Electromagnetism: Maxwell Equations, Wave Propagation and Emission

Chapter 5

Special Techniques and Approximation Methods

A fundamental problem in the application of electrostatics is to determine the potential V, knowing the charge distribution and the dielectric properties of the medium. V obeys Poisson’s equation ΔV = − q_v/ε, whose solution [2.25] contains an arbitrary term V_o(r) that verifies Laplace’s equation ΔV_o = 0. In fact, the expression [2.25] is not always useful, because we do not know the positions of all the charges of the Universe and, even if we know some of them, the solution is often too complicated. On the other hand, the positions of charges on the surface of conductors and the polarization of dielectrics depend on the electric field that we have to determine. Finally, the region, in which we have to determine the potential, is often bounded by surfaces whose potential is given or whose total charge is given. This imposes boundary conditions on the field and the potential. The linearity of electrostatic equations (relating the sources, the field, and the potential) may bring some helpful simplifications to the problem. If a first configuration of charges produces the field E⁽¹⁾ and the potential V⁽¹⁾ and a second configuration produces the field E⁽²⁾ and the potential V⁽²⁾, the configuration formed by the superposition of charges ...