How will the electrostatic interaction between two point charges change if they are shielded from the other by a dielectrical slab? While the physical setting of this electromagnetic problem is relatively simple, it is easy to be wronged, and the correct solution is surprisingly complicated. Here we will show a general answer using the method of images, in which the electrical field is not found by solving the Poisson's equation but by superposing an infinite number of image charges to recurrently satisfy all interfaces' boundary conditions. We also obtain analytical and algebraic results in some special cases.
2. Barcellona, P., R. Bennett, and S. Y. Buhmann, "Manipulating the Coulomb interaction: A Green's function perspective," Journal of Physics Communications, Vol. 2, No. 3, 035027, 2018.
3. Purcell, E. M. and D. J. Morin, Electricity and Magnetism, Cambridge University Press, 2013.
4. Landau, L. D., E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol. 8, Elsevier, 2013.
5. Stratton, J. A., Electromagnetic Theory, Vol. 33, John Wiley & Sons, 2007.
6. Hammond, P., "Electric and magnetic images," Proceedings of the IEE --- Part C: Monographs, Vol. 107, 306, 1960.
7. Jackson, J. D., Classical Electrodynamics, Vol. 31999, Wiley, New York, 1977.
8. Lindell, I. V., "Electrostatic image theory for the dielectric sphere," Radio Science, Vol. 27, No. 1, 1-8, 1992.
9. Lindell, I. V., M. E. Ermutlu, and A. H. Sihvola, "Electrostatic image theory for layered dielectric sphere," IEE Proceedings H (Microwaves, Antennas and Propagation), Vol. 139, No. 2, 186-192, 1992.
10. Nikoshkinen, K. I. and I. V. Lindell, "Image solution for Poisson's equation in wedge geometry," IEEE Transactions on Antennas and Propagation, Vol. 43, No. 2, 179-187, 1995.
11. Johan, C.-E. S. and I. V. Lindell, "Electrostatic image theory for the dielectric sphere with an internal source," Microwave and Optical Technology Letters, Vol. 5, No. 11, 597-602, 1992.
12. Lindell, I. V. and I. N. Keijo, "Electrostatic image theory for the dielectric prolate spheroid," Journal of Electromagnetic Waves and Applications, Vol. 15, No. 8, 1075-1096, 2001.
13. Sometani, T., "Image method for a dielectric plate and a point charge," European Journal of Physics, Vol. 21, No. 6, 549, 2000.
14. Cartier, P. E., B. Julia, P. Moussa, and P. Vanhove, Frontiers in Number Theory, Physics, and Geometry II: on Conformal Field Theories, Discrete Groups and Renormalization, Vol. 2, Springer Science & Business Media, 2007.
15. Loxton, J., "Special values of the dilogarithm function," Acta Arithmetica, Vol. 43, No. 2, 155-166, 1984.
16. Bossa, G. V. and S. May, "Integral representation of electrostatic interactions inside a lipid membrane," Molecules, Vol. 25, No. 17, 3824, 2020.