# Planck charge

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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:[1] [2]

$q_\text{P} = \sqrt{4 \pi\epsilon_0 \hbar c} = \sqrt{2 c h \epsilon_0} = \frac{e}{\sqrt{\alpha}} = 1.875\;5459 \times 10^{-18}$ coulombs,

where:

$c \$ is the speed of light in the vacuum,
$h \$ is Planck's constant,
$\hbar \equiv \frac{h}{2 \pi} \$ is the reduced Planck constant,
$\epsilon_0 \$ is the permittivity of free space
$e \$ is the elementary charge
$\alpha \$ = (137.03599911)−1 is the fine structure constant.

The Planck charge is $\alpha^{-1/2} \approx 11.706$ times larger than the elementary charge e carried by an electron.

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 ($c=1$) they are like the SI system with $\epsilon_0 = \mu_0 = 1$. Therefore it is more appropriate to define the Planck charge as

$q'_\text{P} = \sqrt{\epsilon_0 \hbar c} = \frac{e}{\sqrt{4\pi\alpha}} = 5.291 \times 10^{-19}$ coulombs,

When charges are measured in units of $q'_\text{P}$, i.e., when $q'_\text{P}$ is set equal to 1, we obtain $\alpha = \frac{e^2}{4 \pi}$, which is commonly used in theoretical physics. In contrast, in (non-rationalized) natural cgs units where $q_\text{P}=1$ we have $\alpha = e^2$.

## Notes and references

1. ^ Stock, Michael; Witt, Thomas J (2006). "CPEM 2006 round table discussion 'Proposed changes to the SI'". Metrologia 43 (6): 583. Bibcode:2006Metro..43..583S. doi:10.1088/0026-1394/43/6/014.
2. ^ Pavšič, Matej (2001). The Landscape of Theoretical Physics: A Global View. Dordrecht: Kluwer Academic. pp. 347–352. ISBN 0-7923-7006-6.