Electrostatic “fractional” image methods for perfectly conducting wedges and cones

Engheta (1996) introduced a definition for the electric charge “fractional-order” multipoles using the concept of fractional derivatives and integrals. Here, we utilize that definition to introduce a detailed image theory for the two-dimensional (2-D) electrostatic potential distributions in front of a perfectly conducting wedge with arbitrary wedge angles, and for the three-dimensional potential in front of a perfectly conducting cone with arbitrary cone angles. We show that the potentials in the presence of these structures can be described equivalently as the electrostatic potentials of sets of equivalent “image” charge distributions that effectively behave as “fractional-order” multipoles; hence, the name “fractional” image methods. The fractional orders of these so-called fractional images depend on the wedge angle (for the wedge problem) and on the cone angle (for the cone problem). Special cases where these fractional images behave like the discrete images are discussed, and physical justification and insights into these results are given