Mathematical Analysis of Conducting and Dielectric Sphere in Fractional Space

Mathematical Analysis of Conducting and Dielectric Sphere in Fractional Space


  • Muhammad Imran Shahzad Federal Urdu University of Arts
  • Muhammad Akbar Quaid-i-Azam University
  • Saeed Ahmed Quaid-i-Azam University
  • Sania Shaheen Federal Urdu University of Arts
  • Muhammad Ahmad Raza Federal Urdu University of Arts



Laplacian Equation, Fractional Space, Dielectric Sphere, Separable Method


This paper presents an analytical analysis of a sphere placed in fractional dimensional space. The Laplacian Equation in fractional space describes physics as a complex phenomenon. The general solution of the Laplacian equation in fractional space is obtained by the separable variable technique. We have investigated a close form solution for conducting sphere and dielectric sphere. Further, the electric potential and charge density, induced due to a point charge is calculated in fractional space, and also the energy radiated by the sphere is determined. The results are compared with the classical results by setting the fractional parameter α = 3 which normally lies in the limit 2 < α ≤ 3.


C.G. Bollini, and J.J. Giambiagi. Dimensional renormalization: The number of dimensions as a regularizing parameter. Il Nuovo Cimento B 12: 20-26 (1972).

J.F. Ashmore. On renormalization and complex space-time dimensions. Communications in Mathematical Physics 29: 177-187 (1973).

K.G. Wilson. Quantum field-theory models in less than 4 dimension. Physical Review D 7: 2911-2926 (1973).

F.H. Stillinger. Axiomatic basis for spaces with non integer dimension. Journal of Mathematical Physics 18: 1224-1234 (1977).

X.F. He. Excitons in anisotropic solids: The model of fractional-dimensional space. Physical Review B 43: 2063-2069 (1991).

C.M. Bender, and S. Boettcher. Dimensional expansion for the Ising limit of quantum field theory. Physical Review D 48: 4919-4923 (1993).

C.M. Bender, and K.A. Milton. Scalar Casimir effect for a D-dimension sphere. Physical Review D 50: 6547-6555 (1994).

V.E. Tarasov. Fractional generalization of Liouville equations. Chaos 14: 123-127 (2004).

V.E. Tarasov. Electromagnetic fields on fractals. Modern Physics Letters A 21: 1587-1600 (2006).

A. Zeilinger, and K. Svozil. Measuring the dimension of space-time. Physical Review Letters 54: 2553-2555 (1985).

S. Muslih, and D. Baleanu. Fractional multipoles in fractional space. Nonlinear Analysis: Real World Applications 8: 198-203 (2007).

C. Palmer, and P.N. Stavrinou. Equations of motion in a non integer-dimension space. Journal of Physics A 37: 6987-7003 (2004).

J.D. Jackson. Classical Electrodynamics, 3rd edition. Wiley, New York (1999).

D.J. Griffiths. Introduction to Electrodynamics, 4th Edition. Pearson Education (2012).

T. Myint-U, and L. Debnath. Linear Partial Differential Equations for Scientists and Engineers, 4th Edition. Birkhauser (2007).

V.E. Tarasov. Gravitational field of fractals distribution of particles. Celestial Mechanics and Dynamical Astronomy 94: 1-15 (2006).

J.A. Stratton. Electromagnetic Theory. WileyIEEE Press 640 (1941).

D. Baleanu, A.K. Golmankhaneh, and A.K. Golmankhaneh. On electromagnetic field in fractional space. Nonlinear Analysis: Real World Applications 11: 288-292 (2010).

P.M. Morse, and H. Feshbach. Methods of Theoretical Physics. McGraw-Hill Book Comp., Inc., New York, Toronto, London, Part II 260: 208-209 (1953).




How to Cite

Muhammad Imran Shahzad, Muhammad Akbar, Saeed Ahmed, Sania Shaheen, & Muhammad Ahmad Raza. (2022). Mathematical Analysis of Conducting and Dielectric Sphere in Fractional Space: Mathematical Analysis of Conducting and Dielectric Sphere in Fractional Space. Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 59(4), 61–65.



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