General non-local electrodynamics: Equations and non-local effects

Abstract

The article proposes a formulation of nonlocal electrodynamics by using the integral and integro-differential operators that form a general fractional calculus in the Luchko form. Some basic concepts and effects of general nonlocal electrodynamics are suggested. Among these concepts, the concept of general nonlocal charged medium (GNCM) is suggested, nonlocality of which is described by the pairs of Sonin kernels that belong to the Luchko set. The general fractional vector calculus (GFVC) is used as mathematical tools to take into account nonlocality in space. The general fractional (GF) integral and GF differential equations of nonlocal electrodynamics as generalization of the standard Maxwell’s equations are proposed. General fractional vector operators, such as GF flux, GF circulation, the GF divergence and GF curl operators are defined. To derive GF differential Maxwell’s equations from the GF integral equations, the general fractional divergence (Gauss–Ostrogradsky) theorem, Stokes’s and Titchmarsh theorems are used. The general nonlocal Gauss’s and Ampere’s circuital laws for nonlocal media are considered with details. Examples of using the general nonlocal Gauss’s law to calculate electric fields in nonlocal media are proposed for the case of spherically symmetric nonlocality and charge distribution. Examples of using the general nonlocal Ampere’s circuital law to calculate magnetic fields in nonlocal media are proposed for a cylindrically symmetric nonlocality and current distributions. The nonlocal effects caused by nonlocality in space and time are discussed. Among such effects are described the effects of electric charge screening by spatial nonlocality, screening of electric current by nonlocality in space, field screening by memory (nonlocality in time), violation of local potentiality of electrostatic field by nonlocality, and violation of local solenoidality of magnetic field by nonlocality.