Field of an infinite plane

An infinite plane with surface charge density $\sigma$ (coulombs per square metre) generates a uniform field perpendicular to the plane: surprisingly, the field does not depend on the distance from the plane.

Note

Why doesn’t $E$ depend on the distance from the plane? Because as $r$ grows, the amount of plane “seen” by the point increases, and the two effects — moving further away and the increase in effective surface area — compensate each other exactly.

Calculation with Gauss (pillbox). We choose as the Gaussian surface a small cylinder (pillbox) of cross-section $A$ crossing the plane, with its two bases parallel to it. By symmetry the field is perpendicular to the plane and equal in magnitude on the two faces, while on the lateral surface it is parallel to the normal and does not contribute to the flux. The total flux is thus the sum of the contributions of the two bases:

$\Phi = E \cdot A + E \cdot A = \frac{\sigma A}{\varepsilon_0} \quad\Rightarrow\quad \ev{E = \frac{\sigma}{2\varepsilon_0}}$

Key formula

Infinite plane with surface density $\sigma$: $\ev{E = \frac{\sigma}{2\varepsilon_0}}$ The field is constant, independent of the distance $r$ from the plane.

Infinite plane: pillbox-shaped Gaussian surface (dashed violet cylinder). The flux is non-zero only on the two faces of area $A$ parallel to the plane; the lateral surface gives no contribution ($\vec{E} \perp \uv{n}$). The result $E = \sigma/(2\varepsilon_0)$ is independent of distance.

Warning

Between the two plates of a parallel-plate capacitor each plate contributes $\sigma/(2\varepsilon_0)$, and the two fields add: $E_{\text{tot}} = \frac{\sigma}{\varepsilon_0}$ Outside the plates, instead, the two fields cancel.

Collegamenti

Argomenti: Campo elettrico e potenziale Concetti: Teorema di Gauss Competenze: Applicazione del teorema di Gauss Metodi: Teorema di Gauss Oggetti: Piano infinito carico

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