# Equipotential Lines

Robert P. Munafo, 2023 Mar 20.

The "equipotential lines" are a series of curves, each surrounding the Mandelbrot set without touching it or each other, defined in the following way:

The Mandelbrot set is extruded into a metal bar of infinite length, whose cross-section is everywhere identical to the Mandelbrot set. The finite size of atoms, quantisation of charge, and other real-world limitations are ignored; the bar has perfect conductivity, and the speed of light is infinite. The bar is given an electric charge of +1.0. The charge at infinity is taken to be 0. The electric field, in cross-section, will have lines of equal potential just as for any electric field. There are an infinite number of such lines, one for each electric potential that is greater than or equal to 0, and less than 1. No two equipotential lines have a point in common. The field lines are the curved lines that are everywhere perpendicular to the equipotential lines.

Colloquially, the equipotentials are "loops" that go around the Mandelbrot set without touching it, while the field lines touch the Mandelbrot set at one end and go off to infinity at the other end. They are similar (but not identical) to the lemniscates; the non-similarity is leasst pronounced when a large escape radius is used.

See Connectedness Proof.

revisions: 20230216 first version; 20230320 refer to lemniscates and escape radius.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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