Field between Two Oppositely Charged Parallel Sheets

For two infinite sheets with charge densities +σ and −σ: fields add between the plates and cancel outside. Basis of the parallel plate capacitor.

Setup

Two infinite parallel sheets:

• Sheet 1 (left): surface charge density $+\sigma$, field $E_1 = \sigma/2\varepsilon_0$ pointing away from it on both sides
• Sheet 2 (right): surface charge density $-\sigma$, field $E_2 = \sigma/2\varepsilon_0$ pointing toward it on both sides

Superposition in three regions

Region I (left of both sheets):

$E_1$ points left, $E_2$ points right. They cancel:

$$E = E_1 - E_2 = 0$$

Region II (between the sheets):

$E_1$ points right, $E_2$ also points right (toward $-\sigma$). They add:

$$E = E_1 + E_2 = \frac{\sigma}{2\varepsilon_0} + \frac{\sigma}{2\varepsilon_0} = \frac{\sigma}{\varepsilon_0}$$

Region III (right of both sheets):

$E_1$ points right, $E_2$ points left. They cancel:

$$E = 0$$ $$\boxed{E = \frac{\sigma}{\varepsilon_0} \text{ (between)}, \quad E = 0 \text{ (outside)}}$$

Significance

This result is the physical basis of the parallel plate capacitor. All the field is confined between the plates — there is no fringing for truly infinite plates. The uniform field between the plates is exactly $\sigma/\varepsilon_0$.