The Electrostatic Field of a Point Charge Inside a Cylinder, in Connection with Wave Guide Theory

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...