Homework 12:Electric Potential and Field

Abstract

Electrostatic potential problem is an important problem in classical electrodynamics. In the passive region, electrostatic potential satisfies Laplace equation, so long as under certain boundary conditions in solving Laplace equation can obtain spatial distribution of electrostatic potential. Solving partial differential equation has no general method, but such as Poisson equation, a large class of equations can use so-called relaxation algorithm (relaxation method to solve. The homework after class exercise 5.7, discusses the relaxation algorithm of three specific circumstances (Jacobi, Gauss Seidel, simultaneous over-relaxation (SOR) iterative convergence speed problem.

Background

The solution of the electrostatic potential problem in the passive region can be given by the Maxwell equation, and the electrostatic potential distribution satisfies the Laplace equation：

$$\frac{\partial ^2V}{\partial x^2}+\frac{\partial ^2V}{\partial y^2}+\frac{\partial ^2V}{\partial z^2}=0$$

Main body

Parallel plate problem is a very classical problem, the were numerical solution can verify the stability problem. For simplicity only discuss two-dimensional problems, or said that the parallel plate system along a direction is symmetric. Take the potential of two parallel plates respectively, or other values, take the square boundary of zero potential and calculation system of potential distribution as shown bellow.
Gauss-Seidel Method:

The pictures above show parallel plate capacitor and the outer domain of potential surface (line) and the electric field distribution, for simplicity here that electric field along the direction unchanged. Can be found in the parallel plate electric field has a certain symmetry.

Gauss-Seidel Method

Reference
Nicholas J.Giordano, Hisao Nakanishi.Computational physics.清华大学出版社