Question

Two parallel uniformly charged infinite plane sheets, '1' and '2', have charge densities + `sigma and -2sigma `respectively. Give the magnitude and direction of the net electric field at a point.
(i) in between the two sheets and
(ii) outside near the sheet '1'.

Answer 1

Which is same in both the case so it is independent of the size and shape of the closed surface

Let `hat6` be the unit vector directed from left to right

Let P and Q are two points in the inner and outer region of two plates respectively charge densities on plates are + `sigma  and  -2sigma`

(i) Electric field at point P in the inner region of the plates

`vec"E"_1 = sigma/(2epsilon_0) hat"r" and vec"E"_2 = (2sigma)/(2epsilon_0)hat"r"`

∴ Net electric field in the inner region of the plates (i.e., at P) is

`vec"E" = vec"E"_1 vec"E"_2`

`vec"E" = (sigma/(2epsilon_0)+ sigma/epsilon_0)hat"r"`

`vec"E" = (3sigma)/(2epsilon_0)hat"r"`

(ii) Electric field at point Q in the outer region of plate 1

`vec"E"_1 = sigma/(2epsilon_0)(-hat"r") and vec"E"_2 = (2sigma)/(2epsilon_0)hat"r"`

∴ Net electric field in the outer region of plate 1 (i.e. at Q ) is

`vec"E" = vec"E"_1 + vec"E"_2 = (sigma/epsilon_0 - sigma/(2epsilon_0))hat"r"`

`vec"E" = sigma/(2epsilon_0)hat"r"`