A very thin round plate of radius `R` carrying a uniform surface charge density `sigma` is located in vacumm. Find the electric field potentail and strength along the plate's axis as a function of a distance `l` from its centre. Investigation the obtained expression of `l rarr 0` and `l gt gt R`.

Explanation

Let us consider an elementary ring of of thickness `dy` and rasdius `y` as shown in fig. Then potential at a point `p` at distance `l` from the centre of the disc, is
`d varphi = (sigma 2 pi y dy)/(4pi epsilon_(0) (y^(2) + l^(2))^(1//2))`
Hence potential due to the whole disc,
`varphi = int_(0)^(R ) (sigma 2pi y dy)/(4pi epsilon_(0) (y^(2) + l^(2))^(1//2)) = (sigma l)/(2 epsilon_(0)) (sqrt(1 + (R//l)^(2)) - 1)`
From symmtry
`E = E_(1) = (d varphi)/(dl)`
`= - (sigma)/(2epsilon_(0)) [(2l)/(2 sqrt(R^(2) + l^(2))) - 1] = (sigma)/(2 epsilon_(0)) [1 - (1)/(sqrt(1 + (R//l)^(2)))]`
when `l rarr 0, varphi = (sigma R)/(2epsilon_(0)), E = (sigma)/(2epsilon_(0))` and when `l gt gt R`,
`varphi = (sigma R^(2))/(4 epsilon_(0) l), E = (sigma R^(2))/(4 epsilon_(0) l^(2))`