A current carrying wire takes the form of a plane circular loop of radius a. If the current in the loop is I, find the magnetic firld strength B at a point on the axis of the circle, distance b from the centre.
I think I'm supposed to use the Biot-Savart Law, but I don't know how to integrate everything in the form I have it in.
The Biot-Savart law tells us that the magnetic field is given by $$\mathbf{B} = \frac{\mu_0}{4\pi}\int_O\frac{I\mathrm{d}\mathbf{l}\times\mathbf{r}}{|\mathbf{r}|^3},$$ where $O$ is a circle of radius $a$.
Let's parametrise the circle in the usual way: $\mathbf{r}=(a\sin(\theta),a\cos(\theta),b)$ for $\theta\in[0,2\pi)$. Then the line element becomes $\mathrm{d}\mathbf{l} = (a\cos(\theta),-a\sin(\theta),0)\mathrm{d}\theta$ and $$\mathrm{d}\mathbf{l}\times\mathbf{r}= (-ab\sin(\theta),-ab\cos(\theta),a^2)\mathrm{d}\theta,$$ thus
$$\mathbf{B} = \frac{\mu_0}{4\pi}\int_0^{2\pi}\frac{I}{\sqrt{a^2+b^2}^3}(-ab\sin(\theta),-ab\cos(\theta),a^2)\mathrm{d}\theta.$$ We may evaluate this componentwise to get $$\frac{\mu_0 I}{4\pi\sqrt{a^2+b^2}^3}(ab\cos(\theta),-ab\sin(\theta),a^2\theta)|_0^{2\pi} = \frac{a^2\mu_0 I}{2\sqrt{a^2+b^2}^3}(0,0,1).$$
