Consider an infinite plane sheet of charge with surface charge density σ. Let P be a point at a distance r from the sheet (Fig.) and E be the electric field at P.
Consider a Gaussian surface in the form of cylinder of cross− sectional area A and length 2r perpendicular to the sheet of charge.
By symmetry, the electric field is at right angles to the end caps and away from the plane. Its magnitude is the same at P and at the other cap at P′.
Therefore, the total flux through the closed surface is given by
φ = [∫ E ds] p + [∫ E ds] p′ (∵ θ = 0, cos 0 =1 )
φ = E A + E A = 2 E A
If σ is the charge per unit area in the plane sheet, then the net positive charge q within the Gaussian surface is, q = σA.
Using Gauss’s law,
2 E A = q / εo = σA / εo
∴ E = σ / 2εo
Note :
1. If σ > 0 then electric field is normally outward,
2. If σ < 0 then electric field is normally inward to the sheet.
