Free Maxwell equations in vacuum in orthogonal curvilinear coordinates and some applications: Helmholtz equation, poynting vector, and energy density

In this paper, we present a generalized method of solving the Free Maxwell Equations in Vacuum employing the process of separation of variables, which can be used for different systems of orthogonal curvilinear coordinates. This method leads us to obtain the Helmholtz equation for the distinct syste...

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Bibliographic Details
Published in:European physical journal plus 2024-09, Vol.139 (9), p.792, Article 792
Main Authors: Pérez-Carlos, David A., Gutiérrez-Rodríguez, A., Puga-Candelas, Alejandro
Format: Article
Language:English
Subjects:
Applied and Technical Physics
Atomic
Charged particles
Complex Systems
Condensed Matter Physics
Electrons
Fields (mathematics)
Helmholtz equations
Magnetic fields
Mathematical and Computational Physics
Maxwell's equations
Molecular
Optical and Plasma Physics
Parity
Physics
Physics and Astronomy
Prolate spheroids
Regular Article
Spherical coordinates
Theoretical
Time dependence
Variables
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Summary:In this paper, we present a generalized method of solving the Free Maxwell Equations in Vacuum employing the process of separation of variables, which can be used for different systems of orthogonal curvilinear coordinates. This method leads us to obtain the Helmholtz equation for the distinct systems of curvilinear coordinates. We show the case for the prolate spheroidal coordinates, obtaining the solutions dependent on the spatial and time coordinates. An analysis of the Poynting vector field of the solutions is presented together with the energy density analysis. It is shown that prolate spheroidal coordinates do not allow the existence of closed surfaces of the magnetic field; in addition, the nodal surfaces (geometric places where the energy density does not change in time) are not closed surfaces either. Our results may be helpful to the scientific community because they provide us with a systematic method for solving the system of Free Maxwell Equations in Vacuum for such coordinate systems.
ISSN:2190-5444
2190-5444
DOI:10.1140/epjp/s13360-024-05608-y