Poisson Equation

3 Key excerpts on "Poisson Equation"

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  eBook - ePub

  Models and Modeling

  An Introduction for Earth and Environmental Scientists

  • Jerry P. Fairley(Author)
  • 2017(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  CHAPTER 5 The Poisson Equation

  Chapter summary

  In this chapter, we will consider the Poisson Equation. Poisson’s equation is another PDE in the same family (elliptic PDEs) as Laplace’s equation, and is in fact an extension or generalization of the Laplace equation. Like Laplace’s equation, the derivatives in Poisson’s equation are entirely of second‐order and are usually assumed to apply to a spatial domain. Also like Laplace’s equation, Poisson’s equation describes a steady‐state potential field. Unlike Laplace’s equation, however, the derivatives in Poisson’s equation do not (in general) sum to zero. Instead, the derivatives sum to either a constant or a term that is a function of space only (not of time); this term represents a source/sink, such as infiltration to an aquifer from precipitation, discharge from evapotranspiration, recharge from a lake or stream, and so on. Poisson’s equation is therefore the nonhomogeneous equivalent of Laplace’s equation, which is a special case of Poisson’s equation (the homogeneous, or degenerate, case). In the process of applying Poisson’s equation to an example problem, we will learn about a broader range of boundary conditions and consider several points of practical importance, such as how can a nonhomogeneous PDE be nondimensionalized, when can a term in an equation be neglected, and how to construct 2D and 3D finite difference operators.

  5.1 Poisson’s equation

  As with Laplace’s equation, the Poisson Equation is one of the important equations of mathematical physics, where it is commonly used to represent the potential field that results from some sink/source term. The Poisson Equation is perhaps most familiar from its role in electrostatics, where it describes the electrical potential field that results from a known charge distribution. In groundwater hydrology, Poisson’s equation describes the potentiometric surface in an aquifer that results from some combination of forcing on the domain boundaries and source/sink terms internal to the domain. In three‐dimensions (Cartesian coordinates), Poisson’s equation is written:

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  eBook - ePub

  Classical Electromagnetic Radiation

  • Jerry Marion(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER 3

  The Equations of Laplace and Poisson

  Publisher Summary

  The Laplacian operator occurs in many different types of physical problems probably the most important of which is that of wave propagation. This chapter discusses some of the mathematical functions that arise in the solution of wave equations that are the same as those that result from the solution of Laplace’s equation. It discusses some of the important properties of harmonic functions—that is, functions that satisfy Laplace’s equations. The chapter discusses the solutions of Laplace’s equation in rectangular coordinates, in spherical coordinates, and in cylindrical coordinates. It describes the parallel-plate diode that is a simple example of Poisson’s equation.

  3.1 Introduction

  In Chapter 1 we found that the general problem of the electrostatic field is described by Poisson’s equation, Eq. (1.9) :

  (3.1a)

  In regions not containing charge, this reduces to Laplace’s equation:

  (3.1b)

  The Laplacian operator occurs in many different types of physical problems, * probably the most important of which is that of wave propagation. Although we are interested in this book primarily in electromagnetic wave phenomena rather than in electrostatics, some of the mathematical functions which arise in the solution of wave equations are the same as those that result from the solution of Laplace’s equation. It is somewhat easier to introduce these harmonic functions (Legendre functions, spherical harmonics, and Bessel functions) in connection with electrostatic problems.

  We shall study such problems in some detail in order to become familiar with the functions that will be of use later in discussions of radiation phenomena. This will be the extent of the treatment of electrostatics; we will not discuss the method of images nor the use of conjugate functions in the solution of problems in electrostatics. The interested reader is referred to the list of Suggested References for sources of such material.

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  eBook - ePub

  Fundamentals of Ionized Gases

  Basic Topics in Plasma Physics

  • Boris M. Smirnov(Author)
  • 2012(Publication Date)
  • Wiley-VCH

    (Publisher)

  x satisfies the Poisson Equation

  Multiplication of this equation by E = −d φ /d x provides an integrating factor that makes a simple integration possible. We obtain

  (1.30)

  where E 0 = E (0).

  We need to establish the boundary condition on the cathode. We consider the regime where the current density of the beam is small compared with the electron current density of thermoemission. This means that most of the emitted electrons return to the metallic surface, and the external electric field does not significantly alter the equilibrium between the emitted electrons and the surface. Then the boundary condition on the cathode is the same as in the absence of the external electric field, so E (0) = 0. Formula (1.30) leads to the distribution of the electric potential in the gap, given by

  This can be inverted to obtain the connection between the electron current density and the parameters of the gap [27–30]:

  (1.31)

  This dependence is known as the three-halves power law. This describes the behavior of a nonneutral plasma that is formed in the space between two plane electrodes with different voltages. This voltage difference in this case is determined by the charge that is created by charged atomic particles [29–31] and has a universal character; in particular, in a magnetron discharge [32, 33], where electrons are magnetized and hence reproduction of a charge in a magnetron discharge results from ion flux to the cathode with energy of hundreds of electronvolts that creates secondary electrons. Along with this, metal atoms are sputtered, which determines the applications of this gas discharge.

  We consider one more example of transport of the flux of charged particles through a vacuum: an electron beam of radius a that is fixed by a longitudinal magnetic field inside a cylindrical metal tube of a radius ρ 0 [34]. According to the Gauss theorem, the electric field strength E at a distance ρ

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