电磁场与电磁波第三章.ppt

Company Logo @ 第三章 静电场的边值问题 主 要 内 容 电位微分方程、镜像法、分离变量法。 1. 电位微分方程 2. 镜像法 3. 直角坐标系中的分离变量法 4. 圆柱坐标系中的分离变量法 5. 球坐标系中的分离变量法 1. Differential Equations for Electric Potential The relationship between the electric potential ? and the electric field intensity E is Taking the divergence operation for both sides of the above equation gives In a linear, homogeneous, and isotropic medium, the divergence of the electric field intensity E is The differential equation for the electric potential is which is called Poisson’s equation. In a source-free region, and the above equation becomes which is called Laplace’s equation. 1. Differential Equations for Electric Potential 1. 电位微分方程 已知电位 ? 与电场强度 E 的关系为 对上式两边取散度，得 对于线性各向同性的均匀介质，电场强度E 的散度为 那么，电位满足的微分方程式为 泊松方程 拉普拉斯方程 对于无源区， ，上式变为 1. 电位微分方程 因此，对于导体边界，当边界上的电位，或电位的法向导数给定时，或导体表面电荷给定时，空间的静电场即被惟一地确定。这个结论称为静电场惟一性定理。 For electrostatic fields with conductors as boundaries, the field may be given uniquely when the electric potential , its normal derivative, or the charges is given on the conducting boundaries. That is the uniqueness theorem for solutions to problems on electrostatic fields. Uniqueness of solution of differential equations for electric potential (静电场唯一性定理) 静电场的边值问题 —— 根据给定的边界条件求解静电场的电位分布。 对于线性各向同性的均匀介质，有源区中的电位满足泊松方程方程 在无源区，电位满足拉普拉斯方程 利用格林函数，可以求解泊松方程（了解）。 利用分离变量法可以求解拉普拉斯方程。 （了解） 求解静电场边值问题的另一种简单方法是镜像法。 小结 3. Method of Image Essence: The effect of the boundary is replaced by one or several equivalent charges, and the original inhomogeneous region with a boundary becomes an infinite homogeneous space. Basis：The principle of uniqueness. Therefore, these charges should not change the original boundary conditions. These equivalent charges are at the image positions of the original charges, and are called image charges, and this method is called the method of images. Key：To determine the values and the positi