Flux and Gauss's lemma

The flux of field lines is the unifying tool of all field electrostatics. It is worth starting from the most general form, the one valid for open surfaces, and then arriving at Gauss’s law for closed surfaces as a special case.

Given any surface $\mathcal{S}$ (open or closed), oriented by a normal $\uv{n}$, we define the flux of the electric field through it as the integral of the normal component of the field:

$\Phi(\vec{E},\mathcal{S}) = \int_{\mathcal{S}} \vec{E}\cdot\dd\vec{S} \approx \sum_i \vec{E}_i \cdot \Delta\vec{S}_i$

The geometric meaning is simple and powerful: the flux counts how many field lines pierce the surface, with a positive sign if the lines emerge in the direction of $\uv{n}$ and negative if they cross it in the opposite direction. The net flux is thus a signed measure of the number of lines passing through.

Principle — Gauss's lemma for open surfaces

The flux of the field $\vec{E}$ through any surface $\mathcal{S}$ is proportional to the net number of lines crossing it: $\ev{\Phi(\vec{E},\mathcal{S}) = 4\pi k\,N_{\mathcal{S}}}$ where $N_{\mathcal{S}}$ is the equivalent charge (in coulombs) of the lines piercing $\mathcal{S}$, counted with sign ($+$ in the direction of $\uv{n}$, $-$ in the opposite direction).

Note

Since $4\pi k = 1/\varepsilon_0$, the lemma can also be written in the compact form $\Phi = \frac{N_{\mathcal{S}}}{\varepsilon_0}$

This is the starting point of Gauss’s paradigm: instead of carrying out the laborious integral of $\vec{E}$ over $\mathcal{S}$, it is enough to count the lines crossing it. It is a change of perspective that, in the presence of symmetry, enormously simplifies the calculation of the field.

Gauss’s lemma for open surfaces. Left: all lines pierce the surface in the same direction ($N = 6$, positive flux). Right: some lines cross in the direction of $\uv{n}$ (red, $+$), others in the opposite direction (blue, $-$); the net flux equals $4\pi k(N_+ - N_-)$.

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