ON FRACTIONAL CALCULUS AND FRACTIONAL MULTIPOLES IN ELECTROMAGNETISM

In this paper, using the concept and tools of fractional calculus, we introduce a definition for ''fractional-order'' multipoles of electric -charge densities, and we show that as far as their scalar potential d istributions are concerned, such fractional-order multipoles effective ly behave as ''intermediate'' sources bridging the gap between the cas es of integer-order point multipoles such as point monopoles, point di poles, point quadrupoles, etc, This technique, which involves fraction al differentiation or integration of the Dirac delta function, provide s a tool for formulating an electric source distribution whose potenti al functions can be obtained by using fractional differentiation or in tegration of potentials of integer-order point-multipoles of lower or higher orders, As illustrative examples, the cases of three-dimensiona l (point source) and two-dimensional (line source) problems in electro statics are treated in detail, and extension to time-harmonic case is also addressed, In the three-dimensional electrostatic example, we sug gest an electric-charge distribution which can be regarded as an ''int ermediate'' case between cases of the electric-point monopole (point c harge) and the electric-point dipole (point dipole), and we present it s electrostatic potential which behaves as r(-(1+alpha))P-alpha(-cos t heta) where 0 < alpha < 1 and P-alpha(.) is the Legendre function of n oninteger degree alpha, thus denoting this charge distribution as frac tional 2(alpha)-pole, At the two limiting cases of alpha = 0 and alpha = 1, this fractional 2(alpha)-pole becomes the standard point monopol e and point dipole, respectively, A corresponding intermediate fractio nal-order multipole is also given for the two-dimensional electrostati c case, Potential applications of this treatment to the image method i n electrostatic problems are briefly mentioned, Physical insights and interpretation for such fractional-order 2(alpha)-poles are also given .