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UnitfulGaussian

Documentation for UnitfulGaussian.

Installation

In the REPL, type

julia> ]
pkg> add https://github.com/abhro/UnitfulGaussian.jl

Examples

Coulomb's law

For this example, we're going to calculate the force between a proton and an electron, whose distance is equal to the Bohr radius. You can think of this as an extremely simplified model of the hydrogen atom.

Note that depending on the unit system, we have to use a different version of Coulomb's law:

\[\begin{align*} F_\text{Gauss} &= \frac{q_1 q_2}{r^2}, & F_\text{SI} &= \frac{1}{4π ε_0} \frac{q_1 q_2}{r^2} \end{align*}\]

Since we're only calculating the magnitude of the force, we won't bother ourselves with the vector algebra. And we can also treat the electron charge as also being the elementary charge, since the negative sign of an electron's charge only serves to tell us the direction of the force.

First, let's take a look at what constants the Unitful.jl and UnitfulGaussian.jl packages give us, and which ones we need to define for ourselves:

julia> using Unitful, UnitfulGaussian

julia> u"q"         # Unitful.jl already gives us the elementary charge
1.602176634e-19 C

julia> u"qcgs"      # This package adds the conversion to franklins (aka statcoulombs)
4.803204712570263e-10 Fr

julia> a0 = 5.29772e-11u"m";   # Let's put the Bohr radius in a variable

Now, let's calculate the force (for the Greek letters, type \pi<TAB> and \varepsilon<TAB>):

julia> F_SI = 1/(4π*u"ε0") * u"q"^2/a0^2
9.146244056036602e-25 C^2 H c^2 m^-3

julia> F_Gauss = u"qcgs"^2/a0^2
82.20234211353733 Fr^2 m^-2

Hmm. They don't look like any quantities that talk about force. Let's try converting with the handy |> function.

julia> F_SI |> u"N"
8.220234211353732e-8 N

julia> F_Gauss |> u"dyn"
0.008220234211353735 dyn

Finally! The numbers also look close enough. Let's see if we can confirm that the forces really are equal. Because equality for floats is finicky, we'll instead use the isapprox() or ≈ function.

julia> F_SI ≈ F_Gauss   # type `\approx<TAB>` for ≈
true

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