A steady current $I_{2}$ flows around a loop $\mathcal{C}_{2}$ of a perfectly conducting narrow wire. Assuming that the gauge condition $\nabla \cdot \mathbf{A}=0$ holds, the vector potential at points away from the loop may be taken to be

$$\mathbf{A}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2}}{\left|\mathbf{r}-\mathbf{r}_{2}\right|}$$

First verify that the gauge condition is satisfied here. Then obtain the Biot-Savart formula for the magnetic field

$$\mathbf{B}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2} \times\left(\mathbf{r}-\mathbf{r}_{2}\right)}{\left|\mathbf{r}-\mathbf{r}_{2}\right|^{3}}$$

Next suppose there is a similar but separate loop $\mathcal{C}_{1}$ with current $I_{1}$. Show that the magnetic force exerted on loop $\mathcal{C}_{1}$ by loop $\mathcal{C}_{2}$ is

$$\mathbf{F}_{12}=\frac{\mu_{0} I_{1} I_{2}}{4 \pi} \oint_{\mathcal{C}_{1}} \oint_{\mathcal{C}_{2}} d \mathbf{r}_{1} \times\left(d \mathbf{r}_{2} \times \frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{3}}\right)$$

Is this consistent with Newton's third law? Justify your answer.