Poisson’s and Laplace’s Equations

Abstract

In the previous few chapters, the electric field has been determined using either Gauss’ law or Coulomb’s law. In initial condition, charge distribution or electrostatic potential should be known to apply those laws. There are many practical problems where the charge distribution is not known for every place. There is some complex geometry in high voltage engineering equipment, namely insulators, bushing, surge arrestors, etc. In that case, it is difficult to use Gauss’ law to find their electrostatic potential and electric field intensity distributions. The method of images can be used if the conducting bodies have a boundary with simple geometry. Therefore, some differential equations need to be solved to find the voltage and field distribution around the conductor and air interface of the simple and complex geometry of the electrical engineering equipment. In this chapter, Poisson’s equation, Laplace’s equation, uniqueness theorem, and the solution of Laplace’s equation will be discussed.