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We discuss equilibrium conditions for heterogeneous substances subjectto electrostatic or magnetostatic effects. We demonstrate that theforce-like aleph tensor ℵij and the energy-like beth tensor ℶij for polarizabledeformable substances are divergence-free: ∇iℵij=0 and ∇iℶij=0. We introduce two additional tensors: the divergence-free energy-like gimeltensor 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 Gibbsenergy principle that takes into account polarization effects. While themodel is mathematically rigorous, we caution against the assumption thatit can reliably predict physical phenomena. On the contrary, clear modelsoften lead to conclusions that are at odds with experiment and thereforeshould be treated as physical paradoxes that deserve the attention of thescientific community.

A Variational Approach to Electrostatics of Polarizable Heterogeneous Substances