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Singular Field

ElectromagneticFields.Singular — Module

Singular magnetic field in (x,y,z) coordinates with covariant components of the vector potential given by

\[A (x,y,z) = \frac{B_0}{\sqrt{(x^2 + y^2)}^3} \big( y , \, - x , \, 0 \big)^T\]

resulting in the magnetic field with covariant components

\[B(x,y,z) = B_0 \, \begin{pmatrix} 0 \\ 0 \\ (x^2 + y^2)^{-3/2} \\ \end{pmatrix}\]

Parameters: B₀

source

Constructing the Field

using CairoMakie
using ElectromagneticFields

equ = Singular.init()
Singular Magnetic Field

Plotting

Both the vector potential and the magnetic field diverge as the $z$ axis is approached — the potential like $r^{-2}$, the field like $r^{-3}$ — so linearly spaced contour levels would show nothing but the singularity. The plot therefore uses logarithmically spaced levels:

plot_equilibrium(equ)
Example block output

Evaluating the Field

Singular.@code()

The divergence is steep enough to be worth keeping in mind when this field is used as a test case — an order of magnitude closer to the axis means three orders of magnitude in $|B|$:

[B(0.0, r, 0.0, 0.0) for r in (1.0, 0.1, 0.01)]
3-element Vector{Float64}:
      1.0
   1000.0000000000003
 999999.9999999997