# 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.

The Planck charge is ${\displaystyle \alpha ^{-1/2}\approx 11.706}$ times larger than the elementary charge e carried by an electron. A Planck charge concentrated within a Planck length possesses an electric potential energy that equals (converted to mass) one Planck mass.

As an example, packing together twelve electrons (11.706 is a fractional number and cannot be used) will make them weigh 2.287×10−8 kg, slightly more than the Planck mass (2.176×10−8 kg), and roughly the mass of a flea egg, solely by virtue of their electric potential energy. Note that the rest mass of the electron alone is 9.109×10−31 kg, or 23 orders of magnitude smaller.

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 (ħ=1, 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 ${\displaystyle q'_{\text{P}}}$, i.e., when ${\displaystyle q'_{\text{P}}}$ is set equal to 1, we obtain

${\displaystyle \alpha ={\frac {e^{2}}{4\pi }}~,}$

which is commonly used in quantum field theory, so that e≅0.30282212088.

By contrast, in (non-rationalized) natural cgs units where ${\displaystyle q_{\text{P}}=1}$ we have ${\displaystyle \alpha =e^{2}}$.