The electrostatics of conductors

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Harmonic Functions ◽  
Conformal Transformation ◽  
Early Stage ◽  
Fourier Method ◽  
Relaxation Method ◽  
Numerical Calculations ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Electrostatic Equilibrium

This chapter begins by proving some important properties of (i) conductors in electrostatic equilibrium, and (ii) harmonic functions. These results underpin most of the remaining questions of Chapter 3. The coefficients of capacitance for an arbitrary arrangement of conductors are introduced at an early stage, and numerical calculations then follow in a number of subsequent questions. Some important techniques (both analytical and numerical) for finding solutions to Laplace’s equation are considered. These include: the Fourier method, the relaxation method, themethod of images and the method of conformal transformation. All of these are discussed in some detail, and with appropriate examples.

V.—On Whittaker's Solution of Laplace's Equation

1944 ◽  
Vol 62 (1) ◽  
pp. 31-36
Author(s):  
E. T. Copson
Keyword(s):  
Harmonic Function ◽  
General Solution ◽  
Arbitrary Function ◽  
Harmonic Functions ◽  
Singular Points ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Image Position

In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the formwhere f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.

Spreadsheet Solutions To Laplace's Equation: Seepage And Flow Net

1996 ◽  
pp. 53-67 ◽  
Author(s):  
Mahadzer Mahmud
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Relaxation Method ◽  
Laplace’S Equation ◽  
User Friendliness ◽  
Spreadsheet Program ◽  
Laplace's Equation ◽  
Complex Engineering ◽  
Spreadsheet Modelling

There has been an increasing interest in applying the electronic spreadsheets, traditionally used for the financial and accounting purposes, to solve complex engineering problems. This paper aims to describe the application of spreadsheets to solve the finite difference to Laplace's equation. The simple form of finite difference method also known as the relaxation method will be used for seepage analysis.The spreadsheet program used for this study was chosen based on the power, flexibility and the versatility of this package in various computing environments. Nevertheless any other spreadsheet programs can also perform the same task with varying levels of user friendliness. Keywords: Spreadsheet modelling, flow net, seepage, finite difference method

Reply by author to discussion by Amalendu Roy

Geophysics ◽  
1970 ◽  
Vol 35 (1) ◽  
pp. 159-160
Author(s):  
C. N. G. Dampney
Keyword(s):  
Harmonic Functions ◽  
Potential Functions ◽  
Potential Fields ◽  
Laplace’S Equation ◽  
Potential Equation ◽  
Laplace's Equation ◽  
Physical Functions ◽  
Number Of Authors

Fundamentally, in reply to point (3) of Roy’s discussion, the term “potential” has been used for physical functions which obey Laplace’s equation or the potential equation (Courant and Hilbert, 1962, p. 240) [Formula: see text] where [Formula: see text] (1) for [Formula: see text] and [Formula: see text] (with [Formula: see text]). Courant and Hilbert call the solution of equation (1) potential functions or harmonic functions. A large number of authors including Jeffreys (1956), Kellogg (1953), and Grant and West (1965), have considered the properties of “potential fields” for [Formula: see text]. This is the context in which I have used the term “potential.”

scholarly journals Note on the electric capacity coefficients of spheres

1912 ◽  
Vol 87 (598) ◽  
pp. 485-487
Keyword(s):  
General Solution ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Electric Capacity ◽  
Physical Problems

In the 'Proceedings' of the Society, vol. 87, p. 109, Mr. Jeffery obtains a general solution of Laplace’s equation in a form suitable for physical problems in connection with two spheres. As an illustration he applies his solution to the problem of finding the capacity coefficients of two equal spheres, obtaining a result which he shows to be equivalent to one of Maxwell’s series formulæ. He then computes a table of the numerical values of these coefficients.

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