A Variational Approach to Electrostatics of Polarizable Heterogeneous Substances

We discuss equilibrium conditions for heterogeneous substances subject to electrostatic or magnetostatic effects. We demonstrate that the force-like aleph tensor ℵij and the energy-like beth tensor ℶij for polarizable deformable substances are divergence-free: ∇iℵij=0 and ∇iℶij=0. We introduce two additional tensors: the divergence-free energy-like gimel tensor ℷij for rigid dielectrics and the general electrostatic gamma tensor Γij which is not divergence-free. Our approach is based on a logically consistent extension of the Gibbs energy principle that takes into account polarization effects. While the model is mathematically rigorous, we caution against the assumption that it can reliably predict physical phenomena. On the contrary, clear models often lead to conclusions that are at odds with experiment and therefore should be treated as physical paradoxes that deserve the attention of the scientific community