How to derive the magnetic field at a point a distance $x$ from the center of the axis of a circular loop of radius $R$ carrying current $I$? [closed]

Question

Question: What is the magnetic field at a point a distance $x$ from the center of the axis of a circular loop of radius $R$ carrying current $I$?

I was given the answer $\frac{\mu_0}2 \frac{IR^2}{(x^2+r^2)^{3/2}}.$ I am trying to figure out how to use Biot-Savart Law to derive this. $$dB = \frac{\mu_0}{4\pi} \frac{I dl \times \hat{r}}{r^2}.$$ I tried expanding the cross product as $dl \sin \theta,$ but I could not find a nice way to simplify the $\theta.$

Answer 1

Consider a contribution from an infinitesimal loop element. Decompose the field from it at the point of interest into $x$ and $y$ components. $\vec{r}$ is perpendicular to $\vec{dB}$. The angle $x$ can easily be determined from the dimentions given. Now it's trivial geometry.