A simple and effective method is proposed to derive the three-dimensional electric potential induced by a point singularity of any type in an N-phase dielectric medium composed of N−2 intermediate dielectric layers of equal thickness encased in two semi-infinite dielectric media. The point singularity can include a point charge or a point electric dipole. The original boundary value problem for the N-phase medium is reduced to the determination of a single unknown three-dimensional harmonic function through satisfaction of the continuity conditions across all of the N−1 perfect planar interfaces. The single harmonic function can be completely determined after analytically solving the resulting linear recurrence relations, which are independent of the type and the specific location of the singularity. When the singularity is a point charge, we obtain the self-energy of the point charge expressed in terms of a single function and the Coulomb force on the point charge expressed in terms of the negative derivative of this function.
multilayered dielectric medium point charge point electric dipole linear recurrence relation analytical solution
Chinese Library Classification
O174.3 O411.1 O441.1
2010 Mathematics Subject Classification
31B05 35J08 35J25 65N80 65Q30
This is a preview of subscription content, log in to check access.
KIPP, R. A. and CHAN, C. H. Complex image method for sources in bounded regions of multilayer structures. IEEE Transactions on Microwave Theory and Technology, 42, 860–865 (1994)CrossRefGoogle Scholar
YUAN, M., SARKAR, T. K., and SALAZAR-PALMA, M. A direct discrete complex image method from the closed-form Green’s functions in multilayered media. IEEE Transactions on Microwave Theory and Technology, 54, 1025–1032 (2006)CrossRefGoogle Scholar
KISHI, H., MIZUNO, Y., and CHAZONO, H. Base-metal electrode-multilayer ceramic capacitors: past, present and future perspectives. Japanese Journal of Applied Physics, 42, 1–15 (2003)CrossRefGoogle Scholar
PAN, M. J. and RANDALL, C. A. A brief introduction to ceramic capacitors. IEEE Electrical Insulation Magazine, 26, 44–50 (2010)CrossRefGoogle Scholar
RAHMAN, H. I. and TANG, T. Electric potential of a point charge in multilayered dielectrics evaluated from Hankel transform. Journal of Engineering Mathematics, 110, 63–73 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
ZALOJ, V. and AGMON, N. Electrostatics of multilamellar vesicles: Legendre expansion and reaction-field Brownian dynamics. The Journal of Chemical Physics, 108, 1216–1224 (1998)CrossRefGoogle Scholar
WANG, X., CHEN, L., and SCHIAVONE, P. Three-dimensional Green’s functions for a point heat source in a trimaterial. Zeitschrift für Angewandte Mathematik und Physik, 70(2), 61 (2019)MathSciNetCrossRefzbMATHGoogle Scholar