Applications of Gauss' Theorem > Electric Field due to an Infinite Plane Sheet of Charge

An infinite plane sheet of charge is an idealized, uniformly charged flat surface extending infinitely in all directions, used to model real-world situations such as charged conducting plates or capacitor plates at points close to the surface. Because the sheet is infinite, the field it produces has a special property: it does not weaken with distance, unlike the field of a point charge.

Step 1 - Setup: Consider a thin, infinite sheet with uniform surface charge density σ. By symmetry, the electric field must be perpendicular to the sheet at every point.

Step 2 - Gaussian Surface: Construct a cylindrical Gaussian surface with its axis perpendicular to the sheet, and flat circular ends of area A at points P and P′, one on each side of the sheet, equidistant from it.

Step 3 - Flux through curved surface: Since E is parallel to the curved surface everywhere, the flux through the curved part of the cylinder is zero.

Step 4 - Flux through flat ends: E is perpendicular to both flat circular ends, so the total flux through them is:

Φ_E​ = EA + EA = 2EA

Step 5 - Enclosed charge: The charge enclosed within the Gaussian cylinder is:

Step 6 - Apply Gauss's Law:

Step 7 — Final Result:

The field is independent of the distance r from the sheet — it remains constant at every point near the sheet.

Direction Rule: The field points away from the sheet if σ is positive, and toward the sheet if σ is negative.

Validity Note: This result holds strictly for points close to the sheet, treating it as effectively infinite

A total charge of 17.7 × 10⁻⁴ C is spread uniformly over a sheet of area 200 m². Find the electric field near the sheet.