Electrostatic Field of Point Charges (3D)

What this page shows

Up to eight charged conducting spheres sit on an insulating board. Every number you see comes from adding up the Coulomb field of each sphere, so changing a charge or dragging one changes everything at once: field lines (the direction of the force on a positive test charge), equipotentials (lines of equal voltage), the colour maps, a Gauss sphere that counts flux, and a free test particle that you can release anywhere.

Coulomb's law and superposition

One point charge q makes the field E = k q / r² pointing away from it (towards it if q is negative) and the potential V = k q / r, with k = 1/(4πε0) = 8.988×109 N m²/C². Fields add as vectors and potentials add as numbers: E = Σ k qi (r - ri) / |r - ri|³, V = Σ k qi / |r - ri|. Here charges are in nanocoulombs and lengths in metres, so k = 8.988 V m / nC.

Field lines and equipotentials

A field line is a curve that is everywhere parallel to E. The page integrates dr/ds = E/|E| with a midpoint step, starting on the surface of each positive sphere (the number of lines is proportional to the charge) until a line enters a negative sphere or leaves the picture. Lines never cross, because E has one direction at each point. Equipotentials (V = constant, drawn with marching squares at a spacing you choose) always cross field lines at right angles, because E = -∇V points straight downhill in potential.

Gauss's law

The flux of E through any closed surface equals the enclosed charge divided by ε0: Φ = ∯ E·dA = Qenc/ε0 = 4π k Qenc. The yellow Gauss sphere is integrated numerically with a 24 x 48 Gauss-Legendre and uniform-angle quadrature, so the measured curve in the third graph is a real surface integral, not the formula. It jumps by 4π k q each time the growing sphere swallows a charge and ignores charges outside, even though the field of those charges is not zero on the surface.

Test particle and energy

A released particle of charge qt and mass m feels m a = qt E (a velocity-Verlet step is used). The electrostatic force is conservative, so ½ m v² + qt V stays constant; the fourth graph shows the two parts trading places while the sum is flat. The page also reports the energy needed to assemble the charges, U = Σi<j k qi qj / rij: negative for attracting pairs, positive for repelling ones.

Checks run in the page

1. The Gauss flux through a big sphere around all charges equals 4π k Q and an empty far sphere has zero flux. 2. E equals -∇V by central differences at six points. 3. The energy of a released test particle is conserved. 4. The two-charge field at 30 separations matches the dipole formula V = k p cosθ / r².

Honest simplifications

The spheres are treated as rigidly charged conductors that do not respond to each other (real conductors redistribute their charge when another charge comes close, which the page ignores; this is why the fields are exact point-charge fields outside the spheres and zero inside). Test-particle mass and charge are in nano-units so that motion is visible; real sparks start near 3 MV/m, which the fields next to the larger charges exceed, so treat such values as a teaching model. There is no air, no radiation, no magnetic effect, and the board itself is not part of the field.