1/31 – Robert Eisenberg, Rush University
January 31, 2022
12:00 PM - 1:00 PM
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Speaker: Robert Eisenberg
Department of Physiology & Biophysics
Title: Maxwell Equations Imply Exact and Universal Conservation of Total Current, Independent of the Properties of Matter
Mechanical systems need specific models to describe how matter moves when a force is applied. General models are likely to be too vague to be useful. When an electric force is applied to matter with charge, the matter moves—it is said to ‘polarize’—usually with large effects. Polarization is described by a single dielectric constant〖 ϵ〗_r, a real positive number ϵ_r≥1, in the Maxwell equations of electrodynamics. A single dielectric constant does not adequately describe  the properties of material charge in systems of some importance, including all of biology, electrochemistry, and much of electronics.
Here, we show that the Maxwell equations without a dielectric constant imply a universal and exact conservation of total current (that includes ε_0 ∂E⁄∂t) independent of any properties of matter whatsoever. . For example, in ion channels of biological membranes or components of electronic circuits, the total current is equal in space at every time. Ordinary differential equations in time are enough to describe the total current: a spatial variable is not needed to describe total current flow in one dimensional unbranched systems. The simplification is substantial and significant. Partial differential equations in time and space are not needed.
The Maxwell equations become (in these systems) a perfect low pass filter  converting, for example, the infinite spatial variation of Brownian (≃ thermal) motion to no variation at all, to a spatial constant! Kirchhoff’s law becomes exact [4, 5] (in circuits, under all conditions and times) in striking contrast to its usual derivation as a long time, low frequency, nearly DC approximation.
This ‘infinite’ simplification is possible because the Maxwell equations move atoms (with charge) precisely as needed to conserve total current everywhere, at any time, including in a vacuum (with zero mass density) and inside atoms. Other force fields do not have these properties. Electrodynamics is different from other force fields because
(1) charge is entirely independent of velocity, even velocities close to the speed of light, unlike time and distance and (relativistic) mass that depend on velocity; and
(2) total current exists everywhere, including in a vacuum, where the flux of mass is zero.
1. Eisenberg, R. S. 2019. Dielectric Dilemma. preprint available at https://arxiv.org/abs/1901.10805.
2. Eisenberg, R. s. 2020. Maxwell Equations for Material Systems. doi: 10.20944/preprints202011.0201.v1.
3. Eisenberg, R. S. 2020. Electrodynamics Correlates Knock-on and Knock-off: Current is Spatially Uniform in Ion Channels. Preprint on arXiv at https://arxiv.org/abs/2002.09012.
4. Eisenberg, R. S. 2019. Kirchhoff's Law can be Exact. arXiv preprint available at https://arxiv.org/abs/1905.13574.
5. Eisenberg, B., N. Gold, Z. Song, and H. Huang. 2018. What Current Flows Through a Resistor? arXiv preprint arXiv:1805.04814.
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