# Electric field due to a uniformly charged infinite plane sheet:

Consider a thin infinite plane sheet of charge with uniform surface charge density σ. We wish to calculate its electric field at a point P at distance r from it.
 Gaussian surface for a uniformly charged infinite plane sheet

By symmetry, electric field E points outwards normal to the sheet. Also, it must have the same magnitude and opposite direction at two points P and P' equidistance from the sheet and on opposite sides. We choose the cylindrical Gaussian surface of cross-sectional area A and length 2r with its axis perpendicular to the sheet.
As the line of force is parallel to the curved surface of the cylinder, the flux through the curved surface is zero. The flux through the plane-end faces of the cylinder is

ϕE=EA+EA=2EA

Charge enclosed by the Gaussian surface,

q=σ A

According to Gauss's theorem

ϕE=qϵ0

2EA=σAϵ0,or,E=σ2ϵ0

Clearly , E is independent of r, the distance from the plane sheet.
• If the sheet is positively charged(σ>0), the field is directed away from it.
• If the sheet is negatively charged(σ>0), the field is directed towards it.

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