The vector potential due to a steady current density $\mathbf{J}$ is given by

$$\mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} d^{3} \mathbf{r}^{\prime}$$

where you may assume that $\mathbf{J}$ extends only over a finite region of space. Use $(*)$ to derive the Biot-Savart law

$$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int \frac{\mathbf{J}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} d^{3} \mathbf{r}^{\prime}$$

A circular loop of wire of radius $a$ carries a current $I$. Take Cartesian coordinates with the origin at the centre of the loop and the $z$-axis normal to the loop. Use the BiotSavart law to show that on the $z$-axis the magnetic field is in the axial direction and of magnitude

$$B=\frac{\mu_{0} I a^{2}}{2\left(z^{2}+a^{2}\right)^{3 / 2}}$$