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.

See also[edit]

Notes and references[edit]

  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.