Applications of Gauss' Theorem > Electric Field due to Two Infinite Parallel Sheets of Charge

Consider two infinite, parallel, thin plane sheets A and B, carrying uniform surface charge densities σ₁ and σ₂ respectively, separated by some distance. Using the superposition principle, the resultant field at any point is the vector sum of the fields due to A and B individually.

Three distinct regions exist around the sheets:

• Region I — to the left of both sheets
• Region II — between the two sheets
• Region III — to the right of both sheets

Case 1: Both Sheets Carry Like Charges (σ₁, σ₂ both positive)

Case 2: Sheets Carry Unlike Charges (σ₁ positive, σ₂ negative, σ₁ > σ₂ in magnitude)

Case 3: Equal and Opposite Charges (σ₁ = σ₂ = σ) — The Capacitor Case

This is the standard configuration used to model a parallel plate capacitor

E_outside​ = 0
E_between = \[\frac {σ}{ε_0}\]

The field between the plates is uniform and independent of the separation distance, provided the sheets are much larger than the gap between them.

Think of two sheets with equal and opposite charge as two people pushing a ball from opposite sides with equal force outside their zone of influence — outside, the forces cancel out; between them, the pushes add up, creating a strong, uniform "push zone." This is exactly how a capacitor stores a uniform field between its plates, used in every camera flash and RAM chip.