Maxwell Equations without a Polarization Field, Using a Paradigm from Biophysics

Abstract

When forces are applied to matter, the distribution of mass changes. Similarly, when an electric field is applied to matter with charge, the distribution of charge changes. The change in the distribution of charge (when a local electric field is applied) might in general be called the induced charge. When the change in charge is simply related to the applied local electric field, the polarization field P is widely used to describe the induced charge. This approach does not allow electrical measurements (in themselves) to determine the structure of the polarization fields. Many polarization fields will produce the same electrical forces because only the divergence of polarization enters Maxwell’s first equation, relating charge and electric forces and field. The curl of any function can be added to a polarization field P without changing the electric field at all. The divergence of the curl is always zero. Additional information is needed to specify the curl and thus the structure of the P field. When the structure of charge changes substantially with the local electric field, the induced charge is a nonlinear and time dependent function of the field and P is not a useful framework to describe either the electrical or structural basis-induced charge. In the nonlinear, time dependent case, models must describe the charge distribution and how it varies as the field changes. One class of models has been used widely in biophysics to describe field dependent charge, i.e., the phenomenon of nonlinear time dependent induced charge, called ‘gating current’ in the biophysical literature. The operational definition of gating current has worked well in biophysics for fifty years, where it has been found to makes neurons respond sensitively to voltage. Theoretical estimates of polarization computed with this definition fit experimental data. I propose that the operational definition of gating current be used to define voltage and time dependent induced charge, although other definitions may be needed as well, for example if the induced charge is fundamentally current dependent. Gating currents involve substantial changes in structure and so need to be computed from a combination of electrodynamics and mechanics because everything charged interacts with everything charged as well as most things mechanical. It may be useful to separate the classical polarization field as a component of the total induced charge, as it is in biophysics. When nothing is known about polarization, it is necessary to use an approximate representation of polarization with a dielectric constant that is a single real positive number. This approximation allows important results in some cases, e.g., design of integrated circuits in silicon semiconductors, but can be seriously misleading in other cases, e.g., ionic solutions.

Setting Boundaries for Statistical Mechanics

“…The actual measured polarization properties of liquids can almost never be reasonably approximated by a dielectric constant ε r as documented in an extensive literature [29,. The literature of impedance spectroscopy is reviewed recently with extensive citations of the literature in [108].…”

“…In fact, the description of polarization by a single positive real number is almost never an adequate representation of the properties of real systems [29,[121][122][123]. The reformulation of the Maxwell equations for nonconstant ε r − 1 will produce equations with very different mathematical form, in general requiring convolutions in the time domain.…”

“…They make a contribution to the classical Maxwell polarization field. The previously cited literature on the dielectric properties of matter is devoted to the measurement, description and analysis of this polarization, including [29,[121][122][123]. The polarization cannot be adequately described by a single dielectric constant ε r , by one positive real number ε r > 1.…”

“…A field with zero divergence is by definition a field that is perfectly conserved, that can neither accumulate nor dissipate nor disappear. Thus, the right side of the Maxwell-Ampere law is a perfectly conserved quantity, an incompressible fluid whose flow might be called 'total current' [29,30]. Because the right hand side of the Maxwell Ampere law always includes a term ε 0 ∂E/∂t that is present everywhere, even where charge and its flux J are zero, this term can provide the coupling needed to create radiation.…”

Introduction to the Physics of Ionic Conduction in Narrow Biological and Artificial Channels

“…This approach can be very helpful and revealing, but it involves greater approximation than spatial averaging because, as R.S. Eisenberg points out [ 5 ], the material’s response to the electric field may be both nonlinear and time-dependent. In order to accommodate such phenomena, while simultaneously challenging physicists to review their knowledge of electromagnetism in biological dielectrics, he proposes and discusses an apparently minor change in Maxwell’s first equation.…”