Thin infinite sheet of width w contains uniform charge distribution `sigma`. Find out electric field intensity at following points :
(a) A point which lies in the same plane at a distance d from one of its edge.
(b) A point which is on the symmetry plane of sheet at a perpendicular distance d from it.

Explanation

(a) consider a thin strip of width dx. Linear charge density of strip : `lambda= sigma dx`
So, electric field due to this strip at point P
`dE= (2 k sigma d x)/x`
`E_("net")= underset(d)overset(d+w)(int) sigma/(2 pi epsi_(0)) (dx)/x`
`= sigma/(2 pi epsi_(0)) ln ((d+w)/d)`
(b) Consider a thin strip of width dx. Linear charge density of strip :
`lambda=sigma dx`
`:. E_(p) =int 2d E cos theta`
or `E_(p)=2. int_(0)^(w//2) (sigma dx)/(2 pi epsi_(0) sqrt(d^(2)+x^(2))). d/sqrt(d^(2)+x^(2))`
`=(sigma d)/(pi epsi_(0)) int_(0)^(w//2) (dx)/(d^(2)+x^(2))=sigma/(pi epsi_(0))"tan"^(-1) w/(2d)`