Calculate potential on the axis of a disc of radius R due to a charge Q uniformly distributed on its surface.

Explanation

Let us consider a point P on the axis of the disc at a distance x from the centre of the disc. Let the disc is divided into a numerous charged rings as shown in figure.

Let radius of ring r width dr and charges dq
`:. Sigma dA = sigma2 pi rdr `
Potential at P
`dV = (kdq)/(r)`
charge on the ring dq = `+sigma [ pi (r+dr)^(2)-pir^(2)]`
`:. dq = +sigma pi [ (r+dr)^(2)-r^(2)]`
`= +sigma pi [ r^(2)+2rdr +dr^(2)-r^(2)]`
`= +sigma pi [ 2 rdr +dr^(2)]`
Neglecting `dr^(2)` as dr is very small
dq = `2 pi r sigma dr `
and dV = `(kdq)/(sqrt(r^(2)+x^(2)))`
`= (kxx2pir sigmadr)/(sqrt(r^(2)+x^(2)))` [From equation (2)]
`:. V = 2 pi k sigma int _(0)^(R) (rdr)/(sqrt(r^(2)+x^(2)))=2piksigmaint_(0)^(R)( r^(2)+x^(2))(-1//2)rdr`
`:. V= 2pi ksigma[(r^(2)+x^(2))^(1//2)-x]_(0)^(R)`
`:. V = (2pi sigma)/(4 pi in_(0))[(R^(2)+x^(2))^(1//2)-x]`
`:. V = (2pixxQ)/(4pi in_(0)xxpi R^(2))[sqrt((R^(2)+x^(2))^(1//2))-x]`
`:. V =(2Q)/(4pi inn_(0)R^(2))[sqrt((R^(2)+x^(2))^(1//2))-x]`