Mathematica(@mathemetica):From flux alone, Coulomb’s law emerges. Gauss’s law: Φ_E = ∮ E · dA = q_in / ε₀ Spherical symmetry makes E constant and radial, so E · 4πr² = q_in / ε₀ ⇒ E = q_in / (4π ε₀ r²) = k q_in / r² (k = 1/(4π ε₀)) This inverse-square field shapes atomic forces, transistor design, and particle accelerators. A surface integral becomes the local force law through pure symmetry.

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Mathematica@mathemetica

2026.08.25 08:59

From flux alone, Coulomb’s law emerges. Gauss’s law: Φ_E = ∮ E · dA = q_in / ε₀ Spherical symmetry makes E constant and radial, so E · 4πr² = q_in / ε₀ ⇒ E = q_in / (4π ε₀ r²) = k q_in / r² (k = 1/(4π ε₀)) This inverse-square field shapes atomic forces, transistor design, and particle accelerators. A surface integral becomes the local force law through pure symmetry.