Question

Obtain the expression for electric field due to an charged infinite plane sheet.

Answer 1

Consider an infinite plane sheet of charges with uniform surface charge density o. Let P be a point at a distance of r from the sheet. Since the plane is infinitely large, the electric field should be the same at all points equidistant from the plane and radially directed at all points. A cylindrical-shaped Gaussian surface of length 2r and area A of the flat surfaces is chosen such that the infinite plane sheet passes perpendicularly through the middle part of the Gaussian surface.

Electric field due to charged infinite planar sheet

Applying Gauss law for this cylindrical surface,

`Φ_"E" = oint vec"E"*"d"vec"A"`

`Φ_"E" = int_"curved surface" vec"E"*"d"vec"A" + int_"P" vec"E"*"d"vec"A" + int_"P'" vec"E"*"d"vec"A" = ("Q"_"encl")/ε_0`

The electric field is perpendicular to the are a element at all points on the curved surface and is parallel to the surface areas at P and P’. Then,

`Φ_"E" = int_"P" "EdA" + int_"P'" "EdA" = ("Q"_"encl")/ε_0`    ...(2)

Since the magnitude of the electric field at these two equal surfaces is uniform, E is taken out of the integration and Q_encl is given by Q_encl = σA, we get

`2"E"int_"P" "dA" = (sigma"A")/ε_0`

The total area of surface either at P or P’

`int_"P" "dA"` = A

Hence 2EA = `(sigma"A")/ε_0` or E = `sigma/(2ε_0)`   ...(3)

In vector from, E = `sigma/(2ε_0) hat"n"`   .....(4)

Hence `hat"n"` is the outward unit vector normal to the plane. Note that the electric field due to an infinite plane sheet of charge depends on the surface charge density and is independent of the distance r.

The electric field will be the same at any point farther away from the charged plane. Equation (4) implies that if o > 0 the electric field at any point P is outward perpendicular n to the plane and if σ < 0 the electric field points inward perpendicularly `(hat"n")` to the plane. For a finite charged plane sheet, equation (4) is approximately true only in the middle region of the plane and at points far away from both ends.