Two methods for solving electrostatic problems with azimuthal symmetry

Articles cb Licença Creative Commons Revista Brasileira de Ensino de Física, vol. 42, e20190225 (2020) www.scielo.br/rbef DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2019-0225 Two methods for solving electrostatic problems with azimuthal symmetry T. E. P. Bueno1 , U. Camara da Silva*1 1 Universidade Federal do Espírito Santo, Departamento de Física, Vitória, ES, Brasil Received on August 23, 2019. Revised on October 2, 2019. Accepted on October 4, 2019. The study of electrostatic phenomena is the gateway to the physics described by Classical Electrodynamics. In this paper, we discuss in detail two methods based on the Uniqueness Theorem for solving electrostatic problems with azimuthal symmetry. The first one is the electrostatic potential extension from the axis of symmetry to an arbitrary point. The other consists in the mutual mapping between two potentials through an inversion transformation. We have prepared a list of six examples for which we calculate, completely or partially, the electrostatic potentials for different charge distributions using both methods. The electric field lines are analyzed and presented graphically in all cases. Keywords: Electrostatic, Azimuthal Symmetry, Uniqueness Theorem, Method of Inversion. 1. Introduction Classical Electrodynamics (CED), formulated at the end of the nineteenth century, is one of the greatest triumphs of science. It not only unified the already known electric and magnetic phenomena but also predicted the existence of electromagnetic waves and, being the first relativistic theory developed, CED served as a foundation for our current understanding of space and time. Together with the gravitational interaction, the classical electromagnetic fields are responsible for all the physics we observe in our macroscopic daily life. In CED, the evolution of the electromagnetic (e.m.) ~ x, t) and the magnetic field field — the electric field E(~ ~ B(~x, t) — is determined when we solve the so-called Maxwell’s Equations ~ ~ E(~ ~ x, t) = ρ(~x, t) , ∇× ~ E(~ ~ x, t) + ∂ B(~x, t) = 0, ∇. ε0 ∂t ~ ~ B(~ ~ x, t) = 0, ∇× ~ B(~ ~ x, t) − µ0 ε0 ∂ E(~x, t) = µ0~j(~x, t), ∇. ∂t (1) assuming that we already know the dynamics of the electric charges (sources of the e.m. field) described by the densities of charge, ρ(~x, t), and current, ~j(~x, t). At the same time, an electromagnetic field defined in space creates a Lorentz force on each charge qi given by   d~xi ~ (e.m.) ~ ~ Fi = qi E(~x, t) + × B(~x, t) , (2) dt where we assume a set of point charges, i.e. ρ(~x, t) = P P i (t) x − ~xi (t)) and ~j(~x) = i qi d~xdt δ(~x − ~xi (t)). i qi δ(~ * Correspondence email address: Copyright by Sociedade Brasileira de Física. Printed in Brazil. Therefore it is not difficult to see that the description of a system formed by electric charges and an e.m. field is a difficult task. In general, we have an endless loop: the electric charges create an e.m. field obeying Maxwell’s equations that modifies their dynamics according to the Lorentz force, and so on. Only in simple systems, when we have control over the field configuration or of the charge distribution, there are analytical solutions. Fortunately, in macroscopic scales, a class of simple systems becomes very relevant — the electrostatic phenomena. Electrostatics consists in determining the electric field formed by a previously known macroscopic charge distribution, characterized by the charge density, ρ(x), which does not evolve in time. Textbooks of Basic Physics [1–3] and Classical Electromagnetic Field Theory [4, 5] usually dedicate a substantial part of their text to the analysis of electrostatic physics. This article is devoted to the introduction and implementation of two powerful techniques described subtly in the references [6, 7]. These methods are little explored in undergraduate courses and allow for the resolution (sometimes only in a partial way) of a wide range of electrostatic problems with azimuthal symmetry. The first technique consists in the determination of the electrostatic potential by an explicit calculation done only on the axis of symmetry. The second technique is the inversion method, in which the potential on the outside of a sphere of radius R is mapped to the inside of it and vice versa, maintaining the boundary conditions on the sphere surface intact. In section 2, we have a review of the principal properties of the Poisson and Laplace equations that govern the electrostatic phenomena. In section 3, the two methods are derived using the Uniqueness Theorem in the context of problems with azimuthal symmetry. Section 4 provides e20190225-2 Two methods for solving electrostatic problems with azimuthal symmetry a series of applications that illustrate the advantages and limitations of the two methods. The solved examples are the charged ring, the ring outside/inside of a grounded conducting sphere, the charged hemisphere, the disc, and the rod. Finally, we present our final considerations in section 5. 2. Poisson and Laplace equations In an electrostatic situation, ρ = ρ(~x) and ~j = ~0, and equation (2) does not provide any information, since constraint forces compensate the electromagnetic force in such a way that the charges do not move. In the absence of dynamics, there is no magnetic field and Maxwell’s Eqs. (1) become only ~ E(~ ~ x) = ρ(~x) , ∇. ε0 ~ E(~ ~ x) = 0. ∇× (3) The second equation implies that the electric field is conservative. So it can be rewritten in terms of the scalar ~ x) = −∇φ(~ ~ x). Substituting this new form potential, E(~ into the first equation of (3) we have the following result ∇2 φ(~x) = − ρ(~x) , ε0 (4) the so-called Poisson’s Equation which describes all electrostatic physics and have very particular characteristics. The most important one is the Uniqueness Theorem: it says that if Dirichlet or Neumann boundary conditions are given, respectively φ ∂V or ~ n̂ · ∇φ , ∂V (5) where ∂V is a closed surface that encloses the volume V and n̂ is the unit vector normal to the surface, then the potential φ(x) is unique in all points of interest. The proof of the theorem can found in several books on the subject [4–6]. In this article, we will deal only with Dirichlet boundary conditions. For a localized charge distribution, i.e. when all charges are inside of a sphere of finite radius, we must impose Dirichlet boundary condition, φ ∂R3 = 0, and φ(~x) is determined (by the Uniqueness Theorem) as [4] Z 1 ρ(~x 0 ) 3 0 φ(~x) = d x. (6) 4πε0 R3 |~x − ~x 0 | The solution given by equation (6) has a problem at the practical level: even for simple charge configurations, it can lead to complicated integrals. We will discuss this point in section 3.1. In points of space without charges, we have Laplace’s equation, ∇2 φ(~x) = 0. (7) An important fact is that the scalar potential has no minimum or maximum at the points where equation Revista Brasileira de Ensino de Física, vol. 42, e20190225, 2020 (7) is valid — Earnshaw’s Theorem. As a consequence there is n (...truncated)