Applications

We start with a very simple one. Consider a homogeneous, ohmian conductor, and let the current density $\vec{j}$ vanish at the closed surface which encircles it. We want to inquire as to the possibility that a stationary current may exist in this conductor. We then have $\vec{\nabla}.\vec{j}=0$, as the current is supposed to be stationary, and $\vec{j}=\sigma\vec{E}$, which is Ohm's law. Also, $\vec{\nabla} \times\vec{E}=0$, for a static $\vec{E}$. But then,

$$\vec{\nabla}\times\vec{j}=\sigma\vec{\nabla}\times\vec{E}=0$$ (11)

and so we have both $\vec{\nabla}.\vec{j}=0$ and $\vec{\nabla}\times\vec{j}=0$. Besides, $\vec{j}$ vanishes at the boundaries. It follows, then, from Helmholtz's theorem, that $\vec{j}=0$. So, no stationary current can run under these conditions. Notice that this is true also for a torus and its continuous deformations, so that it applies to any closed circuit, proving the necessity that the condition $\vec{j}=\sigma\vec{E}$ be broken somewhere in the circuit (where the electromotive force is located). Consider now the electromagnetic potentials. The Maxwell equation

$$\vec{\nabla}.\vec{B}=0$$ (12)

states that there exists a vector field $\vec{A}$ such that

$$\vec{B}=\vec{\nabla}\times\vec{A}$$ (13)

this $\vec{A}$ being called the vector potential. We may assume that $\vec{A}$ vanishes at infinity. Now, what we know about $\vec{A}$ is just its $curl$. Therefore, by Helmholtz's theorem, we are free to choose the value of its divergence. For instance, we may take $\vec{\nabla}.\vec{A}=0$, determining completely $\vec{A}$. This is the so-called Coulomb gauge. But we can also put $\vec{\nabla}.\vec{A}=-\frac{1}{c}\frac{\partial\phi} {\partial t}$, $\phi$ being the scalar potential. This is the Lorentz gauge.