On the vector solutions of Maxwell equations with the spin-weighted spherical harmonics

The Maxwell equations for the spherical components of the electromagnetic fields outside sources do not separate into equations for each component alone. We show, however, that general solutions can be obtained by separation of variables in the case of azimuthal symmetry. Boundary conditions are easier to apply to these solutions, and their forms highlight the similarities and differences between the electric and magnetic cases in both time-independent and time-dependent situ...

Variables are separated in Maxwell equations by the Newman-Penrose method of isotropic complex tetrade in the uniformly accelerated spherical coordinate system. Particular solutions are obtained in terms of spin 1 spherical harmonics. PACS: 03.50.De

A scalar field method to obtain transverse solutions of the vector Laplace and Helmholtz equations in spherical coordinates for boundary-value problems with azimuthal symmetry is described. Neither scalar nor vector potentials are used. Solutions are obtained by use of separation of variables instead of dyadic Green's functions expanded in terms of vector spherical harmonics. Applications to the calculations of magnetic fields from steady and oscillating localized current dis...

We outline a regular way for solving Maxwell's equations. We take, as the starting point, the notion of vector potentials. The rationale for introducing this notion in electrodynamics is that the set of Maxwell's equations is seemingly overdetermined. We demonstrate the existence of two fundamental solutions to Maxwell's equations whose linear combinations comprise the whole variety of classical electromagnetic field configurations.

We present in a unified and self-contained manner the coordinate-free approach to spherical harmonics initiated in the mid 19th century by James Clerk Maxwell, William Thomson and Peter Guthrie Tait. We stress the pedagogical advantages of this approach which leads in a natural way to many physically relevant results that students find often difficult to work out using spherical coordinates and associated Legendre functions. It is shown how most physically relevant results of...

We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions which are up to a finite sum of special generalized spherical harmonics in another appropriately weighted Sobolev space. As a byproduct we prove a generalized sphe...

A new representation for solutions of Maxwell's equations is derived. Instead of being expanded in plane waves, the solutions are given as linear superpositions of spherical wavelets dynamically adapted to the Maxwell field and well-localized in space at the initial time. The wavelet representation of a solution is analogous to its Fourier representation, but has the advantage of being local. It is closely related to the relativistic coherent-state representations for the Kle...

We investigate the representations of the solutions to Maxwell's equations based on the combination of hypercomplex function-theoretical methods with quantum mechanical methods. Our approach provides us with a characterization for the solutions to the time-harmonic Maxwell system in terms of series expansions involving spherical harmonics resp. spherical monogenics. Also, a thorough investigation for the series representation of the solutions in terms of eigenfunctions of Lan...

In electrodynamics courses and textbooks, the properties of plane electromagnetic waves in both conducting and non-conducting media are typically studied from the point of view of the prototype case of a monochromatic plane wave. In this note an approach is suggested that starts from more general considerations and better exploits the independence of the Maxwell equations.

An outline is given of recent work concerning the electromagnetic duality properties of Maxwell theory on curved space-times with or without spin structures.