The Response of a Linear, Homogeneous and Isotropic Dielectric and Magnetic Sphere Subjected to an External Field, DC or Low-Frequency AC, of Any Form

Maxwell’s equations epitomize our knowledge of standard electromagnetic theory in vacuums and matter. Here, we report the clearcut results of an extensive, ongoing investigation aiming to mathematically digest Maxwell’s equations in virtually all problems based on the three standard building units, dielectric and magnetic, found in practice (i.e., spheres, cylinders and plates). Specifically, we address the static/quasi-static case of a linear, homogeneous and isotropic dielectric and magnetic sphere subjected to a DC/low-frequency AC external scalar potential, Uext (vector field, Fext), of any form, produced by a primary/free source residing outside the sphere. To this end, we introduce an expansion-based mathematical strategy that enables us to obtain immediate access to the response of the dielectric and magnetic sphere, i.e., to the internal scalar potential, Uint (vector field, Fint), produced by the induced secondary/bound source. Accordingly, the total scalar potential, U = Uext + Uint (vector field, F = Fext + Fint), is immediately accessible as well. Our approach provides ready-to-use expressions for Uint and U (Fint and F) in all space, i.e., both inside and outside the dielectric and magnetic sphere, applicable for any form of Uext (Fext). Using these universal expressions, we can obtain Uint and U (Fint and F) in essentially one step, without the need to solve each particular problem of different Uext (Fext) every time from scratch. The obtained universal relation between Uint and Uext (Fint and Fext) provides a means to tailor the responses of dielectric and magnetic spheres at all instances, thus facilitating applications. Our approach surpasses conventional mathematical procedures that are employed to solve analytically addressable problems of electromagnetism.