Starting from the Lorentz force law acting on a current distribution $\mathbf{J}$ obeying $\boldsymbol{\nabla} \cdot \mathbf{J}=0$, show that the energy of a magnetic dipole $\mathbf{m}$ in the presence of a time independent magnetic field $\mathbf{B}$ is

$$U=-\mathbf{m} \cdot \mathbf{B}$$

State clearly any approximations you make.

[You may use without proof the fact that

$$\int(\mathbf{a} \cdot \mathbf{r}) \mathbf{J}(\mathbf{r}) \mathrm{d} V=-\frac{1}{2} \mathbf{a} \times \int(\mathbf{r} \times \mathbf{J}(\mathbf{r})) \mathrm{d} V$$

for any constant vector $\mathbf{a}$, and the identity

$$(\mathbf{b} \times \boldsymbol{\nabla}) \times \mathbf{c}=\boldsymbol{\nabla}(\mathbf{b} \cdot \mathbf{c})-\mathbf{b}(\boldsymbol{\nabla} \cdot \mathbf{c})$$

which holds when $\mathbf{b}$ is constant.]

A beam of slowly moving, randomly oriented magnetic dipoles enters a region where the magnetic field is

$$\mathbf{B}=\hat{\mathbf{z}} B_{0}+(y \hat{\mathbf{x}}+x \hat{\mathbf{y}}) B_{1},$$

with $B_{0}$ and $B_{1}$ constants. By considering their energy, briefly describe what happens to those dipoles that are parallel to, and those that are anti-parallel to the direction of $\mathbf{B}$.