Physics: Problems and Solutions
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:''See also: [[Wikipedia:Dimensional analysis|Dimensional analysis]] and [[Wikipedia:Nondimensionalization|Nondimensionalization]]''
<div style="color:white; background-color:black;">
 
  +
  +
In physics, [[Wikipedia:natural units|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 ==
  +
  +
A set of fundamental dimensions is a minimal set of units such that every physical quantity can be expressed in terms of this set and where no quantity in the set can be expressed in terms of the others.<ref>[[Wikipedia:Base unit (measurement)]]</ref>
  +
 
Fundamental units:
 
Fundamental units:
*Time
 
 
*Space
 
*Space
  +
*Time (In [[relativity]] time has units of [[:w:c:math:imaginary numbers|imaginary]] distance)
 
*Mass
 
*Mass
 
*Charge
 
*Charge
  +
*Temperature
   
  +
Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy.
Coulomb's law states that:
 
   
  +
== CGS system of units ==
:<math>F=k_e\frac{q_1 q_2}{d^2}</math>
 
   
  +
From [[Wikipedia:Centimetre–gram–second system of units]]:
where:
 
*<math>k_e</math> is the Coulomb constant which is the "stiffness" of space.
 
   
  +
{| class="wikitable" style="text-align: left;"
If the Coulomb constant is <span class="plainlinks">[http://discovermagazine.com/2003/mar/featscienceof/ "negative pressure"]</span> then it has units of Force/area which gives:
 
  +
|-
 
  +
! Quantity
:<math>F=\frac{F}{d^2} \frac{q_1 q_2}{d^2}</math>
 
  +
! Quantity symbol
 
  +
! CGS unit name
Therefore:
 
  +
! Unit<br>symbol
 
  +
! Unit definition
:<math>q=d^2</math>
 
  +
! Equivalent<br>in SI units
 
  +
|-
Charge is therefore no longer a fundamental unit but instead has units of distance squared.
 
  +
! length, position
  +
| style="text-align:center;"| ''L'', ''x''|| [[centimetre]] || style="text-align:center;"| cm || 1/100 of [[metre]] || = 10<sup>−2</sup>&nbsp;m
  +
|-
  +
! mass
  +
| style="text-align:center;"| ''m''|| [[gram]] || style="text-align:center;"|g || 1/1000 of [[kilogram]] || = 10<sup>−3</sup>&nbsp;kg
  +
|-
  +
! time
  +
| style="text-align:center;"| ''t''|| second|| style="text-align:center;"|s|| 1 second || = 1&nbsp;s
  +
|-
  +
! velocity
  +
| style="text-align:center;"| ''v''|| centimetre per second || style="text-align:center;"|cm/s || cm/s || = 10<sup>−2</sup>&nbsp;m/s
  +
|-
  +
! acceleration
  +
| style="text-align:center;"| ''a''|| [[gal (unit)|gal]] || style="text-align:center;"|Gal || cm/s<sup>2</sup> || = 10<sup>−2</sup>&nbsp;m/s<sup>2</sup>
  +
|-
  +
! [[force (physics)|force]]
  +
| style="text-align:center;"| ''F''|| [[dyne]] || style="text-align:center;"|dyn || g⋅cm/s<sup>2</sup> || = 10<sup>−5</sup>&nbsp;[[newton (unit)|N]]
  +
|-
  +
! energy
  +
| style="text-align:center;"| ''E''|| [[erg]] || style="text-align:center;"|erg || g⋅cm<sup>2</sup>/s<sup>2</sup> || = 10<sup>−7</sup>&nbsp;[[joule|J]]
  +
|-
  +
! [[power (physics)|power]]
  +
| style="text-align:center;"| ''P''|| [[erg]] per second|| style="text-align:center;"|erg/s || g⋅cm<sup>2</sup>/s<sup>3</sup> || = 10<sup>−7</sup>&nbsp;[[watt|W]]
  +
|-
  +
! pressure
  +
| style="text-align:center;"| ''p''|| [[barye]] || style="text-align:center;"| Ba|| g/(cm⋅s<sup>2</sup>) || = 10<sup>−1</sup>&nbsp;[[pascal (unit)|Pa]]
  +
|-
  +
! dynamic [[viscosity]]
  +
| style="text-align:center;"| ''μ''|| [[Poise (unit)|poise]] || style="text-align:center;"|P|| g/(cm⋅s) || = 10<sup>−1</sup>&nbsp;[[pascal second|Pa⋅s]]
  +
|-
  +
! kinematic [[viscosity]]
  +
| style="text-align:center;"| ''ν''|| [[stokes (unit)|stokes]] || style="text-align:center;"|St|| cm<sup>2</sup>/s || = 10<sup>−4</sup>&nbsp;m<sup>2</sup>/s
  +
|-
  +
! [[wavenumber]]
  +
| style="text-align:center;"| ''k'' || [[Wavenumber|kayser]] || style="text-align:center;"|cm<sup>−1</sup>|| cm<sup>−1</sup> || = 100 m<sup>−1</sup>
  +
|-
  +
! charge
  +
| style="text-align:center;"| ''q'' || [[Statcoulomb]] || style="text-align:center;"| statC || style="background-color:yellow;" | cm<sup>3/2</sup> g<sup>1/2</sup> s<sup>−1</sup> || =
  +
|}
   
 
== Natural units ==
 
== Natural units ==
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From [[Wikipedia:natural units]]:
 
From [[Wikipedia:natural units]]:
   
  +
The surface area of a sphere <math>4 \pi r^2</math>
In physics, [[Wikipedia:natural units|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.
 
   
 
In '''[[Wikipedia:Lorentz–Heaviside units|Lorentz–Heaviside units]]''' (rationalized units), [[Coulomb's law]] is:
 
In '''[[Wikipedia:Lorentz–Heaviside units|Lorentz–Heaviside units]]''' (rationalized units), [[Coulomb's law]] is:
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*{{math|''e''}} is the [[elementary charge]],
 
*{{math|''e''}} is the [[elementary charge]],
   
  +
==Summary table==
 
  +
===Base units===
  +
  +
{| class="wikitable" style="white-space: nowrap;"
  +
|+Base units
  +
|-
  +
! Dimension
  +
! Planck<br>(L-H)
  +
! Planck<br>(Gauss)
  +
! Stoney
  +
! Hartree
  +
! Rydberg
  +
! Natural<br>(L-H)
  +
! Natural<br>(Gauss)
  +
! QCD<br>(Original)
  +
! QCD<br>(L-H)
  +
! QCD<br>(Gauss)
  +
|-
  +
| [[Length]] (L)
  +
| <math>\sqrt{4 \pi \hbar G \over c^3}</math>
  +
| <math>\sqrt{\frac{\hbar G}{c^3}}</math>
  +
| <math>\sqrt{\frac{G k_\text{e} e^2}{c^4}}</math>
  +
| <math>\frac{\hbar^2 (4 \pi \epsilon_0)}{m_\text{e} e^2}</math>
  +
| <math>\frac{\hbar^2 (4 \pi \epsilon_0)}{m_\text{e} e^2}</math>
  +
| <math>\frac{\hbar c}{1\,\text{eV}} </math>
  +
| <math>\frac{\hbar c}{1\,\text{eV}} </math>
  +
| <math>\frac{\hbar}{m_\text{p} c}</math>
  +
| <math>\frac{\hbar}{m_\text{p} c}</math>
  +
| <math>\frac{\hbar}{m_\text{p} c}</math>
  +
|-
  +
| [[Time]] (T)
  +
| <math>\sqrt{4 \pi \hbar G \over c^5} </math>
  +
| <math>\frac{\hbar}{m_\text{P}c^2} = \sqrt{\frac{\hbar G}{c^5}} </math>
  +
| <math>\sqrt{\frac{G k_\text{e} e^2}{c^6}} </math>
  +
| <math>\frac{\hbar^3 (4 \pi \epsilon_0)^2}{m_\text{e} e^4} </math>
  +
| <math>\frac{2 \hbar^3 (4 \pi \epsilon_0)^2}{m_\text{e} e^4} </math>
  +
| <math>\frac{\hbar}{1\,\text{eV}} </math>
  +
| <math>\frac{\hbar}{1\,\text{eV}} </math>
  +
| <math>\frac{\hbar}{m_\text{p} c^2}</math>
  +
| <math>\frac{\hbar}{m_\text{p} c^2}</math>
  +
| <math>\frac{\hbar}{m_\text{p} c^2}</math>
  +
|-
  +
| [[Mass]] (M)
  +
| <math>\sqrt{\hbar c \over 4 \pi G}</math>
  +
| <math>\sqrt{\frac{\hbar c}{G}}</math>
  +
| <math>\sqrt{\frac{k_\text{e} e^2}{G}}</math>
  +
| <math>m_\text{e} \ </math>
  +
| <math>2 m_\text{e} \ </math>
  +
| <math>\frac{1\,\text{eV}}{c^2}</math>
  +
| <math>\frac{1\,\text{eV}}{c^2}</math>
  +
| <math>m_\text{p} \ </math>
  +
| <math>m_\text{p} \ </math>
  +
| <math>m_\text{p} \ </math>
  +
|-
  +
| [[Electric charge]] (Q)
  +
| <math>\sqrt{\hbar c \epsilon_0}</math>
  +
| <math>\frac{e}{\sqrt{\alpha}} </math>
  +
| <math>e \ </math>
  +
| <math>e \ </math>
  +
| <math>\frac{e}{\sqrt{2}} \ </math>
  +
| <math>\frac{e}{\sqrt{4\pi\alpha}} </math>
  +
| <math>\frac{e}{\sqrt{\alpha}} </math>
  +
| <math>e</math>
  +
| <math>\frac{e}{\sqrt{4\pi\alpha}}</math>
  +
| <math>\frac{e}{\sqrt{\alpha}}</math>
  +
|-
  +
| [[Temperature]] (Θ)<br>{{pad|1em}}with <math>f=2</math>
  +
| <math>\sqrt{\frac{\hbar c^5}{4 \pi G {k_\text{B}}^2}}</math>
  +
| <math>\frac{m_\text{P} c^2}{k_\text{B}} = \sqrt{\frac{\hbar c^5}{G k_\text{B}^2}}</math>
  +
| <math>\sqrt{\frac{c^4 k_\text{e} e^2}{G {k_\text{B}}^2}}</math>
  +
| <math>\frac{m_\text{e} e^4}{\hbar^2 (4 \pi \epsilon_0)^2 k_\text{B}}</math>
  +
| <math>\frac{m_\text{e} e^4}{2 \hbar^2 (4 \pi \epsilon_0)^2 k_\text{B}}</math>
  +
| <math>\frac{1\,\text{eV}}{k_\text{B}}\cdot\frac{2}{f}</math>
  +
| <math>\frac{1\,\text{eV}}{k_\text{B}}\cdot\frac{2}{f}</math>
  +
| <math>\frac{m_\text{p} c^2}{k_\text{B}}</math>
  +
| <math>\frac{m_\text{p} c^2}{k_\text{B}}</math>
  +
| <math>\frac{m_\text{p} c^2}{k_\text{B}}</math>
  +
|}
  +
  +
  +
=== Summary table ===
   
 
From [[Wikipedia:natural units]]:
 
From [[Wikipedia:natural units]]:
   
{| class="wikitable" style="color:white; background-color: #000000; white-space: nowrap; <!-- margin: 1em auto 1em auto; -->"
+
{| class="wikitable" style="white-space: nowrap;"
 
|-
 
|-
! style="color:white; background-color:blue !important;" | Quantity / Symbol
+
! Quantity / Symbol
  +
! Planck<br>([[Lorentz–Heaviside units|L-H]])
! style="color:white; background-color:blue !important;" |
 
! style="color:white; background-color:blue !important;" | Planck<br>([[Lorentz–Heaviside units|L-H]])
+
! Planck<br>([[Gaussian units|Gauss]])
  +
! Stoney
! style="color:white; background-color:blue !important;" | Planck<br>([[Gaussian units|Gauss]])
 
  +
! Hartree
! style="color:white; background-color:blue !important;" | Stoney
 
  +
! Rydberg
! style="color:white; background-color:blue !important;" | Hartree
 
  +
! "Natural"<br>([[Lorentz–Heaviside units|L-H]])
! style="color:white; background-color:blue !important;" | Rydberg
 
! style="color:white; background-color:blue !important;" | "Natural"<br>([[Lorentz–Heaviside units|L-H]])
+
! "Natural"<br>([[Gaussian units|Gauss]])
  +
! [[Quantum chromodynamics|QCD]]<br>(original)
! style="color:white; background-color:blue !important;" | "Natural"<br>([[Gaussian units|Gauss]])
 
! style="color:white; background-color:blue !important;" | [[Wikipedia:Quantum chromodynamics|QCD]]<br>(original)
+
! [[Quantum chromodynamics|QCD]]<br>([[Lorentz–Heaviside units|L-H]])
! style="color:white; background-color:blue !important;" | [[Wikipedia:Quantum chromodynamics|QCD]]<br>([[Lorentz–Heaviside units|L-H]])
+
! [[Quantum chromodynamics|QCD]]<br>([[Gaussian units|Gauss]])
! style="color:white; background-color:blue !important;" | [[Wikipedia:Quantum chromodynamics|QCD]]<br>([[Gaussian units|Gauss]])
 
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Speed of light]] <br> <math>c \,</math>
+
| [[Speed of light]] <br> <math>c \,</math>
|<math>1 \,</math>
 
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
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|<math>1 \,</math>
 
|<math>1 \,</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Reduced Planck constant]] <br> <math>\hbar=\frac{h}{2 \pi}</math>
+
| [[Reduced Planck constant]] <br> <math>\hbar=\frac{h}{2 \pi}</math>
|<math>\frac{1}{4 \pi \alpha}</math>
 
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
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|<math>1 \,</math>
 
|<math>1 \,</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Elementary charge]] <br> <math>e \,</math>
+
| [[Elementary charge]] <br> <math>e \,</math>
|<math>1 \,</math>
 
 
|<math>\sqrt{4\pi\alpha} \,</math>
 
|<math>\sqrt{4\pi\alpha} \,</math>
 
|<math>\sqrt{\alpha} \,</math>
 
|<math>\sqrt{\alpha} \,</math>
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|<math>\sqrt{\alpha} \,</math>
 
|<math>\sqrt{\alpha} \,</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Vacuum permittivity]] <br> <math>\varepsilon_0 \,</math>
+
| [[Vacuum permittivity]] <br> <math>\varepsilon_0 \,</math>
|<math>1 \,</math>
 
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>\frac{1}{4 \pi}</math>
 
|<math>\frac{1}{4 \pi}</math>
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|<math>\frac{1}{4 \pi}</math>
 
|<math>\frac{1}{4 \pi}</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Vacuum permeability]] <br> <math>\mu_0 = \frac{1}{\epsilon_0 c^2} \,</math>
+
| [[Vacuum permeability]] <br> <math>\mu_0 = \frac{1}{\epsilon_0 c^2} \,</math>
|<math>1 \,</math>
 
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>4 \pi</math>
 
|<math>4 \pi</math>
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|<math>4 \pi</math>
 
|<math>4 \pi</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Impedance of free space]] <br> <math>Z_0 = \frac{1}{\epsilon_0 c} = \mu_0 c \,</math>
+
| [[Impedance of free space]] <br> <math>Z_0 = \frac{1}{\epsilon_0 c} = \mu_0 c \,</math>
|<math>1 \,</math>
 
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>4 \pi</math>
 
|<math>4 \pi</math>
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|<math>4 \pi</math>
 
|<math>4 \pi</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Coulomb constant]] <br> <math>k_e=\frac{1}{4 \pi \epsilon_0}</math>
+
|[[Magnetic flux quantum|Josephson constant]] <br> <math>K_\text{J} =\frac{e}{\pi \hbar} \,</math>
|<math>\frac{1}{4 \pi}</math>
+
|<math>\sqrt{\frac{4\alpha}{\pi}} \,</math>
  +
|<math>\frac{\sqrt{\alpha}}{\pi} \,</math>
  +
|<math>\frac{\alpha}{\pi} \,</math>
  +
|<math>\frac{1}{\pi} \,</math>
  +
|<math>\frac{\sqrt{2}}{\pi} \,</math>
  +
|<math>\sqrt{\frac{4\alpha}{\pi}} \,</math>
  +
|<math>\frac{\sqrt{\alpha}}{\pi} \,</math>
  +
|<math>\frac{1}{\pi} \,</math>
  +
|<math>\sqrt{\frac{4\alpha}{\pi}} \,</math>
  +
|<math>\frac{\sqrt{\alpha}}{\pi} \,</math>
  +
|-
  +
|[[Quantum Hall effect|von Klitzing constant]] <br> <math>R_\text{K} =\frac{2 \pi \hbar}{e^2} \,</math>
  +
|<math>\frac{1}{2\alpha} </math>
  +
|<math>\frac{2\pi}{\alpha} \,</math>
  +
|<math>\frac{2\pi}{\alpha} \,</math>
  +
|<math>2\pi \,</math>
  +
|<math>\pi \,</math>
  +
|<math>\frac{1}{2\alpha} </math>
  +
|<math>\frac{2 \pi}{\alpha} </math>
  +
|<math>2\pi \,</math>
  +
|<math>\frac{1}{2\alpha} </math>
  +
|<math>\frac{2\pi}{\alpha} \,</math>
  +
|-
  +
| [[Coulomb constant]] <br> <math>k_e=\frac{1}{4 \pi \epsilon_0}</math>
 
|<math>\frac{1}{4 \pi}</math>
 
|<math>\frac{1}{4 \pi}</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
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|<math>1 \,</math>
 
|<math>1 \,</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Gravitational constant]] <br> <math>G \,</math>
+
| [[Gravitational constant]] <br> <math>G \,</math>
|<math>\frac{1}{4 \pi}</math>
 
 
|<math>\frac{1}{4 \pi}</math>
 
|<math>\frac{1}{4 \pi}</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
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|<math>\mu^2 \alpha_\text{G}</math>
 
|<math>\mu^2 \alpha_\text{G}</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Boltzmann constant]] <br> <math>k_\text{B} \,</math>
+
| [[Boltzmann constant]] <br> <math>k_\text{B} \,</math>
|<math>1 \,</math>
 
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
 
|<math>1 \,</math>
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|<math>1 \,</math>
 
|<math>1 \,</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Proton rest mass]] <br> <math>m_\text{p} \,</math>
+
| [[Proton rest mass]] <br> <math>m_\text{p} \,</math>
|
 
 
|<math>\mu \sqrt{4 \pi \alpha_\text{G}} \,</math>
 
|<math>\mu \sqrt{4 \pi \alpha_\text{G}} \,</math>
 
|<math>\mu \sqrt{\alpha_\text{G}} \,</math>
 
|<math>\mu \sqrt{\alpha_\text{G}} \,</math>
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|<math>1 \,</math>
 
|<math>1 \,</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Electron rest mass]] <br> <math>m_\text{e} \,</math>
+
| [[Electron rest mass]] <br> <math>m_\text{e} \,</math>
|
 
 
|<math>\sqrt{4 \pi \alpha_\text{G}} \,</math>
 
|<math>\sqrt{4 \pi \alpha_\text{G}} \,</math>
 
|<math>\sqrt{\alpha_\text{G}} \,</math>
 
|<math>\sqrt{\alpha_\text{G}} \,</math>
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|<math>\frac{1}{\mu}</math>
 
|<math>\frac{1}{\mu}</math>
 
|<math>\frac{1}{\mu}</math>
 
|<math>\frac{1}{\mu}</math>
|-
 
| style="color:white; background-color:blue !important;" | [[Planck mass]] <br> <math>m_\text{P}=\sqrt{\frac{\hbar c}{G}},</math>
 
|<math>\sqrt{\frac{1}{\alpha}}</math>
 
|
 
|
 
|
 
|
 
|
 
|
 
|
 
|
 
|
 
|
 
 
|}
 
|}
 
where:
 
where:
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*{{math|''µ''}} is dimensionless [[proton-to-electron mass ratio]]
 
*{{math|''µ''}} is dimensionless [[proton-to-electron mass ratio]]
   
  +
=== Fine-structure constant ===
 
  +
==== Fine-structure constant ====
   
 
From [[Wikipedia:Fine-structure constant]]:
 
From [[Wikipedia:Fine-structure constant]]:
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*{{math|''Z''<sub>0</sub>}} is the [[vacuum impedance]] or impedance of free space
 
*{{math|''Z''<sub>0</sub>}} is the [[vacuum impedance]] or impedance of free space
   
  +
=== Gravitational coupling constant ===
 
  +
==== Gravitational coupling constant ====
   
 
From [[Wikipedia:Gravitational coupling constant]]:
 
From [[Wikipedia:Gravitational coupling constant]]:
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* {{math|''m''<sub>P</sub>}} is the [[Planck mass]]
 
* {{math|''m''<sub>P</sub>}} is the [[Planck mass]]
   
=== Boltzmann constant ===
 
 
From [[Wikipedia:Boltzmann constant]]:
 
 
The [[Wikipedia:Boltzmann constant|Boltzmann constant]], {{mvar|k}}, is a [[scale factor|scaling factor]] between macroscopic ([[thermodynamic temperature]]) and microscopic ([[thermal energy]]) physics. <br>
 
Macroscopically, the [[ideal gas law]] states:
 
:<math>p V = N k_B T = N k_B m v^2 = N k_B 2 E</math>
 
where:
 
*{{mvar|p}} is the pressure
 
*{{mvar|V}} is the volume
 
*{{mvar|N}} is the [[Number of particles|number of molecules]] of gas.
 
*{{mvar|k<sub>B</sub>}} is the Boltzmann constant
 
*{{mvar|T}} is the temperature
 
*{{mvar|v}} is the velocity
 
**Twice the velocity means twice as much momentum transferred per collision '''and''' twice as many collisions per unit time.
 
*{{mvar|E}} is the kinetic energy per particle
 
 
Therefore temperature is twice the energy per degree of freedom per particle
 
 
*<math> T = 2 E</math>
 
   
== Electromagnetism ==
+
== Maxwell's equations ==
   
 
From [[Wikipedia:Lorentz–Heaviside units]]:
 
From [[Wikipedia:Lorentz–Heaviside units]]:
   
{| class="wikitable" cellpadding="8" style="color:white; background-color:black;"
+
{| class="wikitable" cellpadding="8"
 
|-
 
|-
  +
! Name
! style="color:white; background-color:blue !important;" | Name
 
  +
! [[SI units]]
! style="color:white; background-color:blue !important;" | [[SI units]]
 
! style="color:white; background-color:blue !important;" | [[Gaussian units]]
+
! [[Gaussian units]]
! style="color:white; background-color:blue !important;" | [[Lorentz–Heaviside units]]
+
! [[Lorentz–Heaviside units]]
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Gauss's law]]
+
| [[Gauss's law]]
 
:(macroscopic)
 
:(macroscopic)
 
| <math>\nabla \cdot \mathbf{D} = \rho_\text{f}</math>
 
| <math>\nabla \cdot \mathbf{D} = \rho_\text{f}</math>
Line 317: Line 424:
 
| <math>\nabla \cdot \mathbf{D} = \rho_\text{f}</math>
 
| <math>\nabla \cdot \mathbf{D} = \rho_\text{f}</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Gauss's law]]
+
| [[Gauss's law]]
 
:(microscopic)
 
:(microscopic)
 
| <math>\nabla \cdot \mathbf{E} = \rho/\epsilon_0</math>
 
| <math>\nabla \cdot \mathbf{E} = \rho/\epsilon_0</math>
Line 323: Line 430:
 
| <math>\nabla \cdot \mathbf{E} = \rho</math>
 
| <math>\nabla \cdot \mathbf{E} = \rho</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Gauss's law for magnetism]]:
+
| [[Gauss's law for magnetism]]:
 
|<math>\nabla \cdot \mathbf{B} = 0</math>
 
|<math>\nabla \cdot \mathbf{B} = 0</math>
 
|<math>\nabla \cdot \mathbf{B} = 0</math>
 
|<math>\nabla \cdot \mathbf{B} = 0</math>
 
|<math>\nabla \cdot \mathbf{B} = 0</math>
 
|<math>\nabla \cdot \mathbf{B} = 0</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Faraday's law of induction|Maxwell–Faraday equation]]:
+
| [[Faraday's law of induction|Maxwell–Faraday equation]]:
 
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Ampère–Maxwell equation]]
+
| [[Ampère–Maxwell equation]]
 
:(macroscopic):
 
:(macroscopic):
 
| <math>\nabla \times \mathbf{H} = \mathbf{J}_{\text{f}} + \frac{\partial \mathbf{D}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{H} = \mathbf{J}_{\text{f}} + \frac{\partial \mathbf{D}} {\partial t}</math>
Line 339: Line 446:
 
| <math>\nabla \times \mathbf{H} = \frac{1}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{H} = \frac{1}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}</math>
 
|-
 
|-
| style="color:white; background-color:blue !important;" | [[Ampère–Maxwell equation]]
+
| [[Ampère–Maxwell equation]]
 
:(microscopic):
 
:(microscopic):
 
| <math>\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \frac{1}{c^2}\frac{\partial \mathbf{E}} {\partial t}</math>
 
| <math>\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \frac{1}{c^2}\frac{\partial \mathbf{E}} {\partial t}</math>
Line 346: Line 453:
 
|}
 
|}
   
  +
==Gravitoelectromagnetism==
 
  +
=== Gravitoelectromagnetism ===
 
:See also: [[Wikipedia:Einstein_field_equations#The_correspondence_principle|Einstein_field_equations]]
 
:See also: [[Wikipedia:Einstein_field_equations#The_correspondence_principle|Einstein_field_equations]]
   
Line 353: Line 461:
 
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:
 
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:
   
{| class="wikitable" border="1" cellpadding="8" cellspacing="0" style="color:white; background-color:black;"
+
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
 
|-
 
|-
  +
!GEM equations
!style="color:white; background-color:blue !important;" | GEM equations
 
!style="color:white; background-color:blue !important;" | Maxwell's equations
+
!Maxwell's equations
 
|-
 
|-
 
|<math> \nabla \cdot \mathbf{E}_\text{g} = -4 \pi G \rho_\text{g} \ </math>
 
|<math> \nabla \cdot \mathbf{E}_\text{g} = -4 \pi G \rho_\text{g} \ </math>
Line 383: Line 491:
 
* ''c'' is the [[Speed of gravity|speed of propagation of gravity]] (which is equal to the [[speed of light]] according to [[general relativity]]) in m⋅s<sup>−1</sup>.
 
* ''c'' is the [[Speed of gravity|speed of propagation of gravity]] (which is equal to the [[speed of light]] according to [[general relativity]]) in m⋅s<sup>−1</sup>.
   
== CGS ==
 
   
From [[Wikipedia:Centimetre–gram–second system of units]]:
 
   
  +
=== Electromagnetism ===
{| class="wikitable" style="text-align: left;" style="color:white; background-color:black;"
 
  +
  +
The total energy in the electric field surrounding a hollow spherical shell of radius {{mvar|r}} and charge {{mvar|q}} is:
  +
  +
:<math>E = k \frac{1}{2} \frac{q^2}{r}</math>
  +
  +
Therefore:
  +
  +
:<math> {\color{red}2E \cdot \frac{r}{q^2} } = k =constant</math>
  +
  +
The constant {{mvar|k}} is a property of space. It is the "[[Wikipedia:Bulk modulus|stiffness]]" of space. (If space were stiffer then c would be faster.)
  +
  +
Coulomb's law states that:
  +
  +
:<math>F=k_e \frac{q_1 q_2}{d^2}</math>
  +
  +
The Coulomb constant has units of Energy * distance/charge<sup>2</sup> which gives:
  +
  +
:<math>F = {\color{red}F \cdot d \frac{d}{q^2}} \frac{q_1 q_2}{d^2}</math>
  +
  +
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:
  +
  +
:<math>F = G \frac{m_1 m_2}{d^2} </math>
  +
  +
where:
  +
  +
:<math>F = F \cdot d \frac{d}{m^2} \frac{m_1 m_2}{d^2}</math>
  +
  +
But its probably better to say that:
  +
  +
:<math>a = \frac{d}{t^2} = G \frac{m}{d^2}\ </math>
  +
  +
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:
  +
  +
:<math>G \frac{m_1 m_2}{d^2} = k_e\frac{q_1 q_2}{d^2}</math>
  +
  +
Solving we get m = <span class="plainlinks">[http://m.wolframalpha.com/input/?i=%281+coulomb+constant+*+%281+electron+charge%29%5E2+%2F+G%29%5E0.5+in+g 1.859 × 10<sup>-6</sup> g] = [http://m.wolframalpha.com/input/?i=%281+coulomb+constant+*+%281+electron+charge%29%5E2+%2F+G%29%5E0.5+in+planck+masses <math>\sqrt{\alpha}</math> planck masses] = [https://www.wolframalpha.com/input/?i=(1+coulomb+constant+*+(1+electron+charge)%5E2+%2F+G)%5E0.5+in+stoney+masses 1 Stoney mass]</span>
  +
  +
The [[Wikipedia:Schwarzschild radius|Schwarzschild radius]] of a Stoney mass is 2 Stoney lengths.
  +
  +
=== Boltzmann constant ===
  +
{| align=right
  +
|
  +
{| class=wikitable style="margin:0px;"
  +
! Gas
  +
! [[Wikipedia:Heat capacity ratio|Specific<br>heat<br>ratio]]
  +
! Degrees<br>of<br>freedom
 
|-
 
|-
  +
| Helium
! style="color:white; background-color:blue !important;" | Quantity
 
  +
| 1.667
! style="color:white; background-color:blue !important;" | Quantity symbol
 
  +
| 3
! style="color:white; background-color:blue !important;" | CGS unit name
 
! style="color:white; background-color:blue !important;" | Unit<br>symbol
 
! style="color:white; background-color:blue !important;" | Unit definition
 
! style="color:white; background-color:blue !important;" | Equivalent<br>in SI units
 
 
|-
 
|-
  +
| Neon
! style="color:white; background-color:blue !important;" | length, position
 
  +
| 1.667
| style="text-align:center;"| ''L'', ''x''|| [[centimetre]] || style="text-align:center;"| cm || 1/100 of [[metre]] || = 10<sup>−2</sup>&nbsp;m
 
  +
| 3
 
|-
 
|-
  +
| Argon
! style="color:white; background-color:blue !important;" | [[mass]]
 
  +
| 1.667
| style="text-align:center;"| ''m''|| [[gram]] || style="text-align:center;"|g || 1/1000 of [[kilogram]] || = 10<sup>−3</sup>&nbsp;kg
 
  +
| 3
 
|-
 
|-
  +
| Hydrogen
! style="color:white; background-color:blue !important;" | time
 
  +
| 1.597<ref>at -181 C</ref>
| style="text-align:center;"| ''t''|| second|| style="text-align:center;"|s|| 1 second || = 1&nbsp;s
 
  +
| 3.35
 
|-
 
|-
  +
| Hydrogen
! style="color:white; background-color:blue !important;" | [[velocity]]
 
  +
| 1.41
| style="text-align:center;"| ''v''|| centimetre per second || style="text-align:center;"|cm/s || cm/s || = 10<sup>−2</sup>&nbsp;m/s
 
  +
| 4.88
 
|-
 
|-
  +
| Nitrogen
! style="color:white; background-color:blue !important;" | [[acceleration]]
 
  +
| 1.4
| style="text-align:center;"| ''a''|| [[gal (unit)|gal]] || style="text-align:center;"|Gal || cm/s<sup>2</sup> || = 10<sup>−2</sup>&nbsp;m/s<sup>2</sup>
 
  +
| 5
 
|-
 
|-
  +
| Oxygen
! style="color:white; background-color:blue !important;" | [[force (physics)|force]]
 
  +
| 1.395
| style="text-align:center;"| ''F''|| [[dyne]] || style="text-align:center;"|dyn || g⋅cm/s<sup>2</sup> || = 10<sup>−5</sup>&nbsp;[[newton (unit)|N]]
 
  +
| 5.06
 
|-
 
|-
  +
| Chlorine
! style="color:white; background-color:blue !important;" | [[energy]]
 
  +
| 1.34
| style="text-align:center;"| ''E''|| [[erg]] || style="text-align:center;"|erg || g⋅cm<sup>2</sup>/s<sup>2</sup> || = 10<sup>−7</sup>&nbsp;[[joule|J]]
 
  +
| 5.88
 
|-
 
|-
  +
| Carbon dioxide
! style="color:white; background-color:blue !important;" | [[power (physics)|power]]
 
  +
| 1.289
| style="text-align:center;"| ''P''|| [[erg]] per second|| style="text-align:center;"|erg/s || g⋅cm<sup>2</sup>/s<sup>3</sup> || = 10<sup>−7</sup>&nbsp;[[watt|W]]
 
  +
| 6.92
 
|-
 
|-
  +
| Methane
! style="color:white; background-color:blue !important;" | [[pressure]]
 
  +
| 1.304
| style="text-align:center;"| ''p''|| [[barye]] || style="text-align:center;"| Ba|| g/(cm⋅s<sup>2</sup>) || = 10<sup>−1</sup>&nbsp;[[pascal (unit)|Pa]]
 
  +
| 6.58
 
|-
 
|-
  +
| Ethane
! style="color:white; background-color:blue !important;" | dynamic [[viscosity]]
 
  +
| 1.187
| style="text-align:center;"| ''μ''|| [[Poise (unit)|poise]] || style="text-align:center;"|P|| g/(cm⋅s) || = 10<sup>−1</sup>&nbsp;[[pascal second|Pa⋅s]]
 
  +
| [https://www.wolframalpha.com/input/?i=2%2F(1.187-1) 10.7]
  +
|}
 
|-
 
|-
  +
| [https://www.engineeringtoolbox.com/specific-heat-capacity-gases-d_159.html Engineering ToolBox (2003)]<ref>Engineering ToolBox, (2003). ''Specific Heat and Individual Gas Constant of Gases.'' [online]<br>Available at: [https://www.engineeringtoolbox.com/specific-heat-capacity-gases-d_159.html https://www.engineeringtoolbox.com/specific-heat-capacity-gases-d_159.html]<br>[Accessed 20-4-2019].</ref>
! style="color:white; background-color:blue !important;" | kinematic [[viscosity]]
 
| style="text-align:center;"| ''ν''|| [[stokes (unit)|stokes]] || style="text-align:center;"|St|| cm<sup>2</sup>/s || = 10<sup>−4</sup>&nbsp;m<sup>2</sup>/s
 
|-
 
! style="color:white; background-color:blue !important;" | [[wavenumber]]
 
| style="text-align:center;"| ''k'' || [[Wavenumber|kayser]] || style="text-align:center;"|cm<sup>−1</sup>|| cm<sup>−1</sup> || = 100 m<sup>−1</sup>
 
|-
 
! style="color:white; background-color:blue !important;" | [[charge]]
 
| style="text-align:center;"| ''q'' || [[Statcoulomb]] || style="text-align:center;"| statC || style="background-color:blue;" | cm<sup>3/2</sup> g<sup>1/2</sup> s<sup>−1</sup> || =
 
 
|}
 
|}
  +
  +
For monatomic gases:
  +
  +
:<math>P V^{\frac{5}{3}} = Constant</math>
  +
  +
From [[Wikipedia:Boltzmann constant]]:
  +
  +
The [[Wikipedia:Boltzmann constant|Boltzmann constant]], {{mvar|k}}, is a [[scale factor|scaling factor]] between macroscopic ([[thermodynamic temperature]]) and microscopic ([[thermal energy]]) physics. <br>
  +
Macroscopically, the [[ideal gas law]] states:
  +
:<math>k_B T = P \frac{V}{n}</math>
  +
where:
  +
*{{mvar|k<sub>B</sub>}} is the Boltzmann constant
  +
*{{mvar|T}} is the temperature
  +
*{{mvar|P}} is the pressure
  +
*{{mvar|V}} is the volume
  +
*{{mvar|n}} is the [[Number of particles|number of molecules]] of gas.
  +
  +
  +
==== Single particle ====
  +
  +
The pressure exerted on one face of a cube of length {{mvar|d}} by a single particle bouncing back and forth perpendicular to the face with mass {{mvar|m}} and velocity <math>v = \sqrt{v_x + v_y + v_z}</math> is:
  +
  +
:<math>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}</math>
  +
  +
where:
  +
  +
*{{mvar|V<sub>0</sub>}} = {{mvar|d<sup>3</sup>}} is the volume occupied by a single particle.
  +
*{{mvar|v<sub>x</sub>}} 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.
  +
*{{mvar|E<sub>x</sub>}} is the kinetic energy per particle
  +
**{{mvar|E}} = {{mvar|E<sub>x</sub>}} +{{mvar|E<sub>y</sub>}} + {{mvar|E<sub>z</sub>}}
  +
  +
Therefore:
  +
  +
:<math>V_0 = \frac{V}{n}</math>
  +
  +
Therefore:
  +
  +
:<math>T = p \frac{V}{n} = p V_0 = m v^2 = 2 E</math>
  +
  +
Therefore temperature is twice the energy per degree of freedom per particle
  +
  +
*<math> T = 2 E</math>
  +
  +
==== Blackbody radiation ====
  +
  +
From [[Wikipedia:Black-body radiation]]:
  +
  +
Planck's law states that
  +
:<math>B_\nu(\nu, T) = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT} - 1},</math>
  +
where
  +
:B''<sub>ν</sub>''(''T'') is the spectral radiance (the [[Power (physics)|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.
  +
  +
From [[Wikipedia:Larmor formula]]:
  +
  +
For velocities that are small relative to the [[speed of light]], the total power radiated is given by the Larmor formula:
  +
  +
:<math> 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)} </math>
  +
  +
=== Uncertainty principle ===
  +
  +
*[https://www.ias.ac.in/article/fulltext/reso/004/02/0020-0023 https://www.ias.ac.in/article/fulltext/reso/004/02/0020-0023]
  +
*[[Wikipedia:Uncertainty principle|Uncertainty principle]]
  +
*[[Wikipedia:Gaussian function|Gaussian function]]
  +
   
 
== References ==
 
== References ==
 
<references/>
 
<references/>
  +
</div>
 
  +
  +
== External links ==
  +
*[https://thespectrumofriemannium.wordpress.com/tag/stoney-units/ The Spectrum of Riemannium]
  +
*[https://www.physics.wisc.edu/undergrads/courses/fall2011/107/handouts/natural-units.pdf wisc.edu]
  +
*[https://pdfs.semanticscholar.org/0390/23425222f02592fbf674ac86c15125f1a600.pdf Stoney Scale and Large Number Coincidences]
 
[[Category:Blog posts]]
 
[[Category:Blog posts]]

Latest revision as of 02:56, 11 June 2019

See also: Dimensional analysis and Nondimensionalization

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

A set of fundamental dimensions is a minimal set of units such that every physical quantity can be expressed in terms of this set and where no quantity in the set can be expressed in terms of the others.[1]

Fundamental units:

Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy.

CGS system of units

From Wikipedia:Centimetre–gram–second 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 second s 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 second erg/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 =

Natural units

From Wikipedia:natural units:

The surface area of a sphere

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

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

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:

  • c is the speed of light,
  • ħ is the reduced Planck constant,
  • G is the gravitational constant,
  • ke is the Coulomb constant,
  • kB is the Boltzmann constant
  • e is the elementary charge,


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)
Time (T)
Mass (M)
Electric charge (Q)
Temperature (Θ)
 with


Summary table

From Wikipedia:natural units:

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
Reduced Planck constant
Elementary charge
Vacuum permittivity
Vacuum permeability
Impedance of free space
Josephson constant
von Klitzing constant
Coulomb constant
Gravitational constant
Boltzmann constant
Proton rest mass
Electron rest mass

where:

  • α is the dimensionless fine-structure constant
  • αG is the dimensionless gravitational coupling constant
  • µ is dimensionless proton-to-electron mass ratio


Fine-structure constant

From Wikipedia:Fine-structure constant:

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

where:

  • e is the elementary charge
  • π is the mathematical constant pi
  • ħ is the reduced Planck constant
  • c is the speed of light in vacuum
  • ε0 is the electric constant or permittivity of free space
  • µ0 is the magnetic constant or permeability of free space
  • ke is the Coulomb constant
  • RK is the von Klitzing constant
  • Z0 is the vacuum impedance or impedance of free space


Gravitational coupling constant

From Wikipedia:Gravitational coupling constant:

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

where:

  • G is the gravitational constant
  • me is the electron rest mass
  • c is the speed of light in vacuum
  • ħ is the reduced Planck constant
  • mP is the Planck mass


Maxwell's equations

From Wikipedia:Lorentz–Heaviside units:

Name SI units Gaussian units Lorentz–Heaviside units
Gauss's law
(macroscopic)
Gauss's law
(microscopic)
Gauss's law for magnetism:
Maxwell–Faraday equation:
Ampère–Maxwell equation
(macroscopic):
Ampère–Maxwell equation
(microscopic):


Gravitoelectromagnetism

See also: Einstein_field_equations

From Wikipedia: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

where:

  • Eg is the static gravitational field (conventional gravity, also called gravitoelectric in analogous usage) in m⋅s−2;
  • E is the electric field;
  • Bg is the gravitomagnetic field in s−1;
  • B is the magnetic field;
  • ρg is mass density in kg⋅m−3;
  • ρ is charge density:
  • Jg is mass current density or mass flux (Jg = ρgvρ, where vρ is the velocity of the mass flow generating the gravitomagnetic field) in kg⋅m−2⋅s−1;
  • J is electric current density;
  • G is the gravitational constant in m3⋅kg−1⋅s−2;
  • ε0 is the vacuum permittivity;
  • c is the speed of propagation of gravity (which is equal to the speed of light according to general relativity) in m⋅s−1.


Electromagnetism

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

Therefore:

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:

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

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:

where:

But its probably better to say that:

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:

Solving we get m = 1.859 × 10-6 g = planck masses = 1 Stoney mass

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[2] 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)[3]

For monatomic gases:

From Wikipedia:Boltzmann constant:

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

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 is:

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:

Therefore:

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

Blackbody radiation

From Wikipedia:Black-body radiation:

Planck's law states that

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.

From Wikipedia:Larmor formula:

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

Uncertainty principle


References

  1. Wikipedia:Base unit (measurement)
  2. at -181 C
  3. Engineering ToolBox, (2003). Specific Heat and Individual Gas Constant of Gases. [online]
    Available at: https://www.engineeringtoolbox.com/specific-heat-capacity-gases-d_159.html
    [Accessed 20-4-2019].


External links