# 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 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 the 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:

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

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

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

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

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

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

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

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