Ciro Santilli @cirosantilli 34

 Incoming links: Speed of light

2019 redefinition of the SI base units  Updated 2024-12-15  +Created 1970-01-01

web.archive.org/web/20181119214326/https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf gives it in raw:

the unperturbed ground state hyperfine transition frequency of the caesium-133 atom $Δ v_{C s}$ is 9 192 631 770 Hz
the speed of light in vacuum c is 299 792 458 m/s
the Planck constant h is 6.626 070 15 × $1 0^{- 34}$ J s
the elementary charge e is 1.602 176 634 × $1 0^{- 19}$ C
the Boltzmann constant k is 1.380 649 × $1 0^{- 23}$ J/K
the Avogadro constant NA is 6.022 140 76 × $1 0^{23}$ mol
the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W,

The breakdown is:

actually use some physical constant:
- the unperturbed ground state hyperfine transition frequency of the caesium-133 atom $Δ v_{C s}$ is 9 192 631 770 Hz
  Defines the second in terms of caesium-133 experiments. The beauty of this definition is that we only have to count an integer number of discrete events, which is what allows us to make things precise.
- the speed of light in vacuum c is 299 792 458 m/s
  Defines the meter in terms of speed of light experiments. We already had the second from the previous definition.
- the Planck constant h is 6.626 070 15 × $1 0^{- 34}$ J s
  Defines the kilogram in terms of the Planck constant.
- the elementary charge e is 1.602 176 634 × $1 0^{- 19}$ C
  Defines the Coulomb in terms of the electron charge.
arbitrary definitions based on the above just to match historical values as well as possible:
- the Boltzmann constant k is 1.380 649 × $1 0^{- 23}$ J/K
  Arbitrarily defines temperature from previously defined energy (J) to match historical values.
- the Avogadro constant NA is 6.022 140 76 × $1 0^{23}$ mol
  Arbitrarily defines the mol to match historical values. In particular, the kilogram is not an exact multiple of the weight of an atom of hydrogen.
- the luminous efficacy of monochromatic radiation of frequency 540 × 1012 Hz, Kcd, is 683 lm/W
  Arbitrarily defines the Candela in terms of previous values to match historical records. The most useless unit comes last as you'd expect.

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Alpha decay  Updated 2024-12-15  +Created 1970-01-01

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Most of the helium in the Earth's atmosphere comes from alpha decay, since helium is lighter than air and naturally escapes out out of the atmosphere.

Wiki mentions that alpha decay is well modelled as a quantum tunnelling event, see also Geiger-Nuttall law.

As a result of that law, alpha particles have relatively little energy variation around 5 MeV or a speed of about 5% of the speed of light for any element, because the energy is inversely exponentially proportional to half-life. This is because:

if the energy is much larger, decay is very fast and we don't have time to study the isotope
if the energy is much smaller, decay is very rare and we don't have enough events to observe at all

Video 1.

Quantum tunnelling and the Alpha particle Paradox by Physics Explained (2022)

Source.

youtu.be/_f8zeEI0oys?t=796 George Gamow and Edward Condon proposed the quantum tunnelling explanation
youtu.be/_f8zeEI0oys?t=1725 worked out example that predicts the half-life of polonium-210 based on its emission energy

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Boltzmann constant  Updated 2024-12-15  +Created 1970-01-01

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This is not a truly "fundamental" constant of nature like say the speed of light or the Planck constant.

Rather, it is just a definition of our Kelvin temperature scale, linking average microscopic energy to our macroscopic temperature scale.

The way to think about that link is, at 1 Kelvin, each particle has average energy:

1 / 2 k T

(1)

per degree of freedom.

This is why the units of the Boltzmann constant are Joules per Kelvin.

For an ideal monatomic gas, say helium, there are 3 degrees of freedom. so each helium atom has average energy:

3 / 2 k_{B} T

(2)

If we have 2 atoms at 1 K, they will have average energy

6 / 2 k_{B} J

, and so on.

Another conclusion is that this defines temperature as being proportional to the total energy. E.g. if we had 1 helium atom at 2 K then we would have about

6 / 2 k_{B} J

energy, 3 K

9 / 2 k_{B} J

and so on.

This energy is of course just an average: some particles have more, and others less, following the Maxwell-Boltzmann distribution.

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Lorentz transform consequence: everyone sees the same speed of light  Updated 2024-12-15  +Created 1970-01-01

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OK, so let's verify the main desired consequence of the Lorentz transformation: that everyone observes the same speed of light.

Observers will measure the speed of light by calculating how long it takes the light going towards

+ x

cross a rod of length

L = x_{2} - x_{1}

laid in the x axis at position

X 1

TODO image.

Each observer will observe two events:

$(t_{1}, x_{1}, y_{1}, z_{1})$ : the light touches the left side of the rod
$(t_{2}, x_{2}, y_{2}, z_{2})$ : the light touches the right side of the rod

Supposing that the standing observer measures the speed of light as

c

and that light hits the left side of the rod at time

T 1

, then he observes the coordinates:

t_{1} x_{1} t_{2} x_{2} = T 1 = X 1 = \frac{L}{c} = X 1 + L

(1)

Now, if we transform for the moving observer:

t_{1}^{'} x_{1}^{'} t_{2}^{'} x_{2}^{'} = γ (t_{1} - \frac{v x _{1}}{c ^{2}}) = γ (x_{1} - v t_{1}) = γ (t_{2} - \frac{v x _{2}}{c ^{2}}) = γ (x_{2} - v t_{2})

(2)

and so the moving observer measures the speed of light as:

c^{'} = \frac{x _{2}^{'} - x _{1}^{'}}{t _{2}^{'} - t _{1}^{'}} = \frac{( x _{2} - v t _{2} ) - ( x _{1} - v t _{1} )}{( t _{2} - \frac{v x _{2}}{c ^{2}} ) - ( t _{1} - \frac{v x _{1}}{c ^{2}} )} = \frac{( x _{2} - x _{1} ) - v ( t _{2} - t _{1} )}{( t _{2} - t _{1} ) - \frac{v}{c ^{2}} ( x _{2} - x _{1} )} = \frac{\frac{x _{2} - x _{1}}{t _{2} - t _{1}} - v}{1 - \frac{v}{c ^{2}} \frac{x _{2} - x _{1}}{t _{2} - t _{1}}} = \frac{c - v}{1 - \frac{v}{c ^{2}} c} = \frac{c - v}{\frac{c - v}{c}} = c

(3)

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Maxwell's equations  Updated 2024-12-15  +Created 1970-01-01

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Unified all previous electro-magnetism theories into one equation.

Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.

The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.

The equations are a system of partial differential equation.

The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:

$E_{x} (t, x, y, z)$ , $E_{y} (t, x, y, z)$ , $E_{z} (t, x, y, z)$ : directions of the electric field $E : R^{4} \to R^{3}$
$B_{x} (t, x, y, z)$ , $B_{y} (t, x, y, z)$ , $B_{z} (t, x, y, z)$ : directions of the magnetic field $B : R^{4} \to R^{3}$

and two known input functions:

$ρ : R^{3} t o R$ : density of charges in space
$J : R^{3} \to R^{3}$ : current vector in space. This represents the strength of moving charges in space.

Due to the conservation of charge however, those input functions have the following restriction:

\frac{\partial ρ}{\partial t} + \nabla \cdot J = 0

(1)

Equation 1.

Charge conservation

Also consider the following cases:

if a spherical charge is moving, then this of course means that $ρ$ is changing with time, and at the same time that a current exists
in an ideal infinite cylindrical wire however, we can have constant $ρ$ in the wire, but there can still be a current because those charges are moving
Such infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.

The goal of finding

E

and

B

is that those fields allow us to determine the force that gets applied to a charge via the Equation "Lorentz force", and then to find the force we just need to integrate over the entire body.

Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations: