Julia set capacitors 2: field lines reflecting the charge density
Channel: Nils Berglund
Date: 2021-06-12

This simulation uses the same principle as the video https://youtu.be/a31FasNlvyU showing fractal capacitors, for a different family of Julia sets. The main difference is that the field lines reflect the charge density on both conductors: field lines are closer where the surface charge is larger. The simulation solves the heat equation, with the interior of the Julia sets and the exterior of the ellipse kept at different constant temperatures. Colors indicate how fast temperature varies in space (norm of the temperature gradient): red means faster, and blue means slower variation. The white lines follow paths of steepest descent. If the Julia set were fixed, the white lines would converge to electric field lines of a capacitor, in which the Julia sets and the exterior of the ellipse are kept at different electric potentials. One expects that since the Julia sets evolve slowly, after some time the white lines indeed give a good approximation of electric field lines. Parameter values follow part of the main cardioïd of the Mandelbrot set, with an additional oscillation on the imaginary part to obtain occasionally disconnected sets. The fact that field lines seem to enter the Julia sets after a while may be due either to round-off errors, or to a coding error (having to do with resetting the gradient inside the sets).

Music: "Staccato" by Vibe Tracks

The simulation solves the heat equation by discretization. 
The C code will be made available on https://github.com/nilsberglund-orleans/YouTube-simulations