A steady magnetic field $\mathbf{B}(\mathbf{x})$ is generated by a current distribution $\mathbf{j}(\mathbf{x})$ that vanishes outside a bounded region $V$. Use the divergence theorem to show that

$$\int_{V} \mathbf{j} d V=0 \quad \text { and } \quad \int_{V} x_{i} j_{k} d V=-\int_{V} x_{k} j_{i} d V$$

Define the magnetic vector potential $\mathbf{A}(\mathbf{x})$. Use Maxwell's equations to obtain a differential equation for $\mathbf{A}(\mathbf{x})$ in terms of $\mathbf{j}(\mathbf{x})$.

It may be shown that for an unbounded domain the equation for $\mathbf{A}(\mathbf{x})$ has solution

$$\mathbf{A}(\mathbf{x})=\frac{\mu_{0}}{4 \pi} \int_{V} \frac{\mathbf{j}\left(\mathbf{x}^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|} d V^{\prime}$$

Deduce that in general the leading order approximation for $\mathbf{A}(\mathbf{x})$ as $|\mathbf{x}| \rightarrow \infty$ is

$$\mathbf{A}=\frac{\mu_{0}}{4 \pi} \frac{\mathbf{m} \times \mathbf{x}}{|\mathbf{x}|^{3}} \quad \text { where } \quad \mathbf{m}=\frac{1}{2} \int_{V} \mathbf{x}^{\prime} \times \mathbf{j}\left(\mathbf{x}^{\prime}\right) d V^{\prime}$$

Find the corresponding far-field expression for $\mathbf{B}(\mathbf{x})$.