Abstract

This exercise is about electric potential and fields.Compared with the Eular-Cromer method applied in former exercises, this time we use relaxation method to solve problems linked to Laplace's equation and its generalization.

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Background

Laplace's Equation

In order to find the distribution of the electric field of the capacitor, we need to solve for the Laplace's equation. In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:

or:

Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.

Useful methods

The method that we use to find the field is relaxiation method. In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems.

Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences.

We have the following equations:

specially, for two dimensional problem

Jacobi method:

Gauss-Seidel method:

Simultaneous over-relaxation method (SOR method):

The best choice for alpha is:

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The Main Body

Problem 5.4

First of all, we investigate that the plate separation as 0.8(m), the equipotential contours、perspective plot of the potential and electric field are shown below:

Then we change the plate separation to 1(m)

Last we change the plate separation to 1.2(m)

click here to see the code 

Problem 5.7

I have show the equation for Jacobi method、Gauss-Seidel method and Simultaneous over-relaxation method (SOR method) in the background part.

Comparing 3 different methods, the figure shows that convergent speeds SOR method > GS method > Jacobi method.

If we match the value with equations, we will find

At the same accuracy, using Jacobi method, the number of iterations N is proportional to L^2;while using SOR method, the number of iterations N is proportional to L.

click here to see the code 

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Reference

I have copied the codes from Xiao Guo