Planck Units — Complete Reference

The Planck units are nature’s own units of length, mass, time, and charge. Their values are calculated from three measured constants: the gravitational constant $G$, the reduced Planck constant $\hbar$, and the speed of light $c$. For electromagnetic quantities, the permittivity of free space $\varepsilon_0$ is also required. Embedded in these four constants is a unique scale at which quantum mechanics, general relativity, and relativistic mechanics all become simultaneously significant.

At the Planck energy, the reduced Compton wavelength of a particle becomes comparable to its Schwarzschild radius — the boundary where quantum field theory and general relativity can no longer be treated separately. This makes the Planck scale not merely a convenient choice of units, but a physically meaningful reference scale.

This page collects every standard Planck unit in one place: the four base units, derived mechanical and electromagnetic units, and the exact relationships connecting them to the fundamental constants. Each physical constant can be expressed as a product of Planck units and dimensionless numbers — a decomposition that reveals the dimensional structure underlying all of physics.

Base Planck units

Four independent Planck units form the base of the system. The values of the first three — length, mass, and time — are customarily calculated from $G$, $\hbar$, and $c$ alone. Calculating the charge unit also requires $\varepsilon_0$.

QuantitySymbolFormulaValue (SI)Char.
Planck length$l_{\mathrm{P}}$$\sqrt{\hbar G/c^{3}}$$1.616\,255 \times 10^{-35}$ mL
Planck mass$m_{\mathrm{P}}$$\sqrt{\hbar c/G}$$2.176\,434 \times 10^{-8}$ kgM
Planck time$t_{\mathrm{P}}$$\sqrt{\hbar G/c^{5}}$$5.391\,247 \times 10^{-44}$ sT
Planck charge$q_{\mathrm{P}}$$\sqrt{4\pi\varepsilon_{0}\hbar c}$$1.875\,546 \times 10^{-18}$ CQ
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Mutual consistency
The three mechanical base units satisfy $l_{\mathrm{P}} = c\, t_{\mathrm{P}}$ and $E_{\mathrm{P}} = m_{\mathrm{P}} c^{2}$. Given any one of $l_{\mathrm{P}}$, $m_{\mathrm{P}}$, or $t_{\mathrm{P}}$ together with $\hbar$ and $c$, the other two follow directly. Planck charge is independent — it requires $\varepsilon_{0}$ (or equivalently $e$ and $\alpha$) in addition to $G$, $\hbar$, and $c$.

Derived Planck units

These units can be expressed in terms of the base units or calculated directly from the fundamental constants.

QuantitySymbolFormulaValue (SI)Char.
Planck energy$E_{\mathrm{P}}$$m_{\mathrm{P}} c^{2} = \sqrt{\hbar c^{5}/G}$$1.956 \times 10^{9}$ JE
Planck momentum$m_{\mathrm{P}} c$$m_{\mathrm{P}} c$$6.525$ kg·m/sMv
Planck force$F_{\mathrm{P}}$$c^{4}/G = E_{\mathrm{P}}/l_{\mathrm{P}}$$1.210 \times 10^{44}$ NF
Planck power$P_{\mathrm{P}}$$c^{5}/G = E_{\mathrm{P}}/t_{\mathrm{P}}$$3.628 \times 10^{52}$ WP
Planck acceleration$a_{\mathrm{P}}$$c/t_{\mathrm{P}} = \sqrt{c^{7}/(\hbar G)}$$5.561 \times 10^{51}$ m/s$^{2}$a
Planck density$\rho_{\mathrm{P}}$$m_{\mathrm{P}}/l_{\mathrm{P}}^{3} = c^{5}/(\hbar G^{2})$$5.155 \times 10^{96}$ kg/m$^{3}$M/L3
Planck pressure$P_{\mathrm{P,p}}$$F_{\mathrm{P}}/l_{\mathrm{P}}^{2} = c^{7}/(\hbar G^{2})$$4.633 \times 10^{113}$ PaF/A
Planck voltage$V_{\mathrm{P}}$$E_{\mathrm{P}}/q_{\mathrm{P}} = \sqrt{c^{4}/(4\pi\varepsilon_{0} G)}$$1.043 \times 10^{27}$ VE/Q
Planck current$I_{\mathrm{P}}$$q_{\mathrm{P}}/t_{\mathrm{P}} = \sqrt{4\pi\varepsilon_{0} c^{6}/G}$$3.479 \times 10^{25}$ AQ/T
Planck impedance$Z_{\mathrm{P}}$$V_{\mathrm{P}}/I_{\mathrm{P}} = 1/(4\pi\varepsilon_{0} c) = m_{\mathrm{P}} l_{\mathrm{P}}^{2}/(t_{\mathrm{P}} q_{\mathrm{P}}^{2})$$29.98$ ΩV/I
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Notable
The Planck force $F_{\mathrm{P}} = c^{4}/G$ and the Planck power $P_{\mathrm{P}} = c^{5}/G$ depend on neither $\hbar$ nor $q_{\mathrm{P}}$ — they are purely classical-gravitational. The Planck force appears as the universal force scale in both Newton’s gravitational law and Coulomb’s law when these are expressed in Planck units.

Fundamental constants as Planck composites

Each fundamental constant can be expressed as a product of Planck units and dimensionless numbers. The following table collects these decompositions.

ConstantPlanck decompositionValue
Speed of light $c$$\dfrac{l_{\mathrm{P}}}{t_{\mathrm{P}}}$$299{,}792{,}458$ m/s
Reduced Planck constant $\hbar$$m_{\mathrm{P}}\, l_{\mathrm{P}}\, c = \dfrac{m_{\mathrm{P}}\, l_{\mathrm{P}}^{2}}{t_{\mathrm{P}}}$$1.054\,571\,817 \times 10^{-34}$ J·s
Gravitational constant $G$$\dfrac{l_{\mathrm{P}}^{3}}{m_{\mathrm{P}}\, t_{\mathrm{P}}^{2}} = \dfrac{l_{\mathrm{P}}}{m_{\mathrm{P}}}\, c^{2}$$6.674\,30 \times 10^{-11}$ m$^{3}$ kg$^{-1}$ s$^{-2}$
Permittivity $\varepsilon_{0}$$\dfrac{t_{\mathrm{P}}^{2} q_{\mathrm{P}}^{2}}{4\pi\, l_{\mathrm{P}}^{3} m_{\mathrm{P}}}$$8.854 \times 10^{-12}$ F/m
Permeability $\mu_{0}$$\dfrac{4\pi\, m_{\mathrm{P}} l_{\mathrm{P}}}{q_{\mathrm{P}}^{2}}$$1.2566 \times 10^{-6}$ H/m
Boltzmann constant $k_{\mathrm{B}}$Unit translator, not a Planck composite: $k_{\mathrm{B}}$ converts kelvin into joules, and temperature enters the framework as the energy $k_{\mathrm{B}}T$. There is no Planck temperature among the base units.$1.380\,649 \times 10^{-23}$ J/K
Elementary charge $e$$\sqrt{\alpha}\cdot q_{\mathrm{P}}$$1.602\,176\,634 \times 10^{-19}$ C
Fine-structure constant $\alpha$$\left(\dfrac{e}{q_{\mathrm{P}}}\right)^{2}$$1/137.035\,999\,18$
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Key observation
The fundamental constants of physics can be expressed systematically in terms of four dimensional scales (the base Planck units) plus a single dimensionless number ($\alpha$). The constants $c$, $\hbar$, and $G$ each carry a coefficient of exactly $1$ — they are pure Planck composites, containing nothing but Planck units. The electromagnetic constants carry factors of $4\pi$ (a geometric convention) and $q_{\mathrm{P}}^{2}$. Only $e$ and $\alpha$ introduce a number that is not determined by the system itself.

Cross-calculation formulas

Given any one of the three mechanical base Planck units, the other two can be recovered using $\hbar$ and $c$.

From Planck length $l_{\mathrm{P}}$
$$m_{\mathrm{P}} = \frac{\hbar}{l_{\mathrm{P}}\, c} \qquad t_{\mathrm{P}} = \frac{l_{\mathrm{P}}}{c}$$
From Planck mass $m_{\mathrm{P}}$
$$l_{\mathrm{P}} = \frac{\hbar}{m_{\mathrm{P}}\, c} \qquad t_{\mathrm{P}} = \frac{\hbar}{m_{\mathrm{P}}\, c^{2}}$$
From Planck time $t_{\mathrm{P}}$
$$l_{\mathrm{P}} = c\, t_{\mathrm{P}} \qquad m_{\mathrm{P}} = \frac{\hbar}{c^{2}\, t_{\mathrm{P}}}$$
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Charge is independent
The Planck charge $q_{\mathrm{P}}$ cannot be recovered from $l_{\mathrm{P}}$, $m_{\mathrm{P}}$, or $t_{\mathrm{P}}$ alone — it requires $\varepsilon_{0}$ (or equivalently $\alpha$ and $e$). Charge enters the system as a fourth base unit; the electromagnetic constants carry the additional information.
Numerical verification — from $l_{\mathrm{P}}$
$m_{\mathrm{P}} = \hbar/(l_{\mathrm{P}}\, c)$$1.0546 \times 10^{-34} / (1.6163 \times 10^{-35} \times 2.9979 \times 10^{8})$$2.176 \times 10^{-8}$ kg
$t_{\mathrm{P}} = l_{\mathrm{P}}/c$$1.6163 \times 10^{-35} / 2.9979 \times 10^{8}$$5.391 \times 10^{-44}$ s

Key relationships

These relationships reveal the structural connections within the Planck unit system.

Length–time link via $c$
$$l_{\mathrm{P}} = c\, t_{\mathrm{P}}$$

Space and time are related by the same conversion factor at the Planck scale as at any other scale.

Mass–energy via $c^{2}$
$$E_{\mathrm{P}} = m_{\mathrm{P}}\, c^{2}$$

The rest energy of one Planck mass. The same mass-energy equivalence that holds for any particle.

Classical Planck unit (no $\hbar$)
$$F_{\mathrm{P}} = \frac{c^{4}}{G}$$

The Planck force depends only on $c$ and $G$ — a purely relativistic-gravitational quantity with no quantum content.

Quantum action
$$\hbar = m_{\mathrm{P}}\, l_{\mathrm{P}}\, c = E_{\mathrm{P}}\, t_{\mathrm{P}}$$

Action can be expressed as Planck energy times Planck time, or Planck mass times Planck length times the speed of light.

Electromagnetic scale
$$q_{\mathrm{P}}^{2} = 4\pi\varepsilon_{0}\, \hbar c = 4\pi\varepsilon_{0}\, E_{\mathrm{P}}\, l_{\mathrm{P}}$$

The Planck charge squared links the electromagnetic coupling ($\varepsilon_{0}$) to the quantum-relativistic scale ($\hbar c$).

Quantum geometric product
$$l_{\mathrm{P}} \cdot m_{\mathrm{P}} = \frac{\hbar}{c}$$

The product of Planck length and Planck mass is a fixed quantity, independent of $G$. The same product appears in the Compton wavelength: $\bar{\lambda}_{C} = l_{\mathrm{P}}\cdot m_{\mathrm{P}}/m$.

Scale context

Where do Planck units sit relative to the scales of everyday physics?

Energy ratios across regimes — the same dimensionless comparison
RegimeTypical energyRatio to $E_{\mathrm{P}}$Physics
Atomic$\sim 1$ eV$\sim 10^{-28}$Quantum mechanics + electromagnetism; gravity negligible
Nuclear$\sim 1$ MeV$\sim 10^{-22}$Strong and weak forces; quantum field theory; gravity still negligible
Particle physics$\sim 100$ GeV$\sim 10^{-17}$Electroweak unification; spacetime still smooth
GUT scale$\sim 10^{16}$ GeV$\sim 10^{-3}$Possible gauge coupling unification
Planck scale$\sim E_{\mathrm{P}} \approx 10^{19}$ GeV$1$Quantum gravity; expected breakdown of classical spacetime; all forces comparable

All observed particle masses and interaction energies lie many orders of magnitude below the Planck scale. Why nature operates so far from the Planck energy — the hierarchy problem — remains one of the deepest unsolved questions in physics.

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Constant profiles
Speed of light · Gravitational constant · Fine-structure constant · Planck’s constant · Elementary charge · Magnetic constant · Rydberg constant · Vacuum impedance · Einstein gravitational constant · Planck charge · Electric constant