Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries

Abstract

We find solutions of Laplace's equation with specific boundary conditions (in which such solutions take either the value zero or unity in each surface) using a generic curvilinear system of coordinates. Such purely geometrical solutions (that we shall call Basic Harmonic Functions BHF's) are utilized to obtain a more general class of solutions for Laplace's equation, in which the functions take arbitrary constant values on the boundaries. On the other hand, the BHF's are also used to obtain the capacitance of many electrostatic configurations of conductors. This method of finding solutions of Laplace's equation and capacitances with multiple symmetries is particularly simple, owing to the fact that the method of separation of variables becomes much simpler under the boundary conditions that lead to the BHF's. Examples of application in complex symmetries are given. Then, configurations of succesive embedding of conductors are also examined. In addition, expressions for electric fields between two conductors and charge densities on their surfaces are obtained in terms of generalized curvilinear coordinates. It worths remarking that it is plausible to extrapolate the present method to other linear homogeneous differential equations.

Random walk on the Poincaré disk induced by a group of Möbius transformations

“…The bipolar coordinates are commonly used to separate variables in equations of mathematical physics when the Laplace operator is the dominated term (the idea goes back to [5], see, for instance, [8,10,16,24]). Lemma 3.1 and (20) together imply the following:…”

A high accurate method for solving an inverse problem of the Laplace equation in detection of a Robin coefficient

“…The Laplace equation, usually named harmonic function, can be considered as a mathematical description for the study of several models describing various phenomena in the applied sciences, such as: temperature distributions, potentials of electrostatic, magneto-static fields, velocity potentials of incompressible irrotational fluid flows, the electrostatic problems, and in-compressible fluid [1,4,6,18,19,24]. Because of the powerful development of computers, considerable efforts have been made to solve Laplace's equation in different shapes and boundary conditions [23].…”

Quantitative Investigation of Surface Charge Distribution and Point Probing Characteristics of Spherical Scattering Electrical Field Probe for Precision Measurement of Miniature Internal Structures with High Aspect Ratios

“…To analyze the point probing characteristics of the SSEP, a quantitative investigation of the spherical scattering electrical field needs to be conducted to illustrate the law of surface charge distribution. The difficulties are: (1) Conventional theoretical analysis methods [21,22] cannot complete the modeling task with complicated boundary conditions, such as the possibility that the surface being probed could be a plane, spherical, cylindrical, or free geometrical shape; (2) On the other hand, the spherical scattering electrical field is also difficult to model with conventional numerical methods due to the problem of balancing modeling accuracy and computational load. This is because the diameter of the probing ball is on a hundreds of microns to millimeter level, while the probing gap is on a micro and sub-micrometer level, and this creates a multi-scale problem, posing a challenge to gridding and computational accuracy; (3) Experimental methods [23][24][25][26][27] are not applicable here due to the lack of micro-/nano probes with the ability to detect the spherical scattering electrical field in the micro probing gap without introducing violent disturbance into the field.…”