Metrology

Yard

• 0.9144 m
• 3 feet

Mile

• 1760 yards.
• It originated from the different unit system, the Roman one.

Pound

• 0.454 kg

Ounce

• oz, oz.
• Alchemical symbol for an ounce ℥, and half an ounce 🝳.
• 28.34 g

Fluid Ounce

• fl oz, fl. oz.
• US Customary: 29.57 mL
• US Food Labelling: 30 mL

Gill

• 4 US fluid ounces
• 5 imperial fluid ounces

Pint

• 4 gills

Quart

• quarter gallon
• 2 pints

Gallon

• gal
• US Gallon: 231 in³ (≈ 3.785 L)
• 4 quarts

• The definition may vary.

• Watt determined that a horse could turn a mill wheel 144 times in an hour, or 2.4 times in a minute. The wheel was 12 feet (3.7 m) in radius, and Watt judged that the horse could pull with a force of 180 pounds-force (800 N).
• \[ 1\,\mathrm{hp} = 180\,\mathrm{lbf}\cdot 2.4\,\mathrm{turn/min}\cdot (2\pi\cdot 12)\,\mathrm{ft/turn} = 32,572\,\mathrm{ft\,lbf/min} \]

• dB (base quantity) is the unit of the level.

\[ L = 10 \log \frac{Q}{Q_0}\ \mathrm{dB} \]

• A reference unit can be provided so that the quantity have a unit.

• Np
• natural logarithm is used.

• Log 2 of probability

• Natural log of probability

• \(\rm statC\), \(\rm Fr\)
• , esu charge

Electrostatic units first define the unit of charge, and derive other units from it.

It is defined by the Coulomb's law: \[ \mathbf{F} = \frac{q_1^\mathsf{ESU}q_2^\mathsf{ESU}}{r^2} \vu{r}. \] The unit of charge is called statcoulomb, and it is the amount of charge that exert one dyne of force when one centimeter apart from the same charge: \[ \rm statC = \sqrt{dyn \cdot cm^2} = cm^{3/2}\cdot g^{1/2}\cdot s^{-1} \] The point is to express charge entirely from mass, length, and time.

Statcoulomb is related to the SI unit coulomb by 1 C ≘ 10⁻¹ c statC, and the dimensions are related by: \[ q^\mathsf{ESU} = \frac{q^\mathsf{I}}{\sqrt{4\pi\varepsilon_0}} \]

Statcoulomb is related to the electromagnetic unit abcoulomb by c statC ≘ 1 abC, and the dimensions are also related by: \[ q^{\mathsf{ESU}} = c q^{\mathsf{EMU}}. \]

The factor of \(c\) between these units stayed mystery until the unification of electricity and magnetism.

\[ \rm statA := \frac{statC}{s} = cm^{3/2}\cdot g^{1/2} \cdot s^{-2} = \sqrt{dyn} \cdot cm \cdot s^{-1} \]

\[ \rm statV := \frac{erg}{statC} = \frac{cm^2\cdot g\cdot s^{-2}}{cm^{3/2}\cdot g^{1/2}\cdot s^{-1}} = cm^{1/2}\cdot g^{1/2}\cdot s^{-1} = \sqrt{dyn} \]

\[ \rm statV \cdot cm^{-1} = \sqrt{dyn} \cdot cm^{-1} \]

The unit of electric filed is defined by the definition of electric field: \[ \mathbf{E}^\mathsf{ESU} = \frac{q^\mathsf{ESU}}{r^2}\mathbf{\hat{r}} = \frac{\mathbf{F}}{q^\mathsf{ESU}}. \]

The relation to the SI system is: \[ \mathbf{E}^\mathsf{ESU} = \sqrt{4\pi\varepsilon_0}\,\mathbf{E}^\mathsf{SI} \]

\[ \rm stat \text{\Omega} := \frac{statV}{statA} = cm^{-1}\cdot s \]

• \(\rm statT\)

It is defined by the Lorentz force: \[ \mathbf{F} = q^{\sf ESU}\mathbf{v}\times \mathbf{B}^{\sf ESU}. \]

\[ \rm statT := \frac{dyn\cdot s}{statC\cdot cm} = cm^{-3/2}\cdot g^{1/2} = \sqrt{dyn} \cdot s \cdot cm^{-2} \]

Stattesla is related to the SI unit tesla by 1T ≘ 10⁻⁴ c⁻¹ statT, and the dimensions are related by: \[ \mathbf{B}^{\sf ESU} = \sqrt{4\pi\varepsilon_0}\,\mathbf{B}^{\sf SI} = \sqrt{\frac{4\pi}{\mu_0}}\frac{\mathbf{B}^{\mathsf{SI}}}{c}. \]

It does not include the factor of \(c\), compared to Gauss.

• The unit of magnetic flux

\[ \rm statWb := statT \cdot cm^2 \]

• \(\rm abA\), \(\rm Bi\)
• Biot, emu current

Electromagnetic units first define the unit of current, and derive other units from it.

It is defined by the force per centimeter \(f\) between two parallel wires of infinite length: \[ \frac{f}{2} =\frac{I_1^{\sf EMU}I_2^{\sf EMU}}{r}. \] The unit of current is called abampere, and it is the amount of current that exert two dyne per centimeter of force for two wires with same current are one centimeter apart from each other.

The factor of two is to cancel the two that arises for the magnetic field when applying the Biot-Savart law \[ \mathbf{B}^{\sf EMU} = \frac{I^{\sf EMU}\dd{\mathbf{l}}\times \mathbf{\hat{r}}}{r^2} \] for infinitely long wires, and applying the Lorentz force \[ \mathbf{F} = I^{\mathsf{EMU}} \dd{\mathbf{l}} \times \mathbf{B}^{\mathsf{EMU}}. \]

When represented with basis units: \[ \rm abA = \sqrt{dyn} = g^{1/2}\cdot cm^{1/2}\cdot s^{-1}. \]

Abampere is related to the SI unit ampere by 1 A ≘ 10⁻¹ abA and the dimensions are related by: \[ I^{\sf EMU} = \frac{I^{\sf SI}}{\sqrt{4\pi / \mu_0}} = \frac{I^{\sf SI}}{c\sqrt{4\pi\varepsilon_0}}. \]

\[ \rm abC := abA \cdot s = g^{1/2}\cdot cm^{1/2} = \sqrt{dyn} \cdot s \]

• Abtesla

Gauss is defined to be consistent with Biot-Savart law and Lorentz force, as explained above in abampere. \[ \rm G = \frac{dyn}{abA \cdot cm} = \frac{abA \cdot cm}{cm^2} = cm^{-1/2}\cdot g^{1/2}\cdot s^{-1} = \sqrt{dyn}\cdot cm^{-1} \]

The relation to the SI unit tesla is: 1 T ≘ 10⁴ G and the dimensions are related by: \[ \mathbf{B}^{\sf EMU} = c\sqrt{4\pi\varepsilon_0}\,\mathbf{B}^{\sf SI} = \sqrt{4\pi/\mu_0}\mathbf{B}^{\mathsf{SI}}. \]

It is related to the electrostatic unit stattesla by: 1 statT ≘ c G while the dimensions are related by: \[ c\mathbf{B}^{\mathsf{ESU}} = \mathbf{B}^{\mathsf{EMU}}. \]

The conversion factor \(10^4\) from gauss to tesla is particularly nice due to the value of permeability of the vacuum. \[ \frac{B^{\mathsf{EMU}}}{B^\mathsf{SI}} = \sqrt{\frac{4\pi}{\mu_0}} \approx \sqrt{\frac{4\pi}{4\pi\times 10^{-7}\ \mathrm{N/A^2}}} = 10^{\frac{7}{2}}\rm\ N^{-1/2}A = \frac{10^4\ G}{1\ T} \]

The digits of permeability of vacuum being so close to \(4\pi\) is in fact not a coincidence. Because the early definition (until 2019) of the unit of magnetic field was:

• 1/(2⋅10⁻⁷) times
• the magnetic field strength produced at one of the two identical infinitely long wire
• with identical electric current flowing
• separated by one meter
• and exert force of 2⋅10⁻⁷ neuton per meter to each other (definition of 1 ampere).

Just like the defintiion of Gauss.

In order to cancel the inclusion of \(4\pi\) in the denominator of Biot-Savart law, \(4\pi\) is included in \(\mu_0\), and in order to reduce the effect of infinite wire to a infinitesimal segment, the force is halved, giving us the final formula for the familiar Biot-Savar law: \[ \mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \dd{\mathbf{l}}\times \vu{\mathbf{r}}}{r^2} \] and the value of permeability of vacuum: \[ \mu_0 = 4\pi \times 10^{-7}\,\rm T\cdot m/A. \]

After 2019, ampere has been decoupled from the unit of force, and how much force it generates is now a matter of measurement.

The unit of auxiliary magnetic field \(\mathbf{H}\) \[ \rm Oe := abA \cdot cm^{-1} \]

The unit of magnetic flux \(\Phi\) \[ \rm Mx := G \cdot cm^2 \]

Spectral intensity in frequency \[ I_{\rm e, \Omega, \nu} := \pdv{I_{\rm e, \Omega}}{\nu} \]

Spectral intensity in frequency \[ I_{\rm e, \Omega, \lambda} := \pdv{I_{\rm e, \omega}}{\lambda} \]

Spectral irradiance in frequency \[ E_{\rm e, \nu} := \pdv{E_{\rm e}}{\nu} \]

Spectral irradiance in wavelength \[ E_{\rm e, \lambda} := \pdv{E_{\rm e}}{\lambda} \]

Spectral exposure in frequency \[ H_{\rm e, \nu} := \pdv{H_{\rm e}}{\nu} \]

Spectral irradiance in wavelength \[ H_{\rm e, \lambda} := \pdv{H_{\rm e}}{\lambda} \]

Spectral exitance in frequency \[ M_{\rm e, \nu} := \pdv{M_{\rm e}}{\nu} \]

Spectral exitance in wavelength \[ M_{\rm e, \lambda} := \pdv{M_{\rm e}}{\lambda} \]

Spectral radiosity in frequency \[ J_{\rm e, \nu} := \pdv{J_{\rm e}}{\nu} \]

Spectral radiosity in wavelength \[ J_{\rm e, \lambda} := \pdv{J_{\rm e}}{\lambda} \]

Spectral radiance in frequency \[ L_{\rm e, \Omega,\nu} := \pdv{L_{\rm e, \Omega}}{\nu} \]

Spectral radiance in wavelength \[ L_{\rm e, \Omega, \lambda} := \pdv{L_{\rm e, \Omega}}{\lambda} \]

• Every quantities are scaled by the factor of plank quantities, yielding dimensionless quantities that corresponds to the normal quantities.

• The units can be thought of as the ratio to the Plank quantities.
  • e.g.
    • \[ [\text{length}] = \frac{\rm m}{l_\text{P}} \]

• The effect is equivalent to setting \(c = G = \hbar = k_\text{B} = 1\)
• \[ F = \frac{m_1m_2}{r^2} \]

• Only the \(\varepsilon_0\) is removed compared to the Gaussian units.
• \[ q^{\sf HL} := \frac{q^{\sf SI}}{\sqrt{\varepsilon_0}} \]
• \[ \mathbf{E}^{\sf HL} := \sqrt{\varepsilon_0}\,\mathbf{E}^{\sf SI} \]
• \[ \mathbf{B}^{\sf HL} := c\sqrt{\varepsilon_0}\,\mathbf{B}^{\sf SI} = \frac{1}{\sqrt{\mu_0}}\mathbf{B}^{\sf SI} \]

• Similar to Planck units, but set \(\hbar = e = m_\text{e} = 4\pi\varepsilon_0 = 1\).