Solución de las ecuaciones de Maxwell en la materia en medios homogéneos e isotrópicos, en una, dos y tres dimensiones, implementando el método de diferencias finitas en el dominio del tiempo

Abstract

When studying the propagation of electromagnetic fields at an arbitrary point in space, Maxwell's equations are used to relate the electric and magnetic fields, considering both temporal and spatial variations. The difficulty lies in the geometry of the problem, making it challenging to find a numerical solution and visually represent the electromagnetic field solution. That's why computational simulation is implemented as an alternative to describe the electromagnetic field in various geometries and media. The Finite Difference Time Domain (FDTD) method is a computational electromagnetics modeling technique. Being a time-dependent method, the solutions cover a wide range of frequencies in a single simulation. The time-dependent Maxwell's equations for the propagation of electromagnetic waves in homogeneous and isotropic media are discretized using the FDTD method, employing central difference approximations for temporal and spatial derivatives. The resulting equations are solved by implementing the Python programming language and developing an algorithm to obtain a numerical solution to the problem, enabling animated visualizations of the physical behavior of electromagnetic waves in one, two, and three dimensions. The set of discretized equations will have boundary conditions of a perfectly electrically conducting conductor and absorbing boundary conditions at the computational limit, aiming to achieve applicability in different types of problems and obtain a numerical solution. This method offers various applications, allowing the resolution of different types of problems in multiple scientific disciplines, where it is possible to study systems and obtain quantifiable solutions for phenomena related to electromagnetic fields.