Planck charge

In physics, the Planck charge, denoted by ${\displaystyle q_{\text{P}}}$, 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[1][2]

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

where

${\displaystyle c}$ is the speed of light in vacuum

${\displaystyle \hbar }$ is the reduced Planck constant

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

${\displaystyle e}$ is the elementary charge

${\displaystyle \alpha }$ is the fine structure constant.

From a classical calculation,[3] 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,

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

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,

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

where

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

${\displaystyle c}$ is the speed of light

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

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

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

${\displaystyle m_{\text{P}}}$ 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 ${\displaystyle \epsilon _{0}}$ without numerical factors, the resulting rationalized Planck charge is

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

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

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