Electric Field Lines from Point Charges

Every charge fills the space around it with an electric field, and the cleanest way to picture that field is to draw its lines: curves that start on positive charges and end on negative ones, always pointing the way a tiny positive test charge would be pushed. Where the lines crowd together the field is strong; where they spread apart it is weak. That is not a drawing convention, it is Gauss's law: the number of lines through any patch is proportional to the field strength there. Drag the charges around and the whole pattern reorganises instantly, two opposite charges link up into a dipole, two like charges push their lines apart and leave a dead spot between them where the field cancels exactly. The scene shows the field lines flowing over a map of the field strength; the diagnostic is the field strength along the horizontal axis, where the spikes mark the charges and the dips mark the points where the field vanishes.

Figure 1. Electric field of point charges. Top: field lines (integral curves of E) flowing over the field-magnitude map; line crowding shows field strength (Gauss's law). Charges are draggable. Bottom: the field magnitude along the horizontal axis y = 0, spiking at the charges and vanishing at the null points. Method: RK4 streamline integration of E(r) = sum q_i (r - r_i) / |r - r_i|^3.

layoutdipole

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WHAT TO TRY

Drag a charge with the mouse and watch every field line reshape in real time.
Switch to two like charges: the lines repel and a dead spot opens between them, a clean zero in the lower plot.
Pull the two opposite charges of the dipole apart and together; the lines stay linked but stretch and bunch.
Raise the line density: more lines per charge, but the pattern (and the physics) is the same.