FANDOM


Line 13: Line 13:
 
*Charge
 
*Charge
   
  +
== CGS system of units ==
   
=== Electromagnetism ===
+
From [[Wikipedia:Centimetre–gram–second system of units]]:
   
The total energy in the electric field surrounding a hollow spherical shell of radius {{mvar|r}} and charge {{mvar|q}} is:
+
{| class="wikitable" style="text-align: left;"
 
:<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
+
! Quantity
| 1.667
+
! Quantity symbol
| 3
+
! CGS unit name
  +
! Unit<br>symbol
  +
! Unit definition
  +
! Equivalent<br>in SI units
 
|-
 
|-
| Neon
+
! 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
+
! 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
+
! 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
+
! 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
+
! 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
+
! [[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
+
! 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
+
! [[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
+
! 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
+
! 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>
+
! 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> || =
 
|}
 
|}
 
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]]
 
   
 
== Natural units ==
 
== Natural units ==
Line 549: Line 479:
   
   
== CGS system of units ==
 
   
From [[Wikipedia:Centimetre–gram–second system of units]]:
+
=== Electromagnetism ===
   
{| class="wikitable" style="text-align: left;"
+
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
 
|-
 
|-
! Quantity
+
| Helium
! Quantity symbol
+
| 1.667
! CGS unit name
+
| 3
! Unit<br>symbol
 
! Unit definition
 
! Equivalent<br>in SI units
 
 
|-
 
|-
! length, position
+
| Neon
| style="text-align:center;"| ''L'', ''x''|| [[centimetre]] || style="text-align:center;"| cm || 1/100 of [[metre]] || = 10<sup>−2</sup>&nbsp;m
+
| 1.667
  +
| 3
 
|-
 
|-
! mass
+
| Argon
| style="text-align:center;"| ''m''|| [[gram]] || style="text-align:center;"|g || 1/1000 of [[kilogram]] || = 10<sup>−3</sup>&nbsp;kg
+
| 1.667
  +
| 3
 
|-
 
|-
! time
+
| Hydrogen
| style="text-align:center;"| ''t''|| second|| style="text-align:center;"|s|| 1 second || = 1&nbsp;s
+
| 1.597<ref>at -181 C</ref>
  +
| 3.35
 
|-
 
|-
! velocity
+
| Hydrogen
| style="text-align:center;"| ''v''|| centimetre per second || style="text-align:center;"|cm/s || cm/s || = 10<sup>−2</sup>&nbsp;m/s
+
| 1.41
  +
| 4.88
 
|-
 
|-
! acceleration
+
| Nitrogen
| 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>
+
| 1.4
  +
| 5
 
|-
 
|-
! [[force (physics)|force]]
+
| Oxygen
| 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]]
+
| 1.395
  +
| 5.06
 
|-
 
|-
! energy
+
| Chlorine
| 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]]
+
| 1.34
  +
| 5.88
 
|-
 
|-
! [[power (physics)|power]]
+
| Carbon dioxide
| 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]]
+
| 1.289
  +
| 6.92
 
|-
 
|-
! pressure
+
| Methane
| 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]]
+
| 1.304
  +
| 6.58
 
|-
 
|-
! dynamic [[viscosity]]
+
| Ethane
| 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]]
+
| 1.187
|-
+
| [https://www.wolframalpha.com/input/?i=2%2F(1.187-1) 10.7]
! 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]]
+
| [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="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> || =
 
 
|}
 
|}
  +
  +
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]]
   
   

Revision as of 13:12, May 6, 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:

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 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 =

Natural units

From Wikipedia: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

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
$ 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

From Wikipedia: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

From Wikipedia: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

From Wikipedia:Lorentz–Heaviside units:

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

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
$ \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:


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 = 1.859 × 10-6 g = $ \sqrt{\alpha} $ 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:

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

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:

$ 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 $

Blackbody radiation

From Wikipedia:Black-body radiation:

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.

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:

$ 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)} $

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

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