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:

$q_{\text{P}}={\sqrt {4\pi \epsilon _{0}\hbar c}}={\sqrt {2ch\epsilon _{0}}}={\frac {e}{\sqrt {\alpha }}}=1.875\;5459\times 10^{-18}$ 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 greater 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 $\alpha =e^{2}$.

See also

• Orders of magnitude (charge)