∇Understand · Electromagnetism

Charges write the field.

Put a charge somewhere. Field lines spread out from positive ones, end on negative ones, and never cross each other. Drop a test particle, it follows the field like a leaf on a stream. The mathematics is Maxwell from 1865; this playground stays in the electrostatic corner of it, where life is simpler and you can see what the equations actually do.

Coulomb's law · numerical streamline integration · 220 test particles · drag and drop charges

Place, drag, double-click

Click an empty spot to add a positive charge. Shift-click adds a negative one. Drag a charge to move it. Double-click flips its sign. Alt double-click removes it. The four presets give you the canonical configurations: a single charge, a dipole, a quadrupole, and two parallel plates.

What the field actually is

A field assigns a vector to every point in space. The electric field tells a test charge which way it would feel pushed and how hard. The lines you see are not real, but they are honest: they follow the field direction everywhere, their density tracks field strength, and their topology encodes the physics.

Coulomb's 1/r²

The force between two point charges falls off as the square of the distance. Coulomb measured this in 1785 with a torsion balance, decades before anyone understood why. The reason: field lines spread radially, so their density on a sphere of radius r scales as 1/r². The same geometry gives Newton's 1/r² gravity. It is the geometry of three-dimensional space speaking.

Gauss's flux

The first Maxwell equation says: the flux of the electric field through any closed surface equals the enclosed charge, divided by a constant. You see it here: every field line that starts on a positive charge has to end on a negative one or go off to infinity. Counting lines through a box gives you the charge inside, without needing to know where exactly it sits.

Linear means stackable

Maxwell's equations are linear. Two charges produce the vector sum of their individual fields, no cross-terms, no interference of forces. That is why this playground works at all: every test particle just sums up contributions from all visible charges. In nonlinear theories, like general relativity for strong gravity, the field of two sources is not the sum of the individual fields.

What is missing here

Maxwell's full theory has four equations and runs in space and time. This playground holds only two of them, in their static form: Gauss's law for the electric field and the rotation-free constraint that follows from it. The other two need moving charges, currents and time-varying fields. Adding those gives you light. Coming as a separate page.