(continued): The magnitude of the electric field dE created by a point charge dq at a distance d = sqrt(r2 + z2) is dE = (1)/(4pi0) (dq)/(d2).

Okay, Aïcha, let's continue with .

Step 1 (continued): The magnitude of the electric field $dE$ created by a point charge $dq$ at a distance $d = \sqrt{r^2 + z^2}$ is $dE = \frac{1}{4\pi\epsilon_0} \frac{dq}{d^2}$.
Due to symmetry, only the component along the z-axis, $dE_z$, contributes to the total field.

$$dE_z = dE \cos\alpha$$

where $\cos\alpha = \frac{z}{d} = \frac{z}{\sqrt{r^2 + z^2}}$.
So, for the infinitesimal ring:

$$dE_z = \frac{1}{4\pi\epsilon_0} \frac{dq}{r^2 + z^2} \frac{z}{\sqrt{r^2 + z^2}} = \frac{z}{4\pi\epsilon_0} \frac{dq}{(r^2 + z^2)^{3/2}}$$

Substitute $dq = \sigma (2\pi r dr)$:

$$dE_z = \frac{z}{4\pi\epsilon_0} \frac{\sigma (2\pi r dr)}{(r^2 + z^2)^{3/2}} = \frac{\sigma z r dr}{2\epsilon_0 (r^2 + z^2)^{3/2}}$$