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:
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:
N
k
B
T
=
p
V
{\displaystyle Nk_{B}T=pV}
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
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 = \sqrt{v_x + v_y + v_z}}
is:
p
r
e
s
s
u
r
e
=
f
o
r
c
e
a
r
e
a
=
m
o
m
e
n
t
u
m
t
i
m
e
d
2
=
2
m
v
x
2
d
/
v
x
d
2
=
m
v
x
2
d
3
=
2
E
x
V
0
{\displaystyle 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}}
where:
V0 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
Therefore:
V
=
N
V
0
{\displaystyle V=NV_{0}}
Therefore:
N
T
=
p
V
=
p
N
V
0
=
N
m
v
2
=
N
2
E
{\displaystyle NT=pV=pNV_{0}=Nmv^{2}=N2E}
Therefore temperature is twice the energy per degree of freedom per particle
T
=
2
E
{\displaystyle T = 2 E}
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
External links