# 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 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 the 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:

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

Or, to put it in different words, the energy required to pile up one Planck charge within a sphere of one Planck length in diameter will make the sphere one Planck mass heavier:

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

where

${\displaystyle E_{\text{P}}}$ is the Planck energy
${\displaystyle k_{\text{e}}}$ is the Coulomb constant
${\displaystyle q_{\text{P}}}$ is the Planck charge
${\displaystyle l_{\text{P}}}$ is the Planck length
${\displaystyle m_{\text{P}}}$ is the Planck mass

The Gaussian cgs units are defined so that ${\displaystyle 4\pi \epsilon _{0}=1}$, in which case ${\displaystyle q_{\text{P}}}$ has the following simple form,

${\displaystyle q_{\text{P}}={\sqrt {\hbar c}}.}$

It is customary in theoretical physics to adopt the Lorentz–Heaviside units (also known as rationalized cgs). When made natural (${\displaystyle \hbar =1}$, ${\displaystyle c=1}$), they are like the SI system with ${\displaystyle \epsilon _{0}=\mu _{0}=1}$. Therefore, it is more appropriate to instead define the Planck charge as

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

When charges are measured in units of ${\displaystyle q'_{\text{P}}}$, which is commonly used in quantum field theory, we have

${\displaystyle e\approx 0.30282212088~q_{\text{P}}'}$.