Useful formulas and physical constants

This page contains useful formulas used in our research and engineering, as well as code for the open-source Numbat scientific calculator that implements the formulas.

Important information

Some of the code implementations of these formulas only work if the functions are placed in a separate file that is interpreted by Numbat; they are not guaranteed to work in REPL/interactive session mode! Thus it is highly recommended to save the functions to a custom module that you can then import into an interactive section. Specifically, create a file <module-path>/custom/functions.nbt, where <module-path> can be found according to the below table:

Platform	Path
Linux	$HOME/.config/numbat or $XDG_CONFIG_HOME/numbat
macOS	$HOME/Library/Application Support/numbat
Windows	C:\Users\<username>\AppData\Roaming\numbat

Then, simply put all the below functions into <module-path>/custom/functions.nbt. You can easily import them in any Numbat interactive session (REPL) by simply running (in Numbat):

use custom::functions

Tip: It is useful to add a variable NUMBAT_LIB by putting export NUMBAT_LIB = <module-path> (Unix/Bash-like shells) or $env:NUMBAT_LIB = <module-path> (Windows Powershell) in your shell config (~/.bashrc, ~/.zshrc etc. for Unix, $profile for Windows Powershell) so that you can simply type <youreditor> $NUMBAT_LIB to quickly edit the file (e.g. notepad $NUMBAT_LIB on Windows and nano $NUMBAT_LIB on Mac/Linux).

For more information, please consult the Numbat documentation for more details.

Electricity & magnetism

Electrical power $P$ transferred of current $I$ across potential $\Delta V$:

$$P=I\Delta V$$

Numbat code:

# Formula definition
fn electrical_power(I: Current, deltaV: Voltage) -> Power = I*deltaV

# Usage
electrical_power(somecurrent, somevoltage)
electrical_power(somecurrent, somevoltage) -> W # Optionally convert to watts

Stefan-Boltzmann law:

$$u=\epsilon \sigma T^4$$

Where $u$ is the intensity (power density, in units of power over area), $\epsilon$ is the emissivity and is a dimensionless number between 0 and 1 ($\epsilon =1$ for a perfect blackbody), $T$ is the absolute temperature (in Kelvin) and $\sigma =5.67037\times 10^{-8}\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot {\mathrm{K}}^{-4}$ is the Stefan-Boltzmann constant.

Relativity

Electron normalized velocity ($\beta =v/c$) as function of electrostatic potential $\Delta V$, electron mass $m_e$, electron charge $e$, and speed of light $c$:

$$\beta =\sqrt{1-{\left(1+\frac{e\Delta V}{m_e{c}^2}\right)}^{-2}}=\sqrt{1-{\left(1+\frac{e\Delta V}{\text{0.511 MeV}}\right)}^{-2}}$$

Calculation code (for Numbat):

# Formula definition
fn relativity_beta(deltaV: Voltage) -> Scalar = 
		sqrt(1 - (1 + ratio)^(-2))
	where electron_mass_energy = electron_mass * c^2
	  and electron_kinetic_energy = elementary_charge * deltaV
	  and ratio = electron_kinetic_energy/electron_mass_energy

# Usage
relativity_beta(somevoltage)
print("{relativity_beta(somevoltage)}c") # pretty-print e.g. "0.5c"

Warning This code only works if you put the function in a separate file (e.g. as a custom module). Please see the top of the page for more details.

Vacuum systems

Density of dry air ${\rho }_{air}$ as function of temperature $T$ and pressure $P$ where $R_{air}$ is the specific gas constant of air¹:

$${\rho }_{air}=\frac{P}{R_{air}T},R_{air}=287.052874\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}\cdot {\mathrm{K}}^{-1}$$

Note that for room temperature air, it is common to set $T=\text{300 K}$ (equivalent to $26.85^{\circ }\text{C}$ or $80.33^{\circ }\text{F}$).

# Formula definition (one-liner)
fn dry_air_density(P: Pressure, T: Temperature) -> MassDensity = P/(T*287.052874 J / (kg*K))

# Equivalent multiline form (more readable)
fn dry_air_density(P: Pressure, T: Temperature) -> MassDensity = P/(R_air * T) 
	where R_air = 287.052874 J / (kg*K)

# Usage
dry_air_density(somepressure, 300K) # for room temperature, arbitrary pressure

Warning The multiline version of the code only works if you put the function in a separate file (e.g. as a custom module). Please see the top of the page for more details. Note that the one-liner version does not suffer from the same issue.

Footnotes

1. See https://en.wikipedia.org/wiki/Gas_constant#Specific_gas_constant ↩