Multipole Expansion of the Scalar Potential on the Basis of Spherical Harmonics: Bridging the Gap Between the Inside and Outside Spaces via Solution of the Poisson Equation

The multipole expansion on the basis of Spherical Harmonics is a multifaceted mathematical tool utilized in many disciplines of science and engineering. Regarding physics, in electromagnetism, the multipole expansion is exclusively focused on the scalar potential, $U\left(\mathit{r}\right)$, defined only in the so-called inside, $U_{in}\left(\mathit{r}\right)$, and outside, $U_{out}\left(\mathit{r}\right)$, spaces, separated by the middle space wherein the source resides, for both dielectric and magnetic materials. Intriguingly, though the middle space probably encloses more physics than the inside and outside spaces, it is never assessed in the literature, probably due to the rather complicated mathematics. Here, we investigate the middle space and introduce the multipole expansion of the scalar potential, $U_{mid}\left(\mathit{r}\right)$, in this, until now, unsurveyed area. This is achieved through the complementary superposition of the solutions of the inside, $U_{in}\left(\mathit{r}\right)$, and outside, $U_{out}\left(\mathit{r}\right)$, spaces when carefully adjusted at the interface of two appropriately defined subspaces of the middle space. Importantly, while the multipole expansion of $U_{in}\left(\mathit{r}\right)$ and $U_{out}\left(\mathit{r}\right)$ satisfies the Laplace equation, the expression of the middle space, $U_{mid}\left(\mathit{r}\right)$, introduced here satisfies the Poisson equation, as it should. Interestingly, this is mathematically proved by using the method of variation of parameters, which allows us to switch between the solution of the homogeneous Laplace equation to that of the nonhomogeneous Poisson one, thus completely bypassing the standard method in which the multipole expansion of ${|\mathit{r}-{\mathit{r}}^{\prime }|}^{-1}$ is used in the generalized law of Coulomb. Due to this characteristic, the notion of $U_{mid}\left(\mathit{r}\right)$ introduced here can be utilized on a general basis for the effective calculation of the scalar potential in spaces wherein sources reside. The proof of concept is documented for representative cases found in the literature. Though here we deal with the static and quasi-static limit of low frequencies, our concept can be easily developed to the fully dynamic case. At all instances, the exact mathematical modeling of $U_{mid}\left(\mathit{r}\right)$ introduced here can be very useful in applications of both homogeneous and nonhomogeneous, dielectric and magnetic materials.