Toroidal Helical Fields

1987 ◽  
Vol 42 (10) ◽  
pp. 1124-1132 ◽  
Author(s):  
M. Y. Kucinski ◽  
I. L. Caldas
Keyword(s):  
Coordinate System ◽  
Infinite Series ◽  
Plasma Column ◽  
Scalar Potential ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Small Curvature ◽  
Partial Sum ◽  
Magnetic Surfaces ◽  
Series Of Functions

Using the conventional toroidal coordinate system Laplace’s equation for the magnetic scalar potential due to toroidal helical currents is solved. The potential is written as a sum of an infinite series of functions. Each partial sum represents the potential within some accuracy. The effect of the winding law is analysed in the case of small curvature. Approximate magnetic surfaces formed by toroidal helical currents flowing around a standard tokamak chamber are determined. Stability of the plasma column in this system against displacements is discussed.

A note on the magnetic scalar potential of an electric current-ring

1982 ◽  
Vol 92 (1) ◽  
pp. 183-191 ◽  
Author(s):  
A. Hulme
Keyword(s):  
Electric Current ◽  
Steady Flow ◽  
Infinite Series ◽  
Scalar Potential ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Toroidal Coordinates ◽  
Current Ring

AbstractThis paper considers the problem of expressing the magnetic scalar potential associated with the steady flow of an electric current around a thin circular wire, in terms of ‘local’ toroidal coordinates. This potential is known to be a multi-valued function of position and so cannot be expressed directly in terms of fundamental (i.e. single-valued) solutions of Laplace's equation. It is shown that the potential can however be expressed quite simply, as an infinite series of multi-valued toroidal harmonics and that this series is rapidly convergent in the neighbourhood of the current-ring.

Methods for the Summation of Infinite Series in Closed Form

1976 ◽  
Vol 13 (3) ◽  
pp. 257-270
Author(s):  
Mohammed Al-Hakkak
Keyword(s):  
Fourier Analysis ◽  
Closed Form ◽  
Infinite Series ◽  
Finite Differences ◽  
Analytical Methods ◽  
Electrical Engineering ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Direct Relevance

The paper discusses four analytical methods for the summation of infinite series in closed form. Many illustrative examples are given including some with direct relevance to electrical engineering subjects like Fourier analysis of circuits, potential solutions of Laplace's equation, and finite differences.

scholarly journals Unfolding the hippocampus: an intrinsic coordinate system for subfield segmentations and quantitative mapping

2017 ◽  
Author(s):  
Jordan DeKraker ◽  
Kayla M. Ferko ◽  
Jonathan C. Lau ◽  
Stefan Köhler ◽  
Ali R. Khan
Keyword(s):  
Coordinate System ◽  
Grey Matter ◽  
Computational Method ◽  
Three Dimensions ◽  
Stratum Radiatum ◽  
Coordinate Space ◽  
Surgical Patient ◽  
List Type ◽  
Laplace’S Equation ◽  
Laplace's Equation

AbstractThe hippocampus, like the neocortex, has a morphological structure that is complex and variable in its folding pattern, especially in the hippocampal head. The current study presents a computational method to unfold hippocampal grey matter, with a particular focus on the hippocampal head where complexity is highest due to medial curving of the structure and the variable presence of digitations. This unfolding was performed on segmentations from high-resolution, T2-weighted 7T MRI data from 12 healthy participants and one surgical patient with epilepsy whose resected hippocampal tissue was used for histological validation. We traced a critical hippocampal component, the hippocampal sulcus and stratum radiatum, lacunosum moleculaire, (SRLM) in these images, then employed user-guided semi-automated techniques to detect and subsequently unfold the surrounding hippocampal grey matter. This unfolding was performed by solving Laplace’s equation in three dimensions of interest (long-axis, proximal-distal, and laminar). The resulting ‘unfolded coordinate space’ provides an intuitive way of mapping the hippocampal subfields in 2D space (long-axis and proximal-distal), such that similar borders can be applied in the head, body, and tail of the hippocampus independently of variability in folding. This unfolded coordinate space was employed to map intracortical myelin and thickness in relation to subfield borders, which revealed intracortical myelin differences that closely follow the subfield borders used here. Examination of a histological sample from a patient with epilepsy reveals that our unfolded coordinate system shows biological validity, and that subfield segmentations applied in this space are able to capture features not seen in manual tracing protocols.Research highlightsSRLM in hippocampal head consistently detected with 7T, T2 isotropic MRIHippocampal grey matter unfolded using Laplace’s equation in 3DIntracortical myelin and thickness mapped in unfolded coordinate spaceUnfolded subfields capture critical structural regularities and agree with histology

Theoretical investigations on rotationally symmetric guns with field-emission source

1978 ◽  
Vol 36 (1) ◽  
pp. 54-55
Author(s):  
D. Kern
Keyword(s):  
Magnetic Field ◽  
Numerical Solution ◽  
Field Emission ◽  
Scalar Potential ◽  
Analytic Solutions ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Theoretical Investigations ◽  
Rotationally Symmetric ◽  
Orthogonal Coordinates

The principal procedure in computing the properties of an electron-optical system consists of two steps, first the calculation of the electric or magnetic field, and then the calculation of trajectories, from which the properties may be obtained.In regions free from current and charge, the electric or magnetic field follows as the gradient of a solution of Laplace's equation for a scalar potential. Analytic solutions are known only for a few specific cases. They are essentially based upon the separation of Laplace's equation in curvilinear orthogonal coordinates; the difficulties arise from boundaries which cannot be represented as coordinate surfaces.With a numerical solution of this problem, using either the finite-element or the finite-difference-method, difficulties arise when dividing the region considered into triangles or meshes, since the dimensions of the electrodes and of the tip in a field emission gun differ by several orders of magnitude. Combining an analytical with a numerical solution, these difficulties can be avoided.

The irrotational flow of a perfect fluid past two spheres

1949 ◽  
Vol 45 (1) ◽  
pp. 81-87 ◽  
Author(s):  
I. N. Sneddon ◽  
J. Fulton
Keyword(s):  
Boundary Value Problem ◽  
Perfect Fluid ◽  
Infinite Series ◽  
Boundary Value ◽  
Laplace’S Equation ◽  
Algebraic Equations ◽  
Electrostatic Problem ◽  
Laplace's Equation ◽  
Linear Algebraic Equations ◽  
Infinite Set

1. The boundary value problem of Laplace's equation for two spheres is a classical one, and has been the subject of discussion by many mathematicians (1). The earliest attempt to solve a boundary value problem of this type is due to Poisson (2), but his analysis is applicable only to the electrostatic problem. The first of the methods which can be successfully applied to both electrostatic arid hydrodynamical problems was developed later by Lord Kelvin (3); this procedure, which is known as the ‘method of images’, was first applied to the problem of the motion of two spheres in a perfect fluid by Hicks (4). Another method of great generality, that of transforming Laplace's equation to bipolar coordinates and studying the solutions in these coordinates, was developed about the same time by Neumann (5) and much later by Jeffery (6). More recently a new method has been developed by Mitra (7) for the solution of the problem of two spheres in a potential field. It makes use of two sets of spherical polar coordinate systems; the solution is expressed in terms of infinite series whose coefficients satisfy an infinite set of linear algebraic equations. The chief interest of Mitra's method lies in the fact that he has found it possible to derive exact solutions of this infinite set of equations. All of these methods suffer from the disadvantage that the potential function is obtained in the form of an infinite series so that any numerical calculations are rendered cumbersome.

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