Electrostatics in Fractal Geometry: Fractional Calculus Approach

TL;DR

This paper develops a fractional calculus approach to analyze electrostatic properties of composite materials with fractal geometries, deriving electric fields in fractal charge distributions.

Contribution

It introduces a fractional calculus method to model electrostatics in fractal geometries, providing a new analytical framework for such complex materials.

Findings

Derived electric field expressions for fractal charge distributions.

Established a fractional calculus-based methodology for fractal electrostatics.

Provided insights into the influence of fractal dimension on electrostatic behavior.

Abstract

The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume $V_d\sim r^d$ with the fractal dimension $d$ and a complementary host volume $V_h=V_3-V_d$. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.