In physics, Planck scale is the fundamental scale, named after the German physicist Max Planck, who first proposed the ‘’Planck mass’’ in 1899. The electric coupling constant at the Planck scale equals to 1:
- is the Planck charge;
- is the elementary charge;
- is the fine structure constant;
- is the Planck constant;
- is the speed of light in vacuum;
- is the electric constant.
The natural units began in 1881, when George Johnstone Stoney derived units of length, time, and mass, now named Stoney units in his honor, by normalizing G, c, and the elementary charge e to 1. (Stoney was also the first to hypothesize that electric charge is quantized and hence to see the fundamental character of e.) Max Planck first set out the base units (qP excepted) later named in his honor, in a paper presented to the Prussian Academy of Sciences in May 1899. That paper also includes the first appearance of the Planck constant named b, and later called h and named after him. The paper gave numerical values for the base units, in terms of the metric system of his day, that were remarkably close to modern values. We are not sure just how Planck came to discover these units because his paper gave no algebraic details. But he did explain why he valued these units as follows:
...ihre Bedeutung fur alle Zeiten und fur alle, auch au?erirdische und au?ermenschliche Kulturen notwendig behalten und welche daher als »naturliche Ma?einheiten« bezeichnet werden konnen...
...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"...
Fundamental units of vacuum
The set of secondary vacuum constants is: The vacuum permeability: ;
The electromagnetic impedance of free space:
The gravitoelectric gravitational constant: ;
The gravitomagnetic gravitational constant: ;
The above fundamental constants define naturally the following relationship between mass and electric charge:
and these values are the base units of the Planck scale.
Primary Planck units
Gravitational Planck units
where is the Stoney mass.
Planck gravitational coupling constant:
Planck fictitious gravitational torsion mass:
Planck scale gravitational torsion coupling constant: 
Planck gravitational impedance quantum:
Electromagnetic Planck units
Planck electric coupling constant:
Planck fictitious magnetic charge:
Planck scale magnetic coupling constant:
where is the magnetic coupling constant.
Planck electrodynamic impedance quantum:
Secondary Planck scale units
All systems of measurement feature is base units: in the International System of Units (SI), for example, the base unit of length is the meter. In the system of Planck units, the Planck base unit of length is known simply as the ‘’ Planck length’’, the base unit of time is the ‘’ Planck time’’, and so on. These units are derived from the presented above primary Planck units, and arranged in Table 1 so as to cancel out the unwanted dimensions, leaving only the dimension appropriate to each unit. (Like all systems of natural units, Planck units are an instance of dimensional analysis.)
|Name||Dimension||Expressions||SI equivalent |
|Planck wavelength||Length (L)||m|
|Planck time||Time (T)||s|
|Planck classical radius||Length (L)||m|
|Planck Schwarzschild radius||Length (L)||m|
|Planck temperature||Temperature (Θ)||K|
Planck scale forces
Planck scale static forces
Electric Planck scale force:
where is the Planck electric coupling constant.
Gravity Planck scale force:
where is the Planck gravitational coupling constant.
Mixed (charge-mass interaction) Planck force:
So, at the Planck scale we have the equality of all static forces which describes interactions between charges and masses:
Planck scale dynamic forces
Magnetic Planck scale force:
where is the magnetic Planck coupling constant.
Gravitational torsion force:
where is the Planck gravitational torsion coupling constant.
Mixed dynamic (charge-mass interaction) force:
So, at the Planck scale we have the equality of all dynamic forces which describes interactions between dynamic charges and masses:
- Stoney scale
- Planck mass
- Selfconsistent gravitational constants
- Selfconsistent electromagnetic constants
- Maxwell-like gravitational equations
- Quantum Gravitational Resonator
- Planck (1899), p. 479.
- Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System," 287-96.
- Latest (2010) values of the constants 
- Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p.91. ISBN 5-11-002309-3. djvu.
- Planck, Max (1899). "Uber irreversible Strahlungsvorgange". Sitzungsberichte der Koniglich Preu?ischen Akademie der Wissenschaften zu Berlin 5: 440–480. http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1899-1&seite:int=454. Pp. 478-80 contain the first appearance of the Planck base units other than the Planck charge, and of Planck's constant, which Planck denoted by b. a and f in this paper correspond to k and G in this entry.