Line 299:
Line 299:
*<math> T = 2 E</math>
*<math> T = 2 E</math>
+
+
The pressure exerted on one face of a cube of length {{mvar|d}} by a single particle of mass {{mvar|m}} and velocity <math>v = v_x + v_y + v_z</math> is:
+
+
:<math>\frac{momentum}{time} = \frac{2 m v_x}{2 d / v_x}</math>
== Electromagnetism ==
== Electromagnetism ==
Revision as of 23:18, 16 April 2019
See also: Dimensional analysis and Nondimensionalization
Fundamental units:
Coulomb's law states that:
F
=
k
e
q
1
q
2
d
2
{\displaystyle F=k_e \frac{q_1 q_2}{d^2}}
where:
k
e
{\displaystyle k_e}
is the Coulomb constant which is the "stiffness " of space.
If the Coulomb constant is "negative pressure" then it has units of Energy/volume = Force/area which gives:
F
=
F
d
2
q
1
q
2
d
2
{\displaystyle F={\frac {F}{d^{2}}}{\frac {q_{1}q_{2}}{d^{2}}}}
Therefore:
q
=
d
2
{\displaystyle q=d^{2}}
Charge is therefore no longer a fundamental unit but instead has units of distance squared.
Natural units
From Wikipedia:natural units :
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.
In Lorentz–Heaviside units (rationalized units), Coulomb's law is:
F
=
q
1
q
2
r
2
1
4
π
{\displaystyle F=\frac{q_1 q_2}{r^2} \frac{1}{4 \pi}}
In Gaussian units (non-rationalized units), Coulomb's law is:
F
=
q
1
q
2
r
2
{\displaystyle F = \frac{q_1 q_2}{r^2} }
Planck units are defined by
c = ħ = G = k e = k B = 1 ,
Stoney units are defined by:
c = G = k e = e = k B = 1 ,
Hartree atomic units are defined by:
e = m e = ħ = k e = k B = 1
c = 1 α
Rydberg atomic units are defined by:
e √2 = 2m e = ħ = k e = k B = 1
c = 2 α
Quantum chromodynamics (QCD) units are defined by:
c = m p = ħ = k B = 1
Natural units generally means:
ħ = c = k B = 1 .
where:
c is the speed of light ,
ħ is the reduced Planck constant ,
G is the gravitational constant ,
k e is the Coulomb constant ,
k B is the Boltzmann constant
e is the elementary charge ,
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
{\displaystyle c \,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
α
{\displaystyle \frac{1}{\alpha} \ }
2
α
{\displaystyle \frac{2}{\alpha} \ }
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
Reduced Planck constant
ℏ
=
h
2
π
{\displaystyle \hbar = \frac{h}{2 \pi}}
1
4
π
α
{\displaystyle \frac{1}{4 \pi \alpha}}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
α
{\displaystyle \frac{1}{\alpha} \ }
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
Elementary charge
e
{\displaystyle e \,}
1
{\displaystyle 1\,}
4
π
α
{\displaystyle \sqrt{4\pi\alpha} \,}
α
{\displaystyle \sqrt{\alpha} \,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
2
{\displaystyle \sqrt{2} \,}
4
π
α
{\displaystyle \sqrt{4\pi\alpha}}
α
{\displaystyle \sqrt{\alpha}}
1
{\displaystyle 1\,}
4
π
α
{\displaystyle \sqrt{4\pi\alpha} \,}
α
{\displaystyle \sqrt{\alpha} \,}
Vacuum permittivity
ε
0
{\displaystyle \varepsilon_0 \,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
{\displaystyle 1\,}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
4
π
α
{\displaystyle \frac{1}{4 \pi \alpha}}
1
{\displaystyle 1\,}
1
4
π
{\displaystyle \frac{1}{4\pi}}
Vacuum permeability
μ
0
=
1
ϵ
0
c
2
{\displaystyle \mu_0 = \frac{1}{\epsilon_0 c^2} \,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
4
π
{\displaystyle 4\pi}
4
π
{\displaystyle 4\pi}
4
π
α
2
{\displaystyle 4 \pi \alpha^2}
π
α
2
{\displaystyle \pi \alpha^2}
1
{\displaystyle 1\,}
4
π
{\displaystyle 4\pi}
4
π
α
{\displaystyle 4 \pi \alpha}
1
{\displaystyle 1\,}
4
π
{\displaystyle 4\pi}
Impedance of free space
Z
0
=
1
ϵ
0
c
=
μ
0
c
{\displaystyle Z_0 = \frac{1}{\epsilon_0 c} = \mu_0 c \,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
4
π
{\displaystyle 4\pi}
4
π
{\displaystyle 4\pi}
4
π
α
{\displaystyle 4 \pi \alpha}
2
π
α
{\displaystyle 2\pi\alpha}
1
{\displaystyle 1\,}
4
π
{\displaystyle 4\pi}
4
π
α
{\displaystyle 4 \pi \alpha}
1
{\displaystyle 1\,}
4
π
{\displaystyle 4\pi}
Coulomb constant
k
e
=
1
4
π
ϵ
0
{\displaystyle k_e=\frac{1}{4 \pi \epsilon_0}}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
{\displaystyle 1\,}
α
{\displaystyle \alpha}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
{\displaystyle 1\,}
Gravitational constant
G
{\displaystyle G \,}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
4
π
{\displaystyle \frac{1}{4\pi}}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
α
G
α
{\displaystyle \frac{\alpha_\text{G}}{\alpha} \,}
8
α
G
α
{\displaystyle \frac{8 \alpha_\text{G}}{\alpha} \,}
α
G
m
e
2
{\displaystyle \frac{\alpha_\text{G}}{{m_\text{e}}^2} \,}
α
G
m
e
2
{\displaystyle \frac{\alpha_\text{G}}{{m_\text{e}}^2} \,}
μ
2
α
G
{\displaystyle \mu^2 \alpha_\text{G}}
μ
2
α
G
{\displaystyle \mu^2 \alpha_\text{G}}
μ
2
α
G
{\displaystyle \mu^2 \alpha_\text{G}}
Boltzmann constant
k
B
{\displaystyle k_\text{B} \,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
Proton rest mass
m
p
{\displaystyle m_\text{p} \,}
μ
4
π
α
G
{\displaystyle \mu \sqrt{4 \pi \alpha_\text{G}} \,}
μ
α
G
{\displaystyle \mu \sqrt{\alpha_\text{G}} \,}
μ
α
G
α
{\displaystyle \mu \sqrt{\frac{\alpha_\text{G}}{\alpha}} \,}
μ
{\displaystyle \mu \,}
μ
2
{\displaystyle \frac{\mu}{2} \,}
938
MeV
{\displaystyle 938 \text{ MeV}}
938
MeV
{\displaystyle 938 \text{ MeV}}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
Electron rest mass
m
e
{\displaystyle m_\text{e} \,}
4
π
α
G
{\displaystyle \sqrt{4 \pi \alpha_\text{G}} \,}
α
G
{\displaystyle \sqrt{\alpha_\text{G}} \,}
α
G
α
{\displaystyle \sqrt{\frac{\alpha_\text{G}}{\alpha}} \,}
1
{\displaystyle 1\,}
1
2
{\displaystyle \frac{1}{2} \,}
511
keV
{\displaystyle 511 \text{ keV}}
511
keV
{\displaystyle 511 \text{ keV}}
1
μ
{\displaystyle \frac{1}{\mu}}
1
μ
{\displaystyle \frac{1}{\mu}}
1
μ
{\displaystyle \frac{1}{\mu}}
Planck mass
m
P
=
ℏ
c
G
,
{\displaystyle m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}},}
1
α
{\displaystyle {\sqrt {\frac {1}{\alpha }}}}
where:
α is the dimensionless fine-structure constant
α G 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 :
α
=
1
4
π
ε
0
e
2
ℏ
c
=
μ
0
4
π
e
2
c
ℏ
=
k
e
e
2
ℏ
c
=
c
μ
0
2
R
K
=
e
2
4
π
Z
0
ℏ
{\displaystyle
\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:
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
ε 0 is the electric constant or permittivity of free space
µ 0 is the magnetic constant or permeability of free space
k e is the Coulomb constant
R K is the von Klitzing constant
Z 0 is the vacuum impedance or impedance of free space
Gravitational coupling constant
From Wikipedia:Gravitational coupling constant :
The Gravitational coupling constant , α G , is typically defined in terms of the gravitational attraction between two electrons. More precisely,
α
G
=
G
m
e
2
ℏ
c
=
(
m
e
m
P
)
2
≈
1.751751596
×
10
−
45
{\displaystyle \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:
G is the gravitational constant
m e is the electron rest mass
c is the speed of light in vacuum
ħ is the reduced Planck constant
m P is the Planck mass
Boltzmann 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:
p
V
=
N
k
B
T
=
N
k
B
m
v
2
=
N
k
B
2
E
{\displaystyle pV=Nk_{B}T=Nk_{B}mv^{2}=Nk_{B}2E}
where:
p is the pressure
V is the volume
N is the number of molecules of gas.
kB is the Boltzmann constant
T is the temperature
v is the velocity
Twice the velocity means twice as much momentum transferred per collision and twice as many collisions per unit time.
E is the kinetic energy per particle
Therefore temperature is twice the energy per degree of freedom per particle
T
=
2
E
{\displaystyle T = 2 E}
The pressure exerted on one face of a cube of length d by a single particle of mass m and velocity
v
=
v
x
+
v
y
+
v
z
{\displaystyle v=v_{x}+v_{y}+v_{z}}
is:
m
o
m
e
n
t
u
m
t
i
m
e
=
2
m
v
x
2
d
/
v
x
{\displaystyle {\frac {momentum}{time}}={\frac {2mv_{x}}{2d/v_{x}}}}
Electromagnetism
From Wikipedia:Lorentz–Heaviside units :
Name
SI units
Gaussian units
Lorentz–Heaviside units
Gauss's law
(macroscopic)
∇
⋅
D
=
ρ
f
{\displaystyle \nabla \cdot \mathbf{D} = \rho_\text{f}}
∇
⋅
D
=
4
π
ρ
f
{\displaystyle \nabla \cdot \mathbf{D} = 4\pi\rho_\text{f}}
∇
⋅
D
=
ρ
f
{\displaystyle \nabla \cdot \mathbf{D} = \rho_\text{f}}
Gauss's law
(microscopic)
∇
⋅
E
=
ρ
/
ϵ
0
{\displaystyle \nabla \cdot \mathbf{E} = \rho/\epsilon_0}
∇
⋅
E
=
4
π
ρ
{\displaystyle \nabla \cdot \mathbf{E} = 4 \pi \rho}
∇
⋅
E
=
ρ
{\displaystyle \nabla \cdot \mathbf{E} = \rho}
Gauss's law for magnetism :
∇
⋅
B
=
0
{\displaystyle \nabla \cdot \mathbf{B} = 0}
∇
⋅
B
=
0
{\displaystyle \nabla \cdot \mathbf{B} = 0}
∇
⋅
B
=
0
{\displaystyle \nabla \cdot \mathbf{B} = 0}
Maxwell–Faraday equation :
∇
×
E
=
−
∂
B
∂
t
{\displaystyle \nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}}
∇
×
E
=
−
1
c
∂
B
∂
t
{\displaystyle \nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}}
∇
×
E
=
−
1
c
∂
B
∂
t
{\displaystyle \nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t}}
Ampère–Maxwell equation
(macroscopic):
∇
×
H
=
J
f
+
∂
D
∂
t
{\displaystyle \nabla \times \mathbf{H} = \mathbf{J}_{\text{f}} + \frac{\partial \mathbf{D}} {\partial t}}
∇
×
H
=
4
π
c
J
f
+
1
c
∂
D
∂
t
{\displaystyle \nabla \times \mathbf{H} = \frac{4\pi}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}}
∇
×
H
=
1
c
J
f
+
1
c
∂
D
∂
t
{\displaystyle \nabla \times \mathbf{H} = \frac{1}{c}\mathbf{J}_{\text{f}} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}}
Ampère–Maxwell equation
(microscopic):
∇
×
B
=
μ
0
J
+
1
c
2
∂
E
∂
t
{\displaystyle \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \frac{1}{c^2}\frac{\partial \mathbf{E}} {\partial t}}
∇
×
B
=
4
π
c
J
+
1
c
∂
E
∂
t
{\displaystyle \nabla \times \mathbf{B} = \frac{4\pi}{c}\mathbf{J} + \frac{1}{c}\frac{\partial \mathbf{E}} {\partial t}}
∇
×
B
=
1
c
J
+
1
c
∂
E
∂
t
{\displaystyle \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
∇
⋅
E
g
=
−
4
π
G
ρ
g
{\displaystyle \nabla \cdot \mathbf{E}_\text{g} = -4 \pi G \rho_\text{g} \ }
∇
⋅
E
=
ρ
ϵ
0
{\displaystyle \nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}}
∇
⋅
B
g
=
0
{\displaystyle \nabla \cdot \mathbf{B}_\text{g} = 0 \ }
∇
⋅
B
=
0
{\displaystyle \nabla \cdot \mathbf{B} = 0 \ }
∇
×
E
g
=
−
∂
B
g
∂
t
{\displaystyle \nabla \times \mathbf{E}_\text{g} = -\frac{\partial \mathbf{B}_\text{g} } {\partial t} \ }
∇
×
E
=
−
∂
B
∂
t
{\displaystyle \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B} } {\partial t} \ }
∇
×
B
g
=
−
4
π
G
c
2
J
g
+
1
c
2
∂
E
g
∂
t
{\displaystyle \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} }
∇
×
B
=
1
ϵ
0
c
2
J
+
1
c
2
∂
E
∂
t
{\displaystyle \nabla \times \mathbf{B} = \frac{1}{\epsilon_0 c^2} \mathbf{J} + \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t} }
where:
E g is the static gravitational field (conventional gravity , also called gravitoelectric in analogous usage) in m⋅s−2 ;
E is the electric field ;
B g is the gravitomagnetic field in s−1 ;
B is the magnetic field ;
ρ g is mass density in kg⋅m−3 ;
ρ is charge density :
J g is mass current density or mass flux (J g = ρ g v ρ , where v ρ 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 ;
ε 0 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 .
CGS
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
=
References