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Learn how to solve electrostatic problems. Overview of solution methods. Simple 1-D problems. Reduce Poisson’s equation to Laplace’s equation. Capacitance. The method of images. Overview. Illustrated below is a fairly general problem in electrostatics. Many practical problems are special cases of this general problem. Where to start?
Examples of Electrostatic Problems with Dielectrics Problem: Find (electric flux density), (electric field intensity), and (polarization) for a metallic sphere (radius a, charge Q), coated by a dielec-tric (radius b), and the charge densities at the interfaces. Solution: Use Gauss’ Law In region 0, In region 1, a < r < b:
Electrostatic Problems with Solutions and Explanations. Electrostatic charges and forces problems are presented along with detailed solutions. Problems. Problem 1: What is the net force and its direction that the charges at the vertices A and C of the right triangle ABC exert on the charge in vertex B?
So for V, we have only one 2 nd order DE to solve, but if we approach the problem using electric field q, we end up with two equations: · q L Ù, H q L Ù In general, there are two major ways to solve the potential problems in electrostatic: (a) Solve as a source problem using integration, (b) Solve as a boundary value problem, using boundary
1.1. Vector calculus. In this problem we recall a number of standard identities of vector calculus which we will frequently use in electrodynamics. De nitions/conventions: We commonly write the well-known vectorial di erentiation op-erators grad, div, rot using the vector ~r of partial derivatives ri := @=@xi as.
Electrostatics can be formulated in an integral form. This is elegant but usually not very useful except in special geometries with simple boundary conditions. Electrostatics can be formulated in differential form. This is much more useful for actually solving problems.
In this chapter we shall solve a variety of boundary value problems using techniques which can be described as commonplace. 1 Method of Images This method is useful given su–ciently simple geometries. It is closely related to the Green’s function method and can be used to flnd Green’s functions for these same simple geometries.
