A ring of radius a is charged and contains q charge uniformly distributed over the ring . Find an expression for electric field at any position on the axis of the ring .
For `x gt gt a` , where x is the axial distance of the experimental point , what is the reduced form of the expression ? Interprete the result .

Explanation

Suppose a ring of radius a carrying a charge q distributed uniformly on it . To find the centre O , consider an element of length dl (figure)
The charge of an element is dq = `(q dl)/(2 pi a)`

The electric field due to element A is given by `d E = (dq)/(4 pi epsi_(0) r^(2))`
There is a similar electric field at P due to element diametrically opposite point B . The x -component of electric fields due to these elements add up while the y-components cancel . Hence ,
`E = int d E cos theta = int (dq cos theta)/(4 pi epsi_(0) r^(2))`
`= (q)/(2pi a) xx (1)/(4 pi epsi_(0)) xx (x)/((a^(2) + x^(2))^(3//2)) int dl`
`implies E = (1)/(4 pi epsi_(0)) [ (qx)/((a^(2) + x^(2))^(3//2))] ( int dl = 2 pi a)`
The direction of E is from O to P , if charge q is positive .
If `x gt gt a` , then neglecting `a^(2)` in comparison of `x^(2)`
`E = (1)/(4 pi epsi_(0)) (qx)/(x^(3))`
`implies E = (q)/(4pi epsi_(0) x^(2)) " "` (along Px)