Gauss for closed surfaces

When the surface $\mathcal{S}$ is closed (like a balloon) and is oriented with the normal $\uv{n}$ pointing outward, something special happens. Every field line generated by an external charge enters and exits the surface exactly once: it contributes $+1$ at the exit point and $-1$ at the entry point, so with a net contribution of zero. Only the lines generated by internal charges give a net contribution to the flux.

Principle — Gauss's theorem for closed surfaces

The flux of the electric field through a closed surface depends only on the enclosed charge: $\ev{\Phi(\vec{E}) = \frac{Q_{\text{int}}}{\varepsilon_0}}$ where $Q_{\text{int}}$ is the total charge enclosed by the surface and $\varepsilon_0 = 8.85\cdot 10^{-12}\;\text{C}^2/(\text{N}\cdot\text{m}^2)$ is the permittivity of free space.

Note

Since $4\pi k = 1/\varepsilon_0$, Gauss’s theorem in closed form is simply Gauss’s lemma applied to a closed surface, with the external charges cancelling in pairs: $\Phi = 4\pi k \cdot N_{\text{net}} = 4\pi k \cdot Q_{\text{int}} = \frac{Q_{\text{int}}}{\varepsilon_0}$

Gauss’s theorem becomes a powerful calculation tool when the geometry of the problem has a symmetry that allows $\abs{\vec{E}}$ to be extracted from the flux integral: in these cases the field has constant magnitude on the Gaussian surface and can be taken outside the integral. The standard situations (sphere, shell, plane, wire) are treated in the following sections.

Gauss’s theorem: the flux depends only on the charge enclosed inside the closed surface. If the charge is external, every line that enters also exits, and the net flux is zero.

Collegamenti

Argomenti: Campo elettrico e potenziale Concetti: Teorema di Gauss Competenze: Applicazione del teorema di Gauss Metodi: Teorema di Gauss

Esercizi collegati: Which graph of E · From the field to the spherical radius · Speculative physics: Coulomb 1 over r cubed