THE COULOMB LAW

Two charges, one line between them, and a force that answers only to the square of the distance. Halve the gap and the push quadruples; unlike signs pull, like signs shove. Electrostatics' first equation, rendered, not quoted — the engine below computes every force live and dares the red team to break the exponent.

source Coulomb, Premier mémoire sur l'électricité et le magnétisme (1785), Hist. Acad. R. Sci., pp. 569–577 — Mémoires sur l'électricité (Internet Archive)

Blue Team · builds & defends
3
THE MODEL

A point charge is an idealised speck holding charge q at a point. Between two such specks separated by r:

F = k·q₁·q₂ / r², with k = 1/(4πε₀) ≈ 8.988×10⁹ N·m²/C².

The force lies along the line joining them. Its sign follows the product q₁q₂: positive (like charges) means repel, negative (unlike) means attract. The field a charge makes is the force per unit test charge, E = F/qₜ — a property of space, not of whatever you drop into it.

For many sources the net force is the vector sum of the individual Coulomb forces: superposition, because vacuum electrostatics is linear.

5
THE LINEAGE

the-coulomb-law → the-gauss-law. Coulomb (1785) measured the force between two charges and found the inverse square. Gauss repackages that same 1/r² as flux through any closed surface: because the field of a point charge thins exactly as the area of a sphere grows, the total flux is source-only and shape-free.

Same physics, two faces — the point-charge force here is the seed whose flux the neighbouring sphere collects for arbitrary surfaces.

7
THE WITNESS

Live re-check of the running engine. It reads the current force law and confirms the inverse-square ratio F(2r)/F(r)=1/4. If window 6 tampers the exponent, this badge flips red on the spot.

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The Machine
4 in ↓
DATA IN

Charges q₁, q₂ (sign & magnitude), separation r, and source positions in the plane. A movable +1 test charge samples the field. Everything below is computed from these — no stored answers.

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0 LIT
THE PANEL
r = 1.00
q₁=+1, q₂=+1 → like · repel
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8 out ↓
DATA OUT

Proven live: force is strictly inverse-square, linear in each charge, sign set by q₁q₂, forces superpose (a charge midway between two equal like charges feels exactly zero), and the field is independent of the test charge.

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Red Team · attacks & breaks
1
THE ADVERSARY

Where the clean law stops being clean:

WALL Point-charge fiction. Real charge has extent; near contact the 1/r² mislead-s and multipoles matter. As r→0 the law diverges to infinite force — unphysical, patched only by quantum electrodynamics.
WALL Static only. Coulomb assumes charges at rest. Move them and magnetic forces, retarded potentials and radiation appear — this is the electrostatic limit of the full Maxwell field.
WALL Vacuum & linearity. In matter, dielectrics screen the force (replace ε₀ by ε); superposition itself leans on linearity, which extreme-field QED vacuum quietly bends.
2
THE GRAVEYARD
  • The force falls off as 1/r.
    → Inverse square, 1/r². Double r and F drops to a quarter, not a half.
  • Force depends only on the distance.
    → It scales with the product q₁q₂ too — double either charge and F doubles.
  • The inverse square is exact at every scale.
    → AMBER: verified experimentally to extraordinary precision (photon-mass / exponent-deviation bounds), but QED corrections enter at subatomic scales and matter screens it. "Exact" only in classical vacuum.
6
THE TAMPER

The disclosed planted void. This button swaps the real 1/r² for an inverse-linear 1/r law. Doubling r would then only halve F. Window 7's witness catches it live.