Intensity of electric field near a charged plane

Intensity of electric field near a charged plane sheet of infinite extension

Suppose that on one surface of a uniformly charged flat sheet of infinite extension whose surface density is δ.

Near a positively charged flat sheet of infinite extension, there is a point P at a distance r from the sheet at which the electric field intensity is to be calculated.

Suppose that, point P’ is the intersection of point P on the other side of the sheet, We imagine a Gaussian cylinder across the sheet whose flat ends are parallel to the sheet and pass through points P and P’.

 If area of each end of this cylinder is A, then electric flux passing through both ends of the cylinder is,

Φₑ=∫ₐ→E.→A+∫ₐ→E.→A

Φₑ=∫ₐE.dA.cos0°+∫ₐE.dA.cos0°

Φₑ=2∫ₐE.dA

Φₑ=2E∫ₐdA

Φₑ=2EA——-(1)

But by Gauss’s theorem,

Φₑ = q/∈₀ 

Where q= is the total charge enclosed by the Gaussian cylinder.

 Hence,

q=δ.A        (δ=q/A)

Φₑ = q/∈₀ 

Φₑ=δ.A/∈₀——-(2)

From equation (1) and (2)

2E.A=δ.A/∈₀

E=δ/2∈₀