Derive from Maxwell's equations the Biot-Savart law

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

giving the magnetic field $\mathbf{B}(\mathbf{r})$ produced by a steady current density $\mathbf{j}(\mathbf{r})$ that vanishes outside a bounded region $V$.

[You may assume that the divergence of the magnetic vector potential is zero.]

A steady current density $\mathbf{j}(\mathbf{r})$ has the form $\mathbf{j}=\left(0, j_{\phi}(\mathbf{r}), 0\right)$ in cylindrical polar coordinates $(r, \phi, z)$ where

$$j_{\phi}(\mathbf{r})= \begin{cases}k r & 0 \leqslant r \leqslant b, \quad-h \leqslant z \leqslant h \\ 0 & \text { otherwise },\end{cases}$$

and $k$ is a constant. Find the magnitude and direction of the magnetic field at the origin.

$$\left[\text { Hint }: \quad \int_{-h}^{h} \frac{d z}{\left(r^{2}+z^{2}\right)^{3 / 2}}=\frac{2 h}{r^{2}\left(h^{2}+r^{2}\right)^{1 / 2}}\right]$$