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

Explanation

Let the point P lies at a distance x from the centre of the disk and take the plane of the disk to be perpendicular to the x-axis. Let the disc is divided into a number of charged rings as shown in figure.

The electric potential of each ring, of radius r and width dr, have charge dq is given by
`sigmadA=sigma2pirdr`
and potential is given by
`dV=(d_(e)dq)/(sqrt(r^(2)+x^(2)))=(k_(e)sigma2pirdr)/(sqrt(r^(2)+x^(2)))`
where `k_(e)=(1)/(4piepsi_(0))` the total electric potential at P, is given by
`V=pik_(e)sigmaint_(0)^(a)(2r" "dr)/(sqrt(r^(2)+x^(2)))=pik_(e)sigmaint_(0)^(a)(r^(2)+x^(2))^(-1//2)3r" "dr`
`V=2pik_(e)sigma[(x^(2)+a^(2))^(1//2)-x]`
So, we have by substring `k_(e)=(1)/(4piepsi_(0))`
`V=(1)/(4piepsi_(0))(2Q)/(a^(2))[sqrt(x^(2)+a^(2))-x]`.