Find the electric ﬁeld a distance along the axis from a disc of radius and uniform charge density .
Hint: a disk can be thought of as a bunch of concentric rings. Begin by ﬁnding the electric field a distance along the axis up from a thin ring of charge and radius .
You may use the integral
Consider the ring problem first. Due to the symmetry of this geometry, there is a cancellation effect in the y-direction. Therefore, all contributions to the electric field in the x-direction. The distance of a point on the x-axis from the ring is .
Hence, the differential element of electric field (along the x-axis) is
For the disk problem, replace the differential charge with because the areal charge density is defined as and for a disk.
Therefore, the electric field is modified to (we are integrating with respect to the variable now, so is replaced with )
Using the given integral yields,