Result

Infinite sheet of charge

(E2.3.3) E2.3 · Application of Gauss's Law to Various Charge Distributions →

E = \frac{σ}{2 ε _{0}}

Statement

A uniform field pointing away from the sheet, the same strength no matter how far away you are.

Symbols and units

$E$	electric field	N/C
$σ$	charge per unit area	C/m²
$ε_{0}$	permittivity of free space	8.85×10⁻¹² C²/(N·m²)

When it applies

Close to a large sheet, far from its edges.

Don't use when

An idealization valid close to a large sheet, far from its edges — a finite sheet's field falls off with distance far from it, unlike this constant result.

Derivation

By symmetry E is perpendicular to the sheet and equal on both sides. Use a small box through the sheet, with flat faces of area A parallel to it.

No flux through the sides; flux leaves through both faces:

Φ_{E} = 2 E A

The box encloses the charge on the area it cuts out:

q_{in} = σ A

Gauss's law, then cancel A:

2 E A = \frac{σ A}{ε _{0}} \Rightarrow E = \frac{σ}{2 ε _{0}}

Worked examples

Sheet carrying 1 µC/m²

E = \frac{1 0 ^{- 6}}{2 ( 8.85 \times 1 0 ^{- 12} )} \approx 5.6 \times 1 0^{4} N/C

Two oppositely charged sheets, σ = 2.0 µC/m²

Between the sheets the two fields add:

E_{between} = \frac{σ}{ε _{0}} = \frac{2 \times 1 0 ^{- 6}}{8.85 \times 1 0 ^{- 12}} \approx 2.3 \times 1 0^{5} N/C

Outside the sheets they cancel:

E_{outside} = 0

This configuration is the parallel-plate capacitor.

Watch out for

The field does not fall off with distance — surprising, but it is because the sheet is infinite: moving away, you see more of it at a poorer angle, and the two effects cancel exactly.

Connects to

Gauss's law Conductor rules Parallel plates