Find the electric field caused by a disc of radius a with a uniform surfce charge density `sigma` (charge per unit area) at a point along the axis of the disc a distance x from its centre.

Explanation

We can assume this charge distribution as a collection of concentration rings of charge.
`dA=(2pir)dr`
`dq=sigma dA=(2pisigmar)dr`
`dE_x=1/(4piepsilon_0).((dq)x)/((x^2+r^2)^(3//2))`
`=(1/(4piepsilon_(0)))((2pisigmar dr) x)/((x^2+r^2)^(3//2))`
`: E_x=int_0^a dE_x`
`=int_0^a((2pisigma dr)x)/(4piepsilon_0(x^2+r^2)^(3/2))`
`=(sigmax)/(2epsilon_0)[1-1/sqrt(a^2/x^2+1)]`
If the charge distribution gets very large i.e. `agtgtx`, the term `1/sqrt(a^/x^2+1)` becomes negligibly small and we get `E=sigma/(2epsilon_0)`
Thus, we can say that electric field produced by an infinite plane sheet of charge in inependent of the distance from the sheet. Thus, the field is uniform, its direction is everywhere perpendicular to the sheet.