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Advances in Mathematical Physics
Volume 2015 (2015), Article ID 659127, 7 pages
Research Article

A Variational Approach to Electrostatics of Polarizable Heterogeneous Substances

1Aberdeen Proving Ground, US Army Research Laboratory, Aberdeen, MD 21005-5066, USA
2Drexel University, Philadelphia, PA 19104, USA

Received 5 December 2014; Revised 2 April 2015; Accepted 8 April 2015

Academic Editor: Giorgio Kaniadakis

Copyright © 2015 Michael Grinfeld and Pavel Grinfeld. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We discuss equilibrium conditions for heterogeneous substances subject to electrostatic or magnetostatic effects. We demonstrate that the force-like aleph tensor and the energy-like beth tensor for polarizable deformable substances are divergence-free: and . We introduce two additional tensors: the divergence-free energy-like gimel tensor for rigid dielectrics and the general electrostatic gamma tensor 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.