## FANDOM

162 Pages

In physics, natural units are physical units of measurement based only on universal physical constants. For example, the elementary charge e is a natural unit of electric charge, and the speed of light c is a natural unit of speed.

## Fundamental units

Fundamental units:

### Electromagnetism

The total energy in the electric field surrounding a hollow spherical shell of radius r and charge q is:

$E = k \frac{1}{2} \frac{q^2}{r}$

Therefore:

${\color{red}2E \cdot \frac{r}{q^2} } = k =constant$

The constant k is a property of space. It is the "stiffness" of space. (If space were stiffer then c would be faster.)

Coulomb's law states that:

$F=k_e \frac{q_1 q_2}{d^2}$

The Coulomb constant has units of Energy * distance/charge2 which gives:

$F = {\color{red}F \cdot d \frac{d}{q^2}} \frac{q_1 q_2}{d^2}$

The factor of 1/2 in the first equation above comes from the fact that the field diminishes to zero as it penetrates the shell.

### Gravity

Newton's law of universal gravitation states that:

$F = G \frac{m_1 m_2}{d^2}$

where:

$F = F \cdot d \frac{d}{m^2} \frac{m_1 m_2}{d^2}$

But its probably better to say that:

$a = \frac{d}{t^2} = G \frac{m}{d^2}\$

The obvious unit of charge is one electron but there is no obvious unit of mass. We can, however, create one by setting the electric force between two electrons equal to the gravitational force between two equal masses:

$G \frac{m_1 m_2}{d^2} = k_e\frac{q_1 q_2}{d^2}$

Solving we get m =

The Schwarzschild radius of a Stoney mass is 2 Stoney lengths.

### Boltzmann constant

Gas Specific
heat
ratio
Degrees
of
freedom
Helium 1.667 3
Neon 1.667 3
Argon 1.667 3
Hydrogen 1.597 3.35
Hydrogen 1.41 4.88
Nitrogen 1.4 5
Oxygen 1.395 5.06
Chlorine 1.34 5.88
Carbon dioxide 1.289 6.92
Methane 1.304 6.58
Ethane 1.187 10.7
Engineering ToolBox (2003)

For monatomic gases:

$P V^{\frac{5}{3}} = Constant$

The Boltzmann constant, k, is a scaling factor between macroscopic (thermodynamic temperature) and microscopic (thermal energy) physics.
Macroscopically, the ideal gas law states:

$k_B T = P \frac{V}{n}$

where:

• kB is the Boltzmann constant
• T is the temperature
• P is the pressure
• V is the volume
• n is the number of molecules of gas.

#### Single particle

The pressure exerted on one face of a cube of length d by a single particle bouncing back and forth perpendicular to the face with mass m and velocity $v = \sqrt{v_x + v_y + v_z}$ is:

$pressure = \frac{force}{area} = \frac{\frac{momentum}{time}}{d^2} = \frac{\frac{2 m v_x}{2 d / v_x}}{d^2} = \frac{m v_x^2}{d^3} = \frac{2 E_x}{V_0} = \frac{2 \frac{E}{3}}{V_0}$

where:

• V0 = d3 is the volume occupied by a single particle.
• vx is the velocity perpendicular to the face
• Twice the velocity means twice as much momentum transferred per collision and twice as many collisions per unit time.
• Ex is the kinetic energy per particle
• E = Ex +Ey + Ez

Therefore:

$V_0 = \frac{V}{n}$

Therefore:

$T = p \frac{V}{n} = p V_0 = m v^2 = 2 E$

Therefore temperature is twice the energy per degree of freedom per particle

• $T = 2 E$

Planck's law states that

$B_\nu(\nu, T) = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT} - 1},$

where

Bν(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T.
h is the Planck constant;
c is the speed of light in a vacuum;
k is the Boltzmann constant;
ν is the frequency of the electromagnetic radiation;
T is the absolute temperature of the body.

Most of the electromagnetic radiation is emitted (and absorbed) during the brief but intense acceleration's during the atomic collisions.

For velocities that are small relative to the speed of light, the total power radiated is given by the Larmor formula:

$P = {2 \over 3} \frac{q^2 a^2}{ 4 \pi \varepsilon_0 c^3}= \frac{q^2 a^2}{6 \pi \varepsilon_0 c^3} \mbox{ (SI units)}$

## Natural units

The surface area of a sphere $4 \pi r^2$

In Lorentz–Heaviside units (rationalized units), Coulomb's law is:

• $F=\frac{q_1 q_2}{r^2} \frac{1}{4 \pi}$

In Gaussian units (non-rationalized units), Coulomb's law is:

• $F=\frac{q_1 q_2}{r^2}$

Planck units are defined by

c = ħ = G = ke = kB = 1,

Stoney units are defined by:

c = G = ke = e = kB = 1,

Hartree atomic units are defined by:

e = me = ħ = ke = kB = 1
c = 1α

Rydberg atomic units are defined by:

e2 = 2me = ħ = ke = kB = 1
c = 2α

Quantum chromodynamics (QCD) units are defined by:

c = mp = ħ = kB = 1

Natural units generally means:

ħ = c = kB = 1.

where:

### Base units

Base units
Dimension Planck
(L-H)
Planck
(Gauss)
Stoney Hartree Rydberg Natural
(L-H)
Natural
(Gauss)
QCD
(Original)
QCD
(L-H)
QCD
(Gauss)
Length (L) $\sqrt{4 \pi \hbar G \over c^3}$ $\sqrt{\frac{\hbar G}{c^3}}$ $\sqrt{\frac{G k_\text{e} e^2}{c^4}}$ $\frac{\hbar^2 (4 \pi \epsilon_0)}{m_\text{e} e^2}$ $\frac{\hbar^2 (4 \pi \epsilon_0)}{m_\text{e} e^2}$ $\frac{\hbar c}{1\,\text{eV}}$ $\frac{\hbar c}{1\,\text{eV}}$ $\frac{\hbar}{m_\text{p} c}$ $\frac{\hbar}{m_\text{p} c}$ $\frac{\hbar}{m_\text{p} c}$
Time (T) $\sqrt{4 \pi \hbar G \over c^5}$ $\frac{\hbar}{m_\text{P}c^2} = \sqrt{\frac{\hbar G}{c^5}}$ $\sqrt{\frac{G k_\text{e} e^2}{c^6}}$ $\frac{\hbar^3 (4 \pi \epsilon_0)^2}{m_\text{e} e^4}$ $\frac{2 \hbar^3 (4 \pi \epsilon_0)^2}{m_\text{e} e^4}$ $\frac{\hbar}{1\,\text{eV}}$ $\frac{\hbar}{1\,\text{eV}}$ $\frac{\hbar}{m_\text{p} c^2}$ $\frac{\hbar}{m_\text{p} c^2}$ $\frac{\hbar}{m_\text{p} c^2}$
Mass (M) $\sqrt{\hbar c \over 4 \pi G}$ $\sqrt{\frac{\hbar c}{G}}$ $\sqrt{\frac{k_\text{e} e^2}{G}}$ $m_\text{e} \$ $2 m_\text{e} \$ $\frac{1\,\text{eV}}{c^2}$ $\frac{1\,\text{eV}}{c^2}$ $m_\text{p} \$ $m_\text{p} \$ $m_\text{p} \$
Electric charge (Q) $\sqrt{\hbar c \epsilon_0}$ $\frac{e}{\sqrt{\alpha}}$ $e \$ $e \$ $\frac{e}{\sqrt{2}} \$ $\frac{e}{\sqrt{4\pi\alpha}}$ $\frac{e}{\sqrt{\alpha}}$ $e$ $\frac{e}{\sqrt{4\pi\alpha}}$ $\frac{e}{\sqrt{\alpha}}$
Temperature (Θ)
with $f=2$
$\sqrt{\frac{\hbar c^5}{4 \pi G {k_\text{B}}^2}}$ $\frac{m_\text{P} c^2}{k_\text{B}} = \sqrt{\frac{\hbar c^5}{G k_\text{B}^2}}$ $\sqrt{\frac{c^4 k_\text{e} e^2}{G {k_\text{B}}^2}}$ $\frac{m_\text{e} e^4}{\hbar^2 (4 \pi \epsilon_0)^2 k_\text{B}}$ $\frac{m_\text{e} e^4}{2 \hbar^2 (4 \pi \epsilon_0)^2 k_\text{B}}$ $\frac{1\,\text{eV}}{k_\text{B}}\cdot\frac{2}{f}$ $\frac{1\,\text{eV}}{k_\text{B}}\cdot\frac{2}{f}$ $\frac{m_\text{p} c^2}{k_\text{B}}$ $\frac{m_\text{p} c^2}{k_\text{B}}$ $\frac{m_\text{p} c^2}{k_\text{B}}$

### Summary table

Quantity / Symbol Planck
(L-H)
Planck
(Gauss)
Stoney Hartree Rydberg "Natural"
(L-H)
"Natural"
(Gauss)
QCD
(original)
QCD
(L-H)
QCD
(Gauss)
Speed of light
$c \,$
$1 \,$ $1 \,$ $1 \,$ $\frac{1}{\alpha} \$ $\frac{2}{\alpha} \$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$
Reduced Planck constant
$\hbar=\frac{h}{2 \pi}$
$1 \,$ $1 \,$ $\frac{1}{\alpha} \$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$
Elementary charge
$e \,$
$\sqrt{4\pi\alpha} \,$ $\sqrt{\alpha} \,$ $1 \,$ $1 \,$ $\sqrt{2} \,$ $\sqrt{4\pi\alpha}$ $\sqrt{\alpha}$ $1 \,$ $\sqrt{4\pi\alpha} \,$ $\sqrt{\alpha} \,$
Vacuum permittivity
$\varepsilon_0 \,$
$1 \,$ $\frac{1}{4 \pi}$ $\frac{1}{4 \pi}$ $\frac{1}{4 \pi}$ $\frac{1}{4 \pi}$ $1 \,$ $\frac{1}{4 \pi}$ $\frac{1}{4 \pi \alpha}$ $1 \,$ $\frac{1}{4 \pi}$
Vacuum permeability
$\mu_0 = \frac{1}{\epsilon_0 c^2} \,$
$1 \,$ $4 \pi$ $4 \pi$ $4 \pi \alpha^2$ $\pi \alpha^2$ $1 \,$ $4 \pi$ $4 \pi \alpha$ $1 \,$ $4 \pi$
Impedance of free space
$Z_0 = \frac{1}{\epsilon_0 c} = \mu_0 c \,$
$1 \,$ $4 \pi$ $4 \pi$ $4 \pi \alpha$ $2 \pi \alpha$ $1 \,$ $4 \pi$ $4 \pi \alpha$ $1 \,$ $4 \pi$
Josephson constant
$K_\text{J} =\frac{e}{\pi \hbar} \,$
$\sqrt{\frac{4\alpha}{\pi}} \,$ $\frac{\sqrt{\alpha}}{\pi} \,$ $\frac{\alpha}{\pi} \,$ $\frac{1}{\pi} \,$ $\frac{\sqrt{2}}{\pi} \,$ $\sqrt{\frac{4\alpha}{\pi}} \,$ $\frac{\sqrt{\alpha}}{\pi} \,$ $\frac{1}{\pi} \,$ $\sqrt{\frac{4\alpha}{\pi}} \,$ $\frac{\sqrt{\alpha}}{\pi} \,$
von Klitzing constant
$R_\text{K} =\frac{2 \pi \hbar}{e^2} \,$
$\frac{1}{2\alpha}$ $\frac{2\pi}{\alpha} \,$ $\frac{2\pi}{\alpha} \,$ $2\pi \,$ $\pi \,$ $\frac{1}{2\alpha}$ $\frac{2 \pi}{\alpha}$ $2\pi \,$ $\frac{1}{2\alpha}$ $\frac{2\pi}{\alpha} \,$
Coulomb constant
$k_e=\frac{1}{4 \pi \epsilon_0}$
$\frac{1}{4 \pi}$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $\frac{1}{4 \pi}$ $1 \,$ $\alpha$ $\frac{1}{4 \pi}$ $1 \,$
Gravitational constant
$G \,$
$\frac{1}{4 \pi}$ $1 \,$ $1 \,$ $\frac{\alpha_\text{G}}{\alpha} \,$ $\frac{8 \alpha_\text{G}}{\alpha} \,$ $\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,$ $\frac{\alpha_\text{G}}{{m_\text{e}}^2} \,$ $\mu^2 \alpha_\text{G}$ $\mu^2 \alpha_\text{G}$ $\mu^2 \alpha_\text{G}$
Boltzmann constant
$k_\text{B} \,$
$1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$ $1 \,$
Proton rest mass
$m_\text{p} \,$
$\mu \sqrt{4 \pi \alpha_\text{G}} \,$ $\mu \sqrt{\alpha_\text{G}} \,$ $\mu \sqrt{\frac{\alpha_\text{G}}{\alpha}} \,$ $\mu \,$ $\frac{\mu}{2} \,$ $938 \text{ MeV}$ $938 \text{ MeV}$ $1 \,$ $1 \,$ $1 \,$
Electron rest mass
$m_\text{e} \,$
$\sqrt{4 \pi \alpha_\text{G}} \,$ $\sqrt{\alpha_\text{G}} \,$ $\sqrt{\frac{\alpha_\text{G}}{\alpha}} \,$ $1 \,$ $\frac{1}{2} \,$ $511 \text{ keV}$ $511 \text{ keV}$ $\frac{1}{\mu}$ $\frac{1}{\mu}$ $\frac{1}{\mu}$

where:

#### Fine-structure constant

The Fine-structure constant, α, in terms of other fundamental physical constants:

$\alpha = \frac{1}{4 \pi \varepsilon_0} \frac{e^2}{\hbar c} = \frac{\mu_0}{4 \pi} \frac{e^2 c}{\hbar} = \frac{k_\text{e} e^2}{\hbar c} = \frac{c \mu_0}{2 R_\text{K}} = \frac{e^2}{4 \pi}\frac{Z_0}{\hbar}$

where:

#### Gravitational coupling constant

The Gravitational coupling constant, αG, is typically defined in terms of the gravitational attraction between two electrons. More precisely,

$\alpha_\mathrm{G} = \frac{G m_\mathrm{e}^2}{\hbar c} = \left( \frac{m_\mathrm{e}}{m_\mathrm{P}} \right)^2 \approx 1.751751596 \times 10^{-45}$

where:

## Maxwell's equations

Name SI units Gaussian units Lorentz–Heaviside units
Gauss's law
(macroscopic)
$\nabla \cdot \mathbf{D} = \rho_\text{f}$ $\nabla \cdot \mathbf{D} = 4\pi\rho_\text{f}$ $\nabla \cdot \mathbf{D} = \rho_\text{f}$
Gauss's law
(microscopic)
$\nabla \cdot \mathbf{E} = \rho/\epsilon_0$ $\nabla \cdot \mathbf{E} = 4\pi\rho$ $\nabla \cdot \mathbf{E} = \rho$
Gauss's law for magnetism: $\nabla \cdot \mathbf{B} = 0$ $\nabla \cdot \mathbf{B} = 0$ $\nabla \cdot \mathbf{B} = 0$
Maxwell–Faraday equation: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$ $\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$ $\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}$
Ampère–Maxwell equation
(macroscopic):
$\nabla \times \mathbf{H} = \mathbf{J}_{\text{f}} + \frac{\partial \mathbf{D}} {\partial t}$ $\nabla \times \mathbf{H} = \frac{4\pi}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}$ $\nabla \times \mathbf{H} = \frac{1}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}$
Ampère–Maxwell equation
(microscopic):
$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \frac{1}{c^2}\frac{\partial \mathbf{E}} {\partial t}$ $\nabla \times \mathbf{B} = \frac{4\pi}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t}$ $\nabla \times \mathbf{B} = \frac{1}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t}$

### Gravitoelectromagnetism

According to general relativity, the gravitational field produced by a rotating object (or any rotating mass–energy) can, in a particular limiting case, be described by equations that have the same form as in classical electromagnetism. Starting from the basic equation of general relativity, the Einstein field equation, and assuming a weak gravitational field or reasonably flat spacetime, the gravitational analogs to Maxwell's equations for electromagnetism, called the "GEM equations", can be derived. GEM equations compared to Maxwell's equations in SI units are:

GEM equations Maxwell's equations
$\nabla \cdot \mathbf{E}_\text{g} = -4 \pi G \rho_\text{g} \$ $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$
$\nabla \cdot \mathbf{B}_\text{g} = 0 \$ $\nabla \cdot \mathbf{B} = 0 \$
$\nabla \times \mathbf{E}_\text{g} = -\frac{\partial \mathbf{B}_\text{g} } {\partial t} \$ $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B} } {\partial t} \$
$\nabla \times \mathbf{B}_\text{g} = -\frac{4 \pi G}{c^2} \mathbf{J}_\text{g} + \frac{1}{c^2} \frac{\partial \mathbf{E}_\text{g}} {\partial t}$ $\nabla \times \mathbf{B} = \frac{1}{\epsilon_0 c^2} \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t}$

where:

## CGS system of units

Quantity Quantity symbol CGS unit name Unit
symbol
Unit definition Equivalent
in SI units
length, position L, x centimetre cm 1/100 of metre = 10−2 m
mass m gram g 1/1000 of kilogram = 10−3 kg
time t seconds 1 second = 1 s
velocity v centimetre per second cm/s cm/s = 10−2 m/s
acceleration a gal Gal cm/s2 = 10−2 m/s2
force F dyne dyn g⋅cm/s2 = 10−5 N
energy E erg erg g⋅cm2/s2 = 10−7 J
power P erg per seconderg/s g⋅cm2/s3 = 10−7 W
pressure p barye Ba g/(cm⋅s2) = 10−1 Pa
dynamic viscosity μ poise P g/(cm⋅s) = 10−1 Pa⋅s
kinematic viscosity ν stokes St cm2/s = 10−4 m2/s
wavenumber k kayser cm−1 cm−1 = 100 m−1
charge q Statcoulomb statC cm3/2 g1/2 s−1 =

Community content is available under CC-BY-SA unless otherwise noted.