Physics: Problems and Solutions
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See also: Dimensional analysis and Nondimensionalization

Fundamental units:

Coulomb's law states that:

where:

  • is the Coulomb constant which is the "stiffness" of space. (If space were stiffer then c would be faster.)

The field surrounding a charge holds energy. The total energy is proportional to charge2 and inversely proportional to the radius(n-2). Therefore for any given amount of charge the quantity E * r(n-2) is constant.

The Coulomb constant has units of (Energy/charge2)*distance(n-2) = (Force*distance/charge2)*distance(n-2) which gives:

Newton's law of universal gravitation states that:

where:

But its probably better to say that:

Setting the force between two electrons equal to the force between two unknown masses we get:

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.

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.

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,

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
Coulomb constant
Gravitational constant
Boltzmann constant
Proton rest mass
Electron rest mass
Planck 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

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 of 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

Electromagnetism

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.

table

Base units
Dimension Planck
(Gauss)
Planck
(L-H)
Length (L)
Mass (M)
Time (T)
Electric charge (Q)
Temperature (Θ)

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

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