Electric Field of a Charged Disk

Problem

Find the electric field a distance $x$ along the axis from a disc of radius $R$ and uniform charge density $\sigma$.

Hint: a disk can be thought of as a bunch of concentric rings. Begin by finding the electric field a distance $x$ along the axis up from a thin ring of charge $dQ$ and radius $a$.

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 $r = \sqrt{x^2+a^2}$.

Hence, the differential element of electric field (along the x-axis) is

$dE_x = dE \cos{(\theta)}$

Failed to parse (unknown function "\fra"): {\displaystyle dE_x = \frac{1}{4\pi {\epsilon}_{0}} \frac{dQ}{x^2 + a^2} \fra{x}{\sqrt{x^2 + a^2}} } .

Integrating yields,

$E_{x}={\frac {1}{4\pi {\epsilon }_{0}}}{\frac {x}{({\sqrt {x^{2}+a^{2}}})^{3}}}\int dQ$

$E={\frac {1}{4\pi {\epsilon }_{0}}}{\frac {xQ}{({\sqrt {x^{2}+a^{2}}})^{3}}}{\textbf {i}}$