On the potential of an infinite dielectric cylinder and a line of charge: Green's function in an elliptic coordinate approach

Abstract

A two-dimensional Laplace equation is separable in elliptic coordinates and leads to a Chebyshev-like differential equation for both angular and radial variables. In the case of the angular variable $\eta$ ($-1\leq\eta \leq1),$ the solutions are the well known first class Chebyshev polynomials. However, in the case of the radial variable $\xi$ ($1\leq\xi<\infty)$ it is necessary to construct another independent solution which, to our knowledge, has not been previously reported in the current literature nor in textbooks; this new solution can be constructed either by a Fröbenius series representation or by using the standard methods through the knowledge of the first solution (first-class Chebyshev polynomials). In any case, either must lead to the same result because of linear independence. Once we know these functions, the complete solution of a two-dimensional Laplace equation in this coordinate system can be constructed accordingly, and it could be used to study a variety of boundary-value electrostatic problems involving infinite dielectric or conducting cylinders and lines of charge of this shape, since with this information, the corresponding Green's function for the Laplace operator can also be readily obtained using the procedures outlined in standard textbooks on mathematical physics. These aspects are dealt with and discussed in the present work and some useful trends regarding applications of the results are also given in the case of an explicit example, namely, the case of a dielectric elliptic cylinder and an infinite line of charge.