Blow-up of Electric Eields between Closely Spaced Spherical Perfect Conductors∗

In the present work, we derive that electric fields blow up as a pair of spherical perfect conductors approach each other, and calculate the blow-up rate by a simple method. In the case of two dimensional space, referring to an anti-plane shear model, we can assume that the conductors and the electric fields represent the cross-sections of parallel stiff fibers and the stresses, respectively. Hence, this blow-up result can also be evidence for high stress concentrations occurring in fiber-reinforced composites. The practical significance has stimulated several approaches related to this. Recently, Bao, Li and Yin have established the optimal blow-up rate of electric fields related to the distance between conductors that is not restricted to two dimensional space. In particular, their result in three dimensional space is something unexpected and significantly different from anticipation of the interested persons. In this paper, we assume that the two conductors are unit spheres in n dimensional space (n ≥ 2). Using Green’s theorem and point charges, we present a new method to easily get a optimal lower bound of the blow-up rate by Bao, Li and Yin. This simple derivation helps to understand the blow-up phenomenon easily. MSC-class: 15A15, 15A09, 15A23