Electrostatic "Fractional" Image Methods for Perfectly Conducting Wedges and Cones

In our earlier work, we introduced a definition for the electric charge "fractional-order" multipoles using the concept of fractional derivatives and integrals [l]. 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. Comments Copyright 1996 IEEE. Reprinted from IEEE Transactions on Antennas and Propagation, Volume 44, Issue 12, December 1996, pages 1565-1574. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/243 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 44, NO. 12, DECEMBER 1996 1565 Electrostatic “Fractional” Image Methods for Perfectly Conducting Wedges and Cones Nader Engheta, Fellow, IEEE Abstract-In our earlier work, we introduced a definition for the electric charge “fractional-order” multipoles using the concept of fractional derivatives and integrals [l]. 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.In our earlier work, we introduced a definition for the electric charge “fractional-order” multipoles using the concept of fractional derivatives and integrals [l]. 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.