Interactive Laplace Solver: Draw Your Own Conductors

In empty space the electric potential obeys Laplace's equation: at every point it is exactly the average of its neighbours, with no bumps or dips of its own. That gives a beautifully simple way to solve for the field of any set of conductors. Start from any guess, then sweep across the grid replacing each free point with the average around it, over and over. The potential ripples and settles until nothing changes, and what remains is the unique solution that matches the conductors you fixed. This is relaxation, the workhorse behind real field solvers. Pick a classic geometry or paint your own electrodes straight onto the grid and watch the field re-solve in real time, the equipotential lines bending into place. The diagnostic tracks the residual, how much the field still changes each sweep, plunging toward zero as it converges.

Figure 1. Interactive Laplace solver by successive over-relaxation. Top: the potential as a diverging colour map with equipotential contours; conductors (Dirichlet cells) are fixed and can be painted. Bottom: the residual (largest change per sweep) on a log scale, decaying toward zero as the solution converges. Method: red-black SOR, phi_ij to the neighbour average, with the grounded box as the outer boundary.

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WHAT TO TRY

Watch a fresh setup solve from flat: the colour map fills in and the equipotential lines snap into place as the residual falls.
Paint your own electrodes onto the grid with the brush and watch the field re-solve around them in real time.
Try the coaxial cable: the equipotentials become evenly spaced rings, the logarithmic potential of a cable.
Read the lower plot: the residual drops by orders of magnitude per few sweeps, then flattens once converged.