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.

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

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

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

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

See also

• Orders of magnitude (charge)