Streamlined Planck units

The rationalized Planck units form a system of units that is closely related to the Planck units . The difference is that the same natural constants do not have the numerical value 1.

In the original Planck units applies: . ${\displaystyle c=\hbar =G=k_{B}=k_{C}=1}$

In the rationalized Planck units: and . ${\displaystyle c=\hbar =k_{B}=1}$ ${\displaystyle G=k_{C}={\frac {1}{4\pi }}}$

While the conversion of physical quantities in the area of relativity theory is easy with the Planck units, the conversion of quantities in electrodynamics is easier with the rationalized Planck units .

Equations in comparison with other unit systems

As with all rationalized units , the -factors are omitted from the rationalized Planck units in the Maxwell equations . In the case of the rationalized Planck units, all of them are omitted or take the numerical value . ${\displaystyle 4\pi }$ ${\displaystyle c}$ ${\displaystyle 1}$

Maxwell's equations
Surname	formula	Constant K (or , ) in the following system of units: ${\displaystyle K_{a}}$ ${\displaystyle K_{b}}$
		SI	Planck units	Streamlined Planck units	Heaviside-Lorentz
Gaussian law	${\displaystyle {\vec {\nabla }}\cdot {\vec {E}}=K\rho }$	${\displaystyle {\frac {1}{\varepsilon _{0}}}}$	${\displaystyle 4\pi }$	${\displaystyle 1}$	${\displaystyle 1}$
Gaussian law for magnetic fields	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$	-	-	-	-
Induction law	${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-K{\frac {\partial {\vec {B}}}{\partial t}}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle c^{-1}}$
Flooding Act	${\displaystyle {\vec {\nabla }}\times {\vec {B}}=K_{a}{\vec {J}}+K_{b}{\frac {\partial {\vec {E}}}{\partial t}}}$	${\displaystyle \mu _{0}}$, ${\displaystyle c^{-2}}$	${\displaystyle 4\pi }$, ${\displaystyle 1}$	${\displaystyle 1}$, ${\displaystyle 1}$	${\displaystyle c^{-1}}$, ${\displaystyle c^{-1}}$
Force equations
Coulomb's law	${\displaystyle K_{C}{\frac {q_{1}q_{2}}{r^{2}}}}$	${\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{4\pi }}}$	${\displaystyle {\frac {1}{4\pi }}}$
Newton's law of gravitation	${\displaystyle K_{N}{\frac {m_{1}m_{2}}{r^{2}}}}$	${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle {\frac {1}{4\pi }}}$	${\displaystyle G}$
Electrical component calculations
resistance	${\displaystyle R=\rho {\frac {l}{A}}}$	-	-	-	-
Disc capacitor	${\displaystyle C=K\varepsilon _{r}{\frac {A}{l}}}$	${\displaystyle \varepsilon _{0}}$	${\displaystyle {\frac {1}{4\pi }}}$	${\displaystyle 1}$	${\displaystyle 1}$
Cylinder coil	${\displaystyle L=K\mu _{r}{\frac {N^{2}A}{l}}}$	${\displaystyle \mu _{0}}$	${\displaystyle 4\pi }$	${\displaystyle 1}$	${\displaystyle 1}$

definition

The rationalized Planck units are defined by the following natural constants :

constant	symbol	dimension	Size in SI units
Speed ​​of Light	${\displaystyle c}$	${\displaystyle LT^{-1}}$	2.99792 · 10 8 m s −1
Gravitational constant	${\displaystyle G}$	${\displaystyle L^{3}M^{-1}T^{-2}}$	6.67430 · 10 −11 m 3 kg −1 s −2
Reduced Planck's quantum of action	${\displaystyle \hbar }$	${\displaystyle L^{2}MT^{-1}}$	1.05457 · 10 −34 J s
Electric field constant	${\displaystyle \varepsilon _{0}}$	${\displaystyle Q^{2}T^{2}M^{-1}L^{-3}}$	8.85419 · 10 9 C 2 s 2 kg −1 m −3
Boltzmann's constant	${\displaystyle k_{B}}$	${\displaystyle L^{2}M\Theta ^{-1}T^{-2}}$	1.38065 · 10 −23 J k −1

Basic sizes

By considering the dimensions of these constants , the basic quantities can be calculated:

Surname	size	dimension	term	Value in SI units
Rational mass	Dimensions	${\displaystyle M}$	${\displaystyle m_{R}={\sqrt {\frac {\hbar c}{(4\pi G)}}}}$	6.1396166 10 −9 kg
Rational length	length	${\displaystyle L}$	${\displaystyle l_{R}={\sqrt {\frac {(4\pi G)\hbar }{c^{3}}}}}$	5.7294657 10 −35 m
Rational time	time	${\displaystyle T}$	${\displaystyle t_{R}={\sqrt {\frac {(4\pi G)\hbar }{c^{5}}}}}$	1.9111441 10 −43 s
Rational charge	charge	${\displaystyle Q}$	${\displaystyle q_{R}={\sqrt {\hbar c\varepsilon _{0}}}}$	5.2908176 10 −19 C
Rational temperature	temperature	${\displaystyle \Theta }$	${\displaystyle T_{R}={\sqrt {\frac {\hbar c^{5}}{(4\pi G)k_{B}^{2}}}}}$	3.9966750 10 31 K

Derived quantities

All other sizes can be calculated from the basic sizes:

Surname	size	dimension	term	Value in SI units
Mechanical units
Rational area	surface	${\displaystyle L^{2}}$	${\displaystyle A_{R}=l_{R}^{2}={\frac {(4\pi G)\hbar }{c^{3}}}}$	3.2826778 10 −69 m 2
Rational volume	volume	${\displaystyle L^{3}}$	${\displaystyle V_{R}=l_{R}^{3}=\left({\sqrt {\frac {(4\pi G)\hbar }{c^{3}}}}\right)^{3}}$	1.8807990 · 10 −103 m 3
Rational density	density	${\displaystyle {\frac {M}{L^{3}}}}$	${\displaystyle d_{R}={\frac {m_{R}}{V_{R}}}={\frac {c^{5}}{(4\pi G)^{2}\hbar }}}$	3.2643661 10 94 kg m −3
Rational frequency	frequency	${\displaystyle {\frac {1}{T}}}$	${\displaystyle f_{R}={\frac {1}{t_{R}}}={\sqrt {\frac {c^{5}}{(4\pi G)\hbar }}}}$	5.2324679 · 10 42 Hz
Speed ​​of Light	speed	${\displaystyle {\frac {L}{T}}}$	${\displaystyle v_{R}={\frac {l_{R}}{t_{r}}}=c}$	2.9979246 · 10 8 m s −1
Rational acceleration	acceleration	${\displaystyle {\frac {L}{T^{2}}}}$	${\displaystyle a_{R}={\frac {v_{R}}{t_{R}}}={\sqrt {\frac {c^{7}}{(4\pi G)\hbar }}}}$	1.56868 × 10 51 m s −2
Rational force	force	${\displaystyle {\frac {ML}{T^{2}}}}$	${\displaystyle F_{R}={\frac {m_{R}l_{R}}{t_{R}^{2}}}={\frac {c^{4}}{4\pi G}}}$	9.6309366 · 10 42 N.
Rational energy	energy	${\displaystyle {\frac {L^{2}M}{T^{2}}}}$	${\displaystyle E_{R}=m_{R}v_{R}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}}$	5.5180121 × 10 8 J
Rational performance	power	${\displaystyle {\frac {L^{2}M}{T^{3}}}}$	${\displaystyle P_{R}={\frac {E_{R}}{t_{R}}}={\frac {c^{5}}{4\pi G}}}$	2.8872822 · 10 51 W.
Rational pressure	pressure	${\displaystyle {\frac {M}{LT^{2}}}}$	${\displaystyle \Pi _{R}={\frac {F_{R}}{A_{R}}}={\frac {c^{7}}{(4\pi G)^{2}\hbar }}}$	2.93404 · 10 111 Pa
Electromagnetic units
Rational electric current	Electrical current	${\displaystyle {\frac {Q}{T}}}$	${\displaystyle i_{R}={\frac {q_{R}}{t_{R}}}={\sqrt {\frac {c^{6}\varepsilon _{0}}{4\pi G}}}}$	2.76840312 · 10 24 A.
Rational electrical potential	Electrical potential	${\displaystyle {\frac {L^{2}M}{QT^{2}}}}$	${\displaystyle U_{R}={\frac {P_{R}}{i_{R}}}={\sqrt {\frac {c^{4}}{(4\pi G)\varepsilon _{0}}}}}$	1.04294138 10 27 V.
Vacuum impedance	Electrical resistance	${\displaystyle {\frac {L^{2}M}{TQ^{2}}}}$	${\displaystyle Z_{R}={\frac {U_{R}}{i_{R}}}=Z_{0}}$	3.7673031 · 10 2 Ω
Rational capacity	capacity	${\displaystyle {\frac {T^{2}Q^{2}}{ML^{2}}}}$	${\displaystyle C_{R}={\frac {q_{R}}{U_{R}}}={\sqrt {\frac {(4\pi G)\hbar \varepsilon _{0}^{2}}{c^{3}}}}}$	5.0729766 · 10 −46 F
Rational inductance	Inductance	${\displaystyle {\frac {L^{2}M}{Q^{2}}}}$	${\displaystyle L_{R}=Z_{R}\cdot t_{R}={\sqrt {\frac {(4\pi G)\hbar }{c^{7}\varepsilon _{0}^{2}}}}}$	7.1998591 · 10 −41 H.
Rational magnetic flux	Magnetic river	${\displaystyle {\frac {L^{2}M}{TQ}}}$	${\displaystyle \Psi _{R}=U_{R}\cdot t_{R}={\sqrt {\frac {\hbar }{c\varepsilon _{0}}}}}$	1.9932114 · 10 −16 Wb
Rational magnetic flux density	Magnetic flux density	${\displaystyle {\frac {M}{TQ}}}$	${\displaystyle B_{R}={\frac {\Psi }{A_{R}^{2}}}={\sqrt {\frac {c^{5}}{(4\pi G)^{2}\hbar \varepsilon _{0}}}}}$	6.07208 · 10 52 T
Thermal units
Rational heat capacity	Heat capacity	${\displaystyle {\frac {L^{2}M}{T^{2}\Theta }}}$	${\displaystyle \Gamma _{R}={\frac {E_{R}}{T_{R}}}=k_{B}}$	1.3806490 · 10 −23 J K −1
Rational thermal resistance	Thermal resistance	${\displaystyle {\frac {T^{3}\Theta }{L^{2}M}}}$	${\displaystyle R_{R}={\frac {T_{R}}{P_{R}}}={\sqrt {\frac {(4\pi G)\hbar }{c^{5}k_{B}^{2}}}}}$	1.3842346 10 −20 W K −1

Values ​​of physical numbers

Physical number	value
Properties of the earth
Day	4.52086 10 47 t R
year	1.65127 · 10 50 t R
Age	7.49677 10 59 t R
Dimensions	9.72741 x 10 32 m R
acceleration	6.25153 · 10 −51 a R
volume	5.75931 · 10 123 V R
Equatorial radius	1.11321 · 10 41 l R
Normal atmosphere	3.45342 · 10 −107 Π R
Surface temperature	7.20598 · 10 −30 T R
Properties of water
Melting point at normal pressure	6.83456 · 10 −30 T R
Boiling point at normal pressure	9.33651 · 10 −30 T R
Temperature at triple point	6.83468 · 10 −30 T R
Pressure at the triple point	2.08469 · 10 −112 Π R
Temperature at the critical point	1.61910 · 10 −29 T R
Pressure at the critical point	7.52001 · 10 −110 Π R
Properties of elementary particles
Elemental charge	3.02822 · 10 −1 q R
Electron mass	1.48371 10 −22 m R
Proton mass	2.72431 · 10 −19 m R
Neutron mass	2.72807 · 10 −19 m R
Compton wavelength of electrons	4.23479 · 10 22 l R
Compton wavelength of protons	2.30634 · 10 19 l R
Compton wavelength of neutrons	2.30316 · 10 19 l R
Properties of various elements
Bohr radius	9.23607 · 10 23 l R
Covalent radius of hydrogen	5.41062 · 10 23 l R
Van der Waals radius of hydrogen	2.09443 · 10 24 l R
Atomic unit of measure	2.70463 · 10 −19 m R
Mass of 1 H	2.72626 · 10 −19 m R
Mean lifetime of a neutron	4.60561 · 10 −45 t R
Half-life of 3 H	2.03433 * 10 51 t R
Half-life of 8 B	3.51726 * 10 26 t R