scholarly journals New relationships between spherical and spheroidal harmonics and applications

2021 ◽  
Author(s):  
◽  
Matt Majic
Keyword(s):  
Spherical Harmonics ◽  
Point Charge ◽  
Coordinate Systems ◽  
Prolate Spheroidal ◽  
Dielectric Sphere ◽  
New Class ◽  
Spheroidal Harmonics ◽  
Physical Problems ◽  
Finite Line ◽  
Spheroidal Geometry

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

scholarly journals New relationships between spherical and spheroidal harmonics and applications

2021 ◽  
Author(s):  
◽  
Matt Majic
Keyword(s):  
Spherical Harmonics ◽  
Point Charge ◽  
Coordinate Systems ◽  
Prolate Spheroidal ◽  
Dielectric Sphere ◽  
New Class ◽  
Spheroidal Harmonics ◽  
Physical Problems ◽  
Finite Line ◽  
Spheroidal Geometry

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

Comment on “Surface-charge distribution on a dielectric sphere due to an external point charge: examples of C60 and C240 fullerenes, Phys. Chem. Chem. Phys., 2013, 15, 20115”

2014 ◽  
Vol 16 (28) ◽  
pp. 14969-14970
Author(s):  
Henning Zettergren ◽  
Fredrik Lindén ◽  
Henrik Cederquist
Keyword(s):  
Dielectric Properties ◽  
Surface Charge ◽  
Charge Distribution ◽  
Point Charge ◽  
Chem Phys ◽  
Dielectric Sphere ◽  
External Point ◽  
Surface Charge Distribution ◽  
Phys Chem Chem Phys

We show that the relative surface charge distribution from classical electrostatics cannot be used to discriminate between different assumptions about the dielectric properties of fullerenes interacting with external charges.

Image theory for a point charge inside a layered dielectric sphere

1994 ◽  
Vol 77 (5) ◽  
pp. 327-335 ◽  
Author(s):  
J. C. -E. Sten ◽  
R. Ilmoniemi
Keyword(s):  
Point Charge ◽  
Image Theory ◽  
Dielectric Sphere ◽  
Layered Dielectric

scholarly journals Using the properties of vehicles in the conceptof the multi-environment multi-modal quantomobile

2020 ◽  
Vol 17 (6) ◽  
pp. 195-205
Author(s):  
Ju. G. Kotikov ◽  
Keyword(s):  
Numerical Modeling ◽  
Research And Development ◽  
Physical Vacuum ◽  
Motion Model ◽  
Single Family ◽  
Coordinate Systems ◽  
Hypothetical Model ◽  
Ground Vehicle ◽  
New Class ◽  
The One

The development of the concept of the quantum engine, that uses the energy of physical vacuum, makes it possible to create a new class of vehicles, namely, the quantomobile, designed as a quantum - powered vehicle. The type of quantum vehicles can be versatile, starting from the simplest version (with the ground vehicle driving modes) to the multi-environment multi-modal quantomobile that can function on land, in the air and in water. To work out a hypothetical model of the multi-environment multi-modal quantomobile, it is necessary to use all the heritage of research and development in the sphere of transport engineering. For 10 variants of the multi-environment multi-modal quantomobile movement - from the air quantum helicopter (quantocraft) to a quantum submarine (quantomarine) - there has been made an analysis of the numerical modeling specifics, the use of coordinate systems, the implementation of the traffic of existing transport vehicles that can be reflected in the concept of multi-environment quantomobile. Two extreme methods of modeling are distinguished: 1) the one based on a single family of coordinate systems and a common (end-to-end for all types of environment) motion model; 2) the one based on models by type of motion with possible switching of coordinate systems.

scholarly journals Polarized radiative transfer equation in some nontrivial coordinate systems

2011 ◽  
Vol 7 (S283) ◽  
pp. 360-361
Author(s):  
Juris Freimanis
Keyword(s):  
Refractive Index ◽  
Differential Operator ◽  
Radiative Transfer ◽  
Euclidean Space ◽  
Coordinate System ◽  
Transfer Equation ◽  
Effective Refractive Index ◽  
Radiative Transfer Equation ◽  
Coordinate Systems ◽  
Prolate Spheroidal

AbstractExplicit expressions for the differential operator of stationary quasi-monochromatic polarized radiative transfer equation in Euclidean space with piecewise homogeneous real part of the effective refractive index are obtained in circular cylindrical, prolate spheroidal, elliptic conical, classic toroidal and simple toroidal coordinate system.

scholarly journals An Analytical Study of the Diffraction of Light by a Circular Aperture Using Spherical Harmonics for n≤1

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Mahmoud Ahmad ◽  
Najah Kabalan ◽  
Samar Omran
Keyword(s):  
Spherical Harmonics ◽  
Analytical Study ◽  
Optical Axis ◽  
Spherical Harmonic Analysis ◽  
Circular Aperture ◽  
Optical Intensity ◽  
Naked Eye ◽  
Physical Problems ◽  
Special Cases ◽  
Polar Radius

Based on the importance of spherical harmonics and their applicability in many physical problems, this research aimed to study the diffraction pattern of light by a circular aperture starting from the first Rayleigh–Sommerfeld diffraction equation and to expand the polar radius of a point on the surface of the circular aperture based on spherical harmonics. We depended on this theoretical framework in our paper. We calculated the optical intensity compounds C00,C10,C11,C1−1 for n=0,1,m=−1,0,1. We studied the intensity distributions in three special cases (along the optical axis, at the geometrical focal plane, and along the boundary of the geometrical shadow). We presented numerical comparative examples to illustrate the variation of the intensity versus a ratio (Z/A is the ratio of the distance between the circular aperture and the observation plane to a radius of the circular aperture), and we used Maple program to represent these results. We noticed that the expansion we made using spherical harmonic analysis led to an increase in the number of fringes bright enough to be visible to the naked eye. We then concluded with a brief discussion of the results.

scholarly journals Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Maria Hadjinicolaou ◽  
Eleftherios Protopapas
Keyword(s):  
Coordinate System ◽  
Fluid Flows ◽  
Scalar Function ◽  
Order Partial Differential Equation ◽  
Coordinate Systems ◽  
Prolate Spheroidal ◽  
Spherical System ◽  
Oblate Spheroidal ◽  
Generalized Eigenfunctions ◽  
Spheroidal Coordinates

When studying axisymmetric particle fluid flows, a scalar function,ψ, is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely,E4ψ=0, whereE2is the Stokes irrotational operator andE4=E2∘E2is the Stokes bistream operator. As it is already known,E2ψ=0in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation acceptsR-separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operatorE4does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, whileE4ψ=0semiseparates variables in the spheroidal coordinate systems and itR-semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equationE′2ψ=0alsoR-separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equationE′4ψ=0  R-semiseparates variables. Since the generalized eigenfunctions ofE′2cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions ofE′2in the modified inverted oblate spheroidal coordinate system.

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