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## Problem

Find the electric ﬁeld a distance along the axis from a disc of radius and uniform charge density .

## Hint

(1) Think of this disk 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 .

(2) You may use the integral

## Solution

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

Integrating the differential yields:

For the disk problem, replace the differential charge with because the areal charge density is defined as and for a disk.

Since we are integrating with respect to the variable now, the variable radii is replaced with . Therefore, the electric field is modified to

Using the given integral yields,

Therefore, the electric ﬁeld a distance along the axis from a disc of radius and uniform charge density is