143 × 10⁷ C/kg — the proton's charge-to-mass ratio, smaller than the electron's by the full factor of 1836.

Proton rms charge radius measured

rp=8.4075×10−16 mr_{\mathrm{p}} = 8.4075 \times 10^{-16}\ \text{m}

mThe root-mean-square radius of the proton's charge distribution, 8.4075 × 10⁻¹⁶ m — 0.841 femtometres, and recently controversial.

Nuclear radius constant

r0=1.2×10−15 mr_{0} = 1.2 \times 10^{-15}\ \text{m}

mThe empirical coefficient in R = r₀A^(1/3), about 1.2 × 10⁻¹⁵ m — the constant that says nuclear matter has a fixed density.

Barn (nuclear cross-section unit) exact

b=1×10−28 m2\mathrm{b} = 1 \times 10^{-28}\ \text{m}^{2}

m²Exactly 10⁻²⁸ m², or 100 fm²: the unit every nuclear cross section is quoted in, and roughly the geometric area of a uranium nucleus.

Planck length measured

ℓP=1.616255×10−35 m\ell_{\mathrm{P}} = 1.616255 \times 10^{-35}\ \text{m}

m√(ħG/c³) = 1.616 255 × 10⁻³⁵ m — the length scale where quantum mechanics and gravity must both apply, and neither alone works.

Planck mass measured

mP=2.176434×10−8 kgm_{\mathrm{P}} = 2.176434 \times 10^{-8}\ \text{kg}

kg√(ħc/G) = 2.176 434 × 10⁻⁸ kg — about 22 micrograms, the only Planck unit on a human scale, and the mass where gravity meets quantum.

Planck time measured

tP=5.391247×10−44 st_{\mathrm{P}} = 5.391247 \times 10^{-44}\ \text{s}

s√(ħG/c⁵) = 5.391 247 × 10⁻⁴⁴ s — the time light takes to cross a Planck length, and the earliest instant physics can describe.

Planck temperature measured

TP=1.416784×1032 KT_{\mathrm{P}} = 1.416784 \times 10^{32}\ \text{K}

K√(ħc⁵/G)/k = 1.416 784 × 10³² K — the temperature at which thermal photons carry the Planck energy and gravity becomes quantum.

Planck energy measured

EP=1,956,081,637 JE_{\mathrm{P}} = 1,956,081,637\ \text{J}

Jm_P c² = 1.956 × 10⁹ J, or 1.221 × 10¹⁹ GeV — the energy scale of quantum gravity, and about the kinetic energy of a car on the motorway.

Fermi coupling constant measured

GF/(ℏc)3=0.000011663787 GeV−2G_{\mathrm{F}}/(\hbar c)^{3} = 0.000011663787\ \text{GeV}^{-2}

GeV⁻²The strength of the weak interaction at low energy, 1.166 378 7 × 10⁻⁵ GeV⁻² — measured from the muon's 2.2 microsecond lifetime.

Electromagnetic 19

Vacuum magnetic permeability measured

μ0=0.00000125663706127 N/A2 (H/m)\mu_{0} = 0.00000125663706127\ \text{N/A}^{2}\text{ (H/m)}

N/A² (H/m)How strongly a current magnetises empty space — the constant in Ampère's law, no longer exactly 4π×10⁻⁷ since the 2019 SI redefinition.

Vacuum electric permittivity measured

ε0=8.8541878188×10−12 F/m\varepsilon_{0} = 8.8541878188 \times 10^{-12}\ \text{F/m}

F/mThe electric constant of free space, setting the strength of Coulomb's law and the capacitance of every parallel-plate geometry.

Characteristic impedance of vacuum measured

Z0=376.730313412 ΩZ_{0} = 376.730313412\ \text{Ω}

ΩThe ratio of electric to magnetic field strength in a plane wave in free space — the 377 ohms every antenna engineer matches to.

Coulomb constant measured

ke=8987551786.2 N⋅m2/C2k_{e} = 8987551786.2\ \text{N}{\cdot}\text{m}^{2}\text{/C}^{2}

N·m²/C²The proportionality constant 1/(4πε₀) in Coulomb's law, fixing the enormous strength of the electrostatic force between charges.

Bohr magneton measured

μB=9.2740100657×10−24 J/T\mu_{\mathrm{B}} = 9.2740100657 \times 10^{-24}\ \text{J/T}

J/TThe natural quantum of magnetic moment for an electron, eħ/2mₑ — the yardstick for atomic magnetism and electron spin.

Nuclear magneton measured

μN=5.0507837393×10−27 J/T\mu_{\mathrm{N}} = 5.0507837393 \times 10^{-27}\ \text{J/T}

J/TThe magnetic-moment unit for nuclei, eħ/2mₚ — smaller than the Bohr magneton by the full proton-to-electron mass ratio of 1836.

Electron magnetic moment measured

μe=−9.2847646917×10−24 J/T\mu_{\mathrm{e}} = -9.2847646917 \times 10^{-24}\ \text{J/T}

J/TThe magnetic moment of a free electron, negative because its charge is, and about 0.116% larger than one Bohr magneton.

Proton magnetic moment measured

μp=1.41060679545×10−26 J/T\mu_{\mathrm{p}} = 1.41060679545 \times 10^{-26}\ \text{J/T}

J/TThe magnetic moment of the proton, 2.79 nuclear magnetons rather than the 1 a point particle would show — evidence of quark structure.

Neutron magnetic moment measured

μn=−9.6623653×10−27 J/T\mu_{\mathrm{n}} = -9.6623653 \times 10^{-27}\ \text{J/T}

J/TA neutral particle with a magnetic moment of −1.913 nuclear magnetons — proof on its own that the neutron has charged internal structure.

Magnetic flux quantum exact

Φ0=2.067833848461929×10−15 Wb\Phi_{0} = 2.067833848461929 \times 10^{-15}\ \text{Wb}

WbThe smallest unit of magnetic flux that can thread a superconducting loop, h/2e — exact since the 2019 SI redefinition fixed h and e.

Josephson constant exact

KJ=4.835978484169836×1014 Hz/VK_{\mathrm{J}} = 4.835978484169836 \times 10^{14}\ \text{Hz/V}

Hz/VThe frequency-to-voltage ratio 2e/h of a Josephson junction — 483.6 THz per volt, and the modern practical realisation of the volt.

von Klitzing constant exact

RK=25812.8074593045 ΩR_{\mathrm{K}} = 25812.8074593045\ \text{Ω}

ΩThe quantum Hall resistance h/e² ≈ 25.813 kΩ, exact since 2019 and the reference by which the ohm is now realised worldwide.

Conductance quantum exact

G0=0.00007748091729863649 SG_{0} = 0.00007748091729863649\ \text{S}

SThe conductance 2e²/h of a single ballistic quantum channel, about 77.5 μS — the step size in nanoscale wires and atomic point contacts.

Inverse conductance quantum exact

G0−1=12906.40372965225 ΩG_{0}^{-1} = 12906.40372965225\ \text{Ω}

ΩHalf the von Klitzing constant, h/2e² ≈ 12.906 kΩ — the resistance of one perfect ballistic channel and the floor for any nanoscale wire.

Proton gyromagnetic ratio measured

γp=267522187.08 s−1⋅T−1\gamma_{\mathrm{p}} = 267522187.08\ \text{s}^{-1}{\cdot}\text{T}^{-1}

s⁻¹·T⁻¹The proton's precession rate per unit magnetic field in angular frequency — the constant that turns a magnet strength into an NMR frequency.

Proton gyromagnetic ratio over 2π measured

γp/2π=42577478.461 Hz/T\gamma_{\mathrm{p}}/2\pi = 42577478.461\ \text{Hz/T}

Hz/TThe proton Larmor frequency per tesla, 42.577 MHz/T — the number every NMR spectroscopist and MRI physicist works in directly.

Electron gyromagnetic ratio measured

γe=1.76085962784×1011 s−1⋅T−1\gamma_{\mathrm{e}} = 1.76085962784 \times 10^{11}\ \text{s}^{-1}{\cdot}\text{T}^{-1}

s⁻¹·T⁻¹The electron's spin precession rate per tesla, 658 times the proton's — the basis of electron spin resonance and of spin-qubit control.

Speed of light in water measured

cH2O=224,900,000 m/sc_{\mathrm{H_2O}} = 224,900,000\ \text{m/s}

m/sLight travels through water at about 225,000 km/s, or c/1.333 — the slowing that bends a straw at the waterline and lets Cherenkov detectors work.

Speed of light in glass measured

cglass=197,230,000 m/sc_{\mathrm{glass}} = 197,230,000\ \text{m/s}

m/sTypical propagation speed in crown glass, c/1.52 or about 197,000 km/s — the delay that makes lenses focus and optical fibres carry data.