Electric field due to an infinite sheet of charge having surface charge density σ is E. Electric field due to an infinite conducting sheet of same surface density of charge is

Given an infinite sheet of charge with a uniform surface charge density denoted by σ, the electric field (E) produced by this sheet can be determined using Gauss's law.

Derivation Using Gauss's Law

1. Gaussian Surface Selection:

   To exploit the symmetry of the problem, we choose a cylindrical Gaussian surface (often referred to as a "pillbox") that intersects the sheet perpendicularly. This cylinder has its flat faces parallel to the sheet, with one face on each side of the sheet.

2. Electric Flux Calculation:

   The electric field due to an infinite sheet of charge is uniform and perpendicular to the surface. Therefore, the flux through the curved side of the cylinder is zero, and only the flat faces contribute to the electric flux.

   The total electric flux (Φ_E) through the Gaussian surface is:

   Φ_E = E × A + E × A = 2EA

   where:

   • E is the magnitude of the electric field.
   • A is the area of one of the flat faces of the cylinder.

3. Applying Gauss's Law:

   Gauss's law states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space (ε₀):

   Φ_E = Q_enc / ε₀

   The charge enclosed (Q_enc) by the Gaussian surface is:

   Q_enc = σ × A

   Substituting into Gauss's law:

   2EA = (σ × A) / ε₀

4. Solving for the Electric Field:

   By simplifying the equation, we find:

   E = σ / (2ε₀)

Electric Field Due to an Infinite Conducting Sheet

For an infinite conducting sheet with the same surface charge density (σ), the electric field just outside the surface is:

E' = σ / ε₀

This result arises because a conductor in electrostatic equilibrium has charges residing on its surface, and the electric field immediately outside is perpendicular to the surface with a magnitude of σ / ε₀.

Comparison of Electric Fields

Comparing the electric fields of the non-conducting and conducting sheets:

• Non-conducting infinite sheet: E = σ / (2ε₀)
• Conducting infinite sheet: E' = σ / ε₀

Thus, the electric field due to the conducting sheet is twice that of the non-conducting sheet:

E' = 2E

Therefore, the electric field due to an infinite conducting sheet with the same surface charge density is twice that of an infinite non-conducting sheet. Given that the electric field due to the non-conducting sheet is E, the field due to the conducting sheet is 2E.