A dipole in a dielectric: Intriguing results and shape dependence of the distant electric field

The field of a point electric dipole in an infinite dielectric is obtained by placing the dipole at the center of a spherical cavity of radius $R$ inside the dielectric and then letting $R\to 0$. The result disagrees with the elementary answer found in textbooks. The mathematical and physical reasons for the disagreement are discussed. The discrepancy is confirmed by the same limiting procedure applied to a uniformly polarized sphere embedded in the dielectric. We next solve the same problem for a polarized spheroid immersed in an infinite dielectric and find that the asymptotic potential shows an unexpected shape dependence, even after taking the limit of an arbitrarily small spheroid. By considering both oblate and prolate spheroids and taking appropriate limits, we recover either the elementary textbook answer or the previous result found for the polarized sphere.