Mathematical Analysis on Spherical Shell of Permeable Material in NID Space

: In this paper, we have studied the magnetic shielding effect of a spherical shell analytically in fractional dimensional space (FDS). The Laplacian equation in fractional space predicts the complex phenomena of physics. This is a boundary value problem that has been solved by the separation variable method mathematically by taking low frequency ω = 0. Electric potential is obtained in fractional dimensional space for the three regions, namely outside the spherical shell, between the shell and hollow sphere and inside the sphere. Also, the induced dipole moment has been derived. We obtain a general solution that reduces to the classical results by setting fractional parameter α = 3 which takes its value (2 < α ≤ 3).


INTRODUCTION
The novel idea of fractional-dimensional space (FDS) is essential in different disciplines of physics worked by numerous researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].Like the researcher, Wilson [3] has investigated quantum field theory (QFT) in FDS.Furthermore, the FDS can be employed as an indicator in the Ising limit of the QFT [6].Stillinger [4] has defined an axiomatic basis for this idea for the development of Schrödinger wave mechanics and Gibbsian statistical mechanics in the α-dimensional space.The runtime operational category of space-time dimension shown by Zeilinger and Svozil [10] provides a likelihood of determination of spacetime dimension empirically.It is also acknowledged that the fractional dimension of space-time should be less than 4. The α-dimensional fractional space has also been modelled in the last few decades [11].
We have extended the problem of a spherical shell of highly permeable material which is derived by Baleanu et al. [17].We have solved it in fractional dimensional space analytically.The primary aim is to use the Laplacian equation to find electric potential and induced dipole moment in FDS.For the integer order α = 3, the original solution is reproduced.

MATERIALS AND METHODS
We consider here a spherical shell of permeable material which is placed in fractional space shown in Figure 1.We have studied the spherical shell of the inner radius 'a' and the outer radius 'b' for the phenomenon of magnetic shielding.This problem has been extended from Jackson [13].The core is made of material of permeability, µ, and placed in a fractional space.B 0 is the uniform magnetic field applied on the surface.We need to discover the fields В and H everywhere in space, but most specifically in the cavity (r < a) as a function of µ.The magnetic field H is determined from a scalar potential H = -∇Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional α-dimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: (1) where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle φ can be expressed as: (2) Eq (3) is separable and suppose. (3) The differential equation (3) followed by the published article [17], can be decoupled into two different parts namely angular and radial which are written as: (5) Therefore, the combined solutions of Ψ (r, θ) in α-dimensional fractional space, can be expressed as (6) Here, the unknown constants a l and b l can be determined by using the boundary conditions (B.Cs.) on Ψ (r, θ).We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.For the outer region r > b, the potential must be of the form, (7) where H = H 0 is the uniform field, at large distance.For the inner regions, a < r < b the potential can be written as: (8) For r<a (9) All coefficients for l ≠ 1 vanish.Then we can construct the solutions for different regions given below: (10) (12) The boundary conditions, at r = a and r = b, are that H θ and B r be continuous for l = 1, the coefficients satisfy the four simultaneous equations.From the above four boundary conditions, we find four simplified equations: Where α 1 =α-1 and κ=µ/µ 0 .
[13].The core is made of material of permeability, µ, and placed in a fractional space.B0 is the uniform magnetic field applied on the surface.We need to discover the fields В and H everywhere in space, but most specifically in the cavity (r < a) as a function of µ.The magnetic field H is determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: ( where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as:
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: (, ) = ( − 2)(),  < .
[13].The core is made of material of permeability, µ, and placed in a fractional space.B0 is the uniform magnetic field applied on the surface.We need to discover the fields В and H everywhere in space, but most specifically in the cavity (r < a) as a function of µ.The magnetic field H is determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as: The differential equation (3) followed by the published article [17], can be decoupled into two different parts namely angular and radial which are written as: 2 + ( − 2)   + ( +  − 2)] () = 0 (4) ] () = 0 (5) Therefore, the combined solutions of Ψ (r, θ) in αdimensional fractional space, can be expressed as Here, the unknown constants al and bl can be determined by using the boundary conditions (B.Cs.) on Ψ (r, θ).
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: [13].The core is made of material of permeability, µ, and placed in a fractional space.B0 is the uniform magnetic field applied on the surface.We need to discover the fields В and H everywhere in space, but most specifically in the cavity (r < a) as a function of µ.The magnetic field H is determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: ( where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as:

𝛹𝛹(𝑟𝑟, 𝑐𝑐) = 𝑅𝑅(𝑟𝑟)𝛩𝛩(𝑐𝑐)
(3) The differential equation (3) followed by the published article [17], can be decoupled into two different parts namely angular and radial which are written as: Therefore, the combined solutions of Ψ (r, θ) in αdimensional fractional space, can be expressed as Here, the unknown constants al and bl can be determined by using the boundary conditions (B.Cs.) on Ψ (r, θ).
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: [13].The core is made of material of permeability, µ, and placed in a fractional space.B0 is the uniform magnetic field applied on the surface.We need to discover the fields В and H everywhere in space, but most specifically in the cavity (r < a) as a function of µ.The magnetic field H is determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as:
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: The differential equation (3) followed by the published article [17], can be decoupled into two different parts namely angular and radial which are written as: Therefore, the combined solutions of Ψ (r, θ) in αdimensional fractional space, can be expressed as Here, the unknown constants al and bl can be determined by using the boundary conditions (B.Cs.) on Ψ (r, θ).
We construct here the solution for three different regions permeability, µ, and e uniform magnetic o discover the fields st specifically in the magnetic field H is = -Ψ, as there are tial Ψ satisfies the ing fractional αate systems" which (1) The differential equation (3) followed by the published article [17], can be decoupled into two different parts namely angular and radial which are written as: Therefore, the combined solutions of Ψ (r, θ) in αdimensional fractional space, can be expressed as Here, the unknown constants al and bl can be determined by using the boundary conditions (B.Cs.) on Ψ (r, θ).
We construct here the solution for three different regions determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: (1) where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as:
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as: (2)

Fig. 1. Spherical Shell of Highly Permeable Material
Placed in FDS Eq (3) is separable and suppose.
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as: (2)

Fig. 1. Spherical Shell of Highly Permeable Material
Placed in FDS Eq (3) is separable and suppose.
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: determined from a scalar potential H = -Ψ, as there are no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: (1) where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as:
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: (, ) = ( − 2)(),  < .
no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: (1) where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as:
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: (1) where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as: no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as: (3) ] () = 0 Therefore, the combined solutions of Ψ (r, θ) in αdimensional fractional space, can be expressed as Here, the unknown constants al and bl can be determined by using the boundary conditions (B.Cs.) on Ψ (r, θ).
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below: no currents present.Thus, the potential Ψ satisfies the Laplacian in fractional space having fractional αdimension in "spherical polar coordinate systems" which is described by Baleanu et al. [17]: where the fractional parameter α lies in the range (2 < α ≤ 3) In this case, the Laplace equation for the potential independent of angle  can be expressed as:
We construct here the solution for three different regions by satisfying the B.Cs., at r = a and r = b.
For the outer region r > b, the potential must be of the form, where H = H0 is the uniform field, at large distance.
For the inner regions, a  r  b the potential can be written as: For ra All coefficients for l  1 vanish.Then we can construct the solutions for different regions given below:

Specia
The po dipole of high field pa the dip