T-21 · Electrostatics

Charges at rest.

Long line charges seen end on: each field falls as 1/r, so the flux out of any closed curve counts the charge inside. The potential and the field lines are the two halves of one complex function, and cross at right angles everywhere. A conductor is an equipotential, and an image charge, placed by Pythagoras, makes it one.

At the probe

The field from Coulomb's law, and again as the downhill slope of the potential, measured by central differences. Its strength is a hypotenuse: |E|² = Eₓ² + E_y².

E	(0.60976, 0)
\|E\| = √(Eₓ² + E_y²)	0.60976
φ	0
E = −∇φ	residual …

Lines and levels at right angles

In the plane the potential has a partner, the stream function ψ = Σq·arg(z − zᵢ): together they are the real and imaginary parts of one complex function, −Σq log(z − zᵢ). The field lines are the level curves of ψ, so they cross the equipotentials at right angles, everywhere. Each line here is traced by Runge–Kutta, and ψ is measured along it.

field lines traced	10
ψ along each, largest wander	residual …