Analysis of the potential decay of an insulator after static electrical charge deposit

What is it about?

It is about a quite classical problem in electrostatics: the rate at which a charged insulator will discharge. This is part of a more general topic which is "relaxation", which may be what happens when a solid is subjected to a given pressure, or a photoconductive material to a pulse of light, etc. and that the response is not instantaneous but requires a given time The simplest hypothesis about this relaxation is assuming an evolution towards equilibrium by a decreasing exponential. However in condensed matter relaxation usually follows time power laws, which means that it may be very fast at the beginning, but presents a very long tail (the decay becomes slower and slower with time). This paper presents a mathematical overview of this problem for a dielectric in open circuit, ie with a bare surface that could be electrostatically charged, and a calculation is given for the response (potential with time) of a dielectric following a quite classical behaviour, which is a "Cole-Cole" response in the frequency domain, when studied between electrodes with a dielectric impedance analyzer. The solution of this problem involves "Mittag Leffler" functions, from the name of an eminent Swedish mathematician living one century ago.

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Photo by Thomas Kelley on Unsplash

Photo by Thomas Kelley on Unsplash

Why is it important?

Dielectric relaxation has been widely studied, but the open-circuit case has been quite neglected, despite it is a very common situation. As underlined in the introduction of the paper, the static charge is a nuisance in countless areas, from grain silos to satellites. It is also a problem for the development of high voltage DC systems which are being now largely developed to allow the long distance power transmission which is required for the connexion of renewable energy sources. Charge decay control is also critical in many applications, from energy harvesting to electret microphones or loudspeakers. Hence, the interest of this paper is that it establishes, for the first time, an analytical formula for the potential decay in a very common situation, which is a relaxation process following the Cole-Cole law in the frequency domain. It provides also examples of how to implement the result (which is a Mittag-Leffler function) to fit to experimental measurements.

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