Deriving the Biot-Savart Law from the Helmholtz theorem?

Question

The Helmholtz theorem tells us that any vector function $\vec{F(\vec{r})}$ that goes to zero sufficiently fast can be expressed as $$\vec{F(\vec{r})}=\nabla(\frac{-1}{4\pi}\int\frac{\nabla'\cdot\vec{F(\vec{r}')}}{R}dV')+\nabla\times(\frac{1}{4\pi}\int\frac{\nabla'\times\vec{F(\vec{r}')}}{R}dV')$$

where $R=|\vec{r}-\vec{r'}|$ is the magnitude of the separation vector. Now from Maxwell's equations (assuming magnetostatics) we have that $\nabla \cdot \vec{B}=0$ and $\nabla \times \vec{B}=\mu_0\vec{J}$. Combining these facts with the Helmholtz theorem above, we get that the magnetic field (within the domain of magnetostatics) should take the form $$\vec{B(\vec{r})}=\nabla\times\frac{\mu_o}{4\pi}\int\frac{\vec{J(\vec{r}')}}{R}dV' \tag{$1$}$$ Now the above should be equivalent to the Biot-Savart law for volume currents which is $$\vec{B(\vec{r})}=\frac{\mu_0}{4\pi}\int\frac{\vec{J(\vec{r}')\times\hat{R}}}{R^2}dV'\tag{$2$}$$ But these two equations are noticeably different (at least in their present form). Is there any way to manipulate equation (1) such that we end up with equation (2) ?

Any help would be most appreciated!

Answer 1

Figured it out.

Using the product rule $\nabla \times(f\vec{J})=f(\nabla \times \vec{J})-\vec{J}\times (\nabla f)$ with $f=1/R$ we get $$\vec{B(\vec{r})}=\nabla\times\frac{\mu_o}{4\pi}\int\frac{\vec{J(\vec{r}')}}{R}dV'$$ $$\vec{B(\vec{r})}=\frac{\mu_o}{4\pi}\int\nabla\times(\frac{\vec{J(\vec{r}')}}{R})dV'$$ $$\vec{B(\vec{r})}=\frac{\mu_o}{4\pi}\int \frac{1}{R}(\nabla \times \vec{J(r')})-\vec{J(r')}\times (\nabla \frac{1}{R})dV'$$ But now the curl operator is with respect to the un-primed $\vec{r}$ coordinates, not the primed $\vec{r'}$ coordinates. So the first term in the above is zero. So then we have $$\vec{B(\vec{r})}=\frac{-\mu_o}{4\pi}\int \vec{J(r')}\times (\nabla \frac{1}{R})dV'$$ Now using the identity $\nabla \frac{1}{R}=\nabla \frac{1}{|\vec{r}-\vec{r'}|}=-\frac{\hat{R}}{R^2}$ , we get that $$\vec{B(\vec{r})}=\frac{\mu_o}{4\pi}\int\frac{\vec{J(\vec{r}')\times \hat{R}}}{R^2}dV'$$ which is precisely the Biot-Savart Law as required