Electrostatics as a Stress Theory in $\mathbb R^3$ : Free and Bound Charge, Gauge Invariance, and Mechanical Interactions
We formulate electrostatics in \rthree as the simplest instance of the theory of stresses: the virtual velocity field of continuum mechanics is simply replaced by a scalar potential field. A charge distribution is then a linear functional on potential fields whose value is the potential energy, and it is represented by a volume charge density and by surface charge densities on the boundaries of subbodies. Cauchy's theorem gives a vector field $\boldsymbol P$, the analog of the traction stress, and a scalar field $r$, so that the energy is $\int(rφ+\boldsymbol P\cdot\nablaφ)dV$. The pair $(r,\boldsymbol P)$, the analog of a variational stress, splits every charge distribution uniquely into free charge, with density $r$, and a dipole distribution, with density $\boldsymbol P$, which carries the bound charges $-\nabla\cdot\boldsymbol P$ and $\boldsymbol P\cdot\boldsymbolν$; the fields $r$ and $\boldsymbol P$ are the densities of charge and of dipole moment. We do not impose gauge invariance of the energy. Instead, we show that it is the analog of the balance of forces on every subbody, and that it singles out the neutral charge distributions. For charges embedded in matter, the forces exerted by the electric field are derived. They are shown to be gauge invariant for every charge distribution, and they include an asymmetric active stress $\boldsymbol E\boldsymbol\otimes\boldsymbol P$ whose skew-symmetric part is the couple density $\boldsymbol P\times\boldsymbol E$.