Can someone provide a derivation of the Biot-Savart law for electromagnetic induction? To be clear, $$ d\vec{B}~=~\frac{\mu_0}{4\pi}\frac{I d\vec{\ell}\times \vec{r}}{r^3}. $$
Is there a simple way to compute the magnetic field at a point between two Helmholtz coils, if the radii of the coils are the same and the current through each coil is the same?
In the static case you can solve Maxwell equations using a vector potential via the poisson equatuion for the magnetic potential.
$\Delta \vec A(\vec r)=-\mu_0 \vec J(\vec r)$
Using the Greens function for the Laplace operator yields the solution of this differential equation.
$\vec A(\vec r)=\frac{\mu_0}{4 \pi}\int d^3r' \frac{\vec J(\vec r')}{|{\vec r-\vec r'}|}$
Now we can calculate the B field via $\vec B = \vec\nabla \times \vec A$ and use the identity $\vec\nabla\times(\phi\vec A)=\phi(\vec\nabla\times(\vec A))-\vec A\times\vec\nabla\phi $. Additionally we have to calculate the gradient of the scalar function 1/|(r-r')|. This gives the Bio Savart law.
