Laplace's Equation

程熹 2013301020038 物基一班
Homework

Abstract

This article investigate the problem that schematic cross-section of a hollow metallic prism with a solid,metallic inner conductor and electric potetial and field inside the prism by using the Laplace's equation at different

Key Word

Laplace's equation,boundary conditon,electric field

Introduction

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.
The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain D such that φ on the boundary of D is equal to some given function. Since the Laplace operator appears in the heat equation, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain doesn't change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.
The Neumann boundary conditions for Laplace's equation specify not the function φ itself on the boundary of D, but its normal derivative. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of D alone.
Solutions of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition, is very useful, e.g., solutions to complex problems can be constructed by summing simple solutions.

Main Features

In the space with no electric, the distribution of potential satisfies the well-known Laplace equation, that is:

$$\frac{{\partial}^2 V}{{\partial}x^2}+\frac{{\partial}^2 V}{{\partial}y^2}+\frac{{\partial}^2 V}{{\partial}z^2}=0$$
with approximation $$\begin{eqnarray}{} \frac{{\partial}^2 V}{{\partial}x^2}&=&\frac{1}{{\Delta}x}[\frac{V(i+1,j,k)-V(i,j,k)}{{\Delta}x}-\frac{V(i,j,k)-V(i-1,j,k)}{{\Delta}x}]\notag\\ &=&\frac{V(i+1,j,k)+V(i-1,j,k)-2V(i,j,k)}{{\Delta}x^2}\notag\\ \end{eqnarray}$$
we get $$V(i,jk)=\frac{1}{6}[V(i+1,j,k)+V(i-1,j,k)+V(i,j+1,k)+V(i,j-1,k)+V(i,j,k+1)+V(i,j,k-1)]$$
Then we use the jocpbi method to stimulate the equation above in numerical way

Result

1.The potential disstribution of the point electric

2.The potential disstribution of the parallel-plate capacitor

Conclusion

1. These diagrams match the image we have in our mind due to our eletromagnetism knowledge
2. Using SOR method is much more efficient than the Gauss-Seidel method.

Acknowledgement

I have used the origin code of 陈天懿，and run his program to get some of the upper result. So I acknowledge him here.

Reference

Computational Physics(second edition) Nicholas J.Giordano and Hisao Nakanishi
Wikipedia Laplace's equation
https://en.wikipedia.org/wiki/Laplace%27s_equation