Dielectric and magnetic spherical hollow shells are employed in many applications as standard building units. These structures are commonly subjected to size reduction to obtain a high surface area/volume ratio, a property that is in favor of specific applications. However, the size reduction enhances the importance of physical mechanisms that originate from surfaces, such as the depolarization effect. Here we tackle the problem of dielectric and magnetic spherical hollow shells, consisting of a linear, homogeneous and isotropic parent material, subjected to an external potential, Uextr, of any spatial form (either dc (static) or ac of low-frequency (quasistatic limit)). By applying the method-of-linear-recursive-solution (MLRS) to the Laplace equation, we calculate analytically the internal, Uintr, and total, Utotr, potentials in respect to the external one, Uextr. From Uintr and Utotr we calculate all relevant scalar and vector physical entities of interest. The MLRS unveils straightforwardly the existence of two distinct depolarization factors, Nl=l/(2l+1) and Nl+1=(l+1)/(2l+1), both depending on the degree, l, however not on the order, m, of the mode of the external potential, Uext(l,m)r. These depolarization factors, Nl and Nl+1, originate from the outer, r=b, and inner, r=a, surfaces and are accompanied by two extrinsic susceptibilities, χe,lext=χe /(1+Nlχe ) and χe,l+1ext=χe /(1+Nl+1χe ), respectively. Importantly, Nl+Nl+1=1, irrespective of the degree, l, as it should. The properties of spherical hollow shells are investigated through analytical modeling and detailed simulations, with emphasis on application-relevant scenarios including resonance phenomena in scattering, quantitative materials characterization, and shielding/distortion. The generic MLRS strategy provides a flexible and reliable route for analyzing depolarization processes in other dielectric and magnetic building-unit geometries encountered in practice.
Moraitis et al. (2026) studied this question.