Universal Expressions for the Polarization and the Depolarization Factor in Homogeneous Dielectric and Magnetic Spheres Subjected to an External Field of Any Form

Spherical structures of dielectric and magnetic materials are studied intensively in basic research and employed widely in applications. The polarization, (P for dielectric and M for magnetic materials), is the parent physical vector of all relevant entities (e.g., moment, , and force, F), which determine the signals recorded by an experimental setup or diagnostic equipment and configure the motion in real space. Here, we use classical electromagnetism to study the polarization, , of spherical structures of linear and isotropic—however, not necessarily homogeneous—materials subjected to an external vector field, (Eext for dielectric and Hext for magnetic materials), dc (static), or even ac of low frequency (quasistatic limit). We tackle an integro-differential equation on the polarization, , able to provide closed-form solutions, determined solely from , on the basis of spherical harmonics, Ylm. These generic equations can be used to calculate analytically the polarization, , directly from an external field, , of any form. The proof of concept is studied in homogeneous dielectric and magnetic spheres. Indeed, the polarization, , can be obtained by universal expressions, directly applicable for any form of the external field, . Notably, we obtain the relation between the extrinsic, , and intrinsic, , susceptibilities (χeext and χeint for dielectric and χmext and χmint for magnetic materials) and clarify the nature of the depolarization factor, , which depends on the degree l—however, not on the order m of the mode (l,m) of the applied . Our universal approach can be useful to understand the physics and to facilitate applications of such spherical structures.