Two infinite planes sepated by a distance `l` carrying a uniform surface charge of densities `sigma` and `-sigma`. The planes have round coaxial holes of radius `R`, with `l lt lt R`. Taking the origin `O` and the `x` cordinates axis as shown in the figure, find the potentail of the electric field and the projection of its strength vector `E_(x)` on the area of the system as functions of the x cordiante. Draw the appoximate plot `varphi (x)`.

Explanation

The field of a pair of oppositely charged sheets with holes can by superposition be reduced to that fo a pair of uniform opposite charged sheets and discs with opposite chagres. Now the charged sheets do not contribute any field outside them. Thus using the result of the previous problem.
`varphi = int_(0)^(R) ((-sigma) l 2 pi r dr x)/(4pi epsilon_(0) (r^(2) + x^(2))^(3//2))`
`= (sigma xl)/(4 epsilon_(0)) int_(x^(2))^(R^(2) + x^(2)) (dy)/(y^(3//2)) = (sigma xl)/(2epsilon_(0) sqrt(R^(2) + x^(2)))`
`E_(x) = (del varphi)/(del x) = (sigma l )/(2 epsilon_(0)) [(1)/(sqrt(R^(2) + x^(2))) - (x^(2))/((R^(2) + x^(2))^(3//2))] = (sigma l R^(2))/(2epsilon_(0) (R^(2) + x^(2))^(3//2))`
The plot is shwon in teh answersheet.