THE GAUSS LAW

The total electric flux through any closed surface equals the charge trapped inside, divided by ε₀ — independent of the surface’s shape, size, or bulges. Rendered, not quoted: the engine computes the flux live and proves that two spheres of different radius around the same charge report the same number.

source J. C. Maxwell, A Treatise on Electricity and Magnetism, vol. 1 (Oxford, 1873) — Gauss’s law as the first field equation · archive.org/details/electricandmagne01maxwrich · AMBER: page image, not a canonical equation id.

Blue Team · builds & defends
3

THE MODEL

Gauss’s law in integral form:

∮ E · dA = Q_enc / ε₀

Field lines are conserved: each line that starts on a charge must pierce the surface once. So the net count — the flux Φ — counts only the enclosed charge. Bulge the surface, shrink it, dent it: the same lines still cross it. For a spherically symmetric charge the field outside is E = kQ/r²; inside a uniform shell the interior field is exactly 0 (the shell theorem).

5

THE LINEAGE

Flux = enclosed charge is the same fact as Coulomb’s inverse square, recast. Integrate kQ/r² over a sphere: the r² in the field cancels the r² in the area, leaving 4πkQ = Q/ε₀, with no r left.

→ neighbour: the-coulomb-law — the point-charge force, promoted to the first of Maxwell’s four equations.

7

THE WITNESS

Live re-check: does the flux still ignore the surface radius? Two Gaussian spheres, radii r₁ and r₂, same enclosed charge — the witness re-computes both and compares. Green while they agree; it flips red the instant the Tamper (window 6) makes flux depend on r.

witness idle
The Machine
4

DATA IN in ↓

Enclosed charge Q = 1.60×10⁻⁹ C at the centre.
Two closed surfaces: sphere r₁ = 0.05 m and sphere r₂ = 7.30 m.
One extra charge placed outside both surfaces.

↓
0

THE PANEL lit

computing…
↓

DATA OUT out ↓

Proven result — the flux is set by the charge alone, not the surface:

booting…
Red Team · attacks & breaks
1

THE ADVERSARY wall

“A bigger surface must catch more flux — it has more area, so it sees more field.”

True that area grows as r² — but the field it samples falls as 1/r². The two cancel exactly. The wall is the inverse square: only because the field is precisely 1/r² does flux come out shape-independent. In a universe with 1/r³ forces, Gauss’s law would fail.

2

THE GRAVEYARD

“Flux depends on how close the surface hugs the charge.”

→ No. Only the enclosed charge counts; the geometry cancels.

“A charge just outside the surface leaks a little net flux in.”

→ Zero. Every line an external charge sends in comes back out — net flux is exactly 0.

“Inside a charged shell the field points inward toward the wall.”

→ The interior field of a uniform shell is 0 everywhere (shell theorem).

6

THE TAMPER

Disclosed planted void: force the flux to depend on the surface radius (Φ ∝ Q/r). Two spheres of different radius will then disagree — and the Witness in window 7 catches it live.