The electrostatic field of a point charge inside a cylinder, in connection with wave guide theory

Abstract

In this paper the field of a point charge inside a hollow, infinitely long circular cylinder is studied. The case of an axial point charge is treated in detail. Three different methods are developed. The first method shows how to calculate the induced charges at the surface of the cylinder without explicit knowledge of the potential itself. The surface charge-density function is obtained as the solution of a Fourier-type integral equation. Then the potential caused by these charges is calculated. The second method works in the opposite direction. Here the potential is obtained as solution of a boundary value problem, followed by the calculation of the corresponding charges at the surface of the cylinder. The integral, obtained for the surface charge density, is transformed by contour integration. Although the resulting series is very useful for numerical purposes, a stronger method is necessary, in order to calculate the charge density just opposite the point source. Figure 1 shows the calculated values of the induced surface charge density. Several approximations for the charge-density function are considered, in connection with recent work of Weber (cf. Table I). The same is done for the field inside the cylinder. Various formulas are given which allow of numerical calculations. In Table II some calculated values of the potential are shown. The third method is based on the theory of Fourier-Bessel-Dini series. The potential is developed in terms of discrete normal solutions of the potential equation in cylindrical coordinates. The coefficients in this development can be derived from the behavior of the potential in the immediate neighborhood of the primary source. Furthermore, it is emphasized that the study of the above potential problem can serve as a guide in questions of wave propagation in hollow circular cylinders. In this connection the third method is shown to be extremely useful, as it enables us to calculate directly the fields in the far zone from that in the immediate neighborhood of the exciting source. This new method is demonstrated in case of acoustic waves inside a cylinder, caused by a harmonically vibrating point source.