**Problem**

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

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**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

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Integrating yields,

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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 )

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Using the given integral yields,

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Therefore,

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