Gauss's theoremtheorem
In mathematics, this is the divergence theorem, independently formulated by Carl Friedrich Gauss (for gravity), Mikhail Ostrogradsky (in a more general form), and George Green. In physics, it became the basis of Gauss's law for electricity (one of Maxwell's equations) and for gravity. The idea: the total 'outflow' of a field inside a volume equals the flux through its boundary. For electricity: ∮ E·dA = Q/ε₀ — a charge creates a field, and its flux through any closed surface enclosing the charge is the same.
How it works
Example: around a charged sphere, the electric field decreases as 1/r², and the flux through any sphere around it is the same. If there is no charge inside the sphere, then as many field lines enter as leave — flux is zero. Lightning rods work on this principle: the sharp tip creates a high field, and excess charge 'drains' from it, preventing a lightning strike.
💡 The theorem works even in the curved space-time of general relativity, where it enters Einstein's equations for gravity.
\oint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \int_V (\nabla \cdot \mathbf{F}) dV
∮_{∂V} — integral over the closed surface ∂V bounding volume V; F — vector field; dS — vector area element (pointing outward); ∇·F — divergence of field F; dV — volume element.
\oint_{\partial V} \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}
∮_{∂V} — integral over a closed surface; E — electric field strength; dA — vector area element; Q_enc — electric charge enclosed within the surface; ε₀ — electric constant (ε₀ ≈ 8.8541878188×10⁻¹² F/m).