Physics: Problems and Solutions
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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

Fundamental units:

Electromagnetism

The field surrounding a charge holds energy. The total energy 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:

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 force between two electrons equal to the force between two unknown 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

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.

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 is the volume occupied by a single particle.
    • V0 = d3
  • 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

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.

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 =

References


External links

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