Planck charge

In physics, the Planck charge, denoted by $q_{\text{P}}$, is one of the base units in the system of natural units called Planck units. It is a quantity of electric charge defined in terms of fundamental physical constants.

The Planck charge is the only base Planck unit that does not depend on the gravitational constant; it is defined as

$q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}={\frac {e}{\sqrt {\alpha }}}\approx 1.875\;5459\times 10^{-18}$ coulombs,

where

$c$ is the speed of light in vacuum

$\hbar$ is the reduced Planck constant

$\epsilon _{0}$ is the permittivity of free space

$e$ is the elementary charge

$\alpha$ is the fine structure constant.

From a classical calculation, the electric potential energy of one Planck charge on the surface of a sphere that is one Planck length in diameter is one Planck energy,

$E_{\text{P}}=k_{\text{e}}{\frac {q_{\text{P}}^{2}}{l_{\text{P}}}}.$

In other words, the energy required to accumulate one Planck charge on a sphere one Planck length in diameter will make the sphere one Planck mass heavier,

$m_{\text{P}}=k_{\text{e}}{\frac {q_{\text{P}}^{2}}{c^{2}l_{\text{P}}}},$

where

$k_{\text{e}}$ is the Coulomb constant

$c$ is the speed of light

$E_{\text{P}}$ is the Planck energy

$q_{\text{P}}$ is the Planck charge

$l_{\text{P}}$ is the Planck length

$m_{\text{P}}$ is the Planck mass

Rationalized units: If, instead, a rationalized form of Planck units is chosen, in which units are defined in terms of ℏ, c and $\epsilon _{0}$ without numerical factors, the resulting rationalized Planck charge is

$q'_{\text{P}}={\sqrt {\epsilon _{0}\hbar c}}={\frac {e}{\sqrt {4\pi \alpha }}}\approx 5.291\times 10^{-19}$ coulombs.

When charges are measured in units of $q'_{\text{P}}$, used in quantum field theory, one has

$e={\sqrt {4\pi \alpha }}\approx 0.30282212~q_{\text{P}}'$.

Physical significance

The Planck charge is the maximum amount of charge that a black hole the size of one Planck length can possess, and adding more charge would make the black hole inevitably larger. In particular, Reissner–Nordström metric (the solution for a non-rotating charged black hole) tends to the Planck length for a mass that tends to zero and a charge that equals the Planck charge.