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

The element on disc will be a circular ring of radius `r` and thickness `dr`.
Area of ring `= pi (r + dr)^(2) - pi r^(2) = 2 pi rdr`
The change on ring `dq = sigma d A = sigma .2 pi r dr`
Small electric field due to ring at distance `x` on the plate axis
`dE_(p) = (1)/(4 pi in_(0)) .(dqx)/((r^(2) + x^(2))^((3)/(2))) = (1)/( 4 pi in_(0)) .(sigma . 2 pi r dr x)/((r^(2) + x^(2))^((3)/(2)))`
`E_(p) = (sigma x)/(2 in_(0)) int_(0)^(R) (r dr)/((r^(2) + x^(2))^((3)/(2)))`
`= (sigma )/(2 in_(0)) [ 1 - (1)/(sqrt(a^(2))/(x^(2)) + 1)] = (sigma)/(2 in_(0)) [ 1 - (x)/(sqrt(a^(2) + x^(2)))]`
If `a gt gt x` , i.e. plate is very large
`E_(p) = (sigma)/(2 in_(0))`
`dV_(p) = (1)/(4 pi in_(0)) . (dq)/(sqrt(r^(2) + x^(2))) = (1)/(4 pi in_(0)) . (sigma 2 pi r dr)/(sqrt(r^(2) + x^(2)))`
`V_(p) = (sigma)/(2 in_(0)) int_(0)^(R) (r dr)/(sqrt (r^(2) + x^(2)))`
` = (sigma)/(2 in_(0)) [ sqrt(R^(2) + x^(2)) - x]`