Time detector design for MeV particles I – the electric field in a triangular domain

We describe in this paper the methods used in the calculation of the electric potential and field in a triangular domain. These calculations are needed in the design of a time detector for high-energy particles. The electric field is created by three metal plates forming a triangular prism with different voltage applied to each of the plates. An analytical solution to the electric potential and field within the triangle is introduced. The method is based on the conformal mapping principle. The discovery of the analytical mapping functions which convert a triangle to a parallel band is of key importance to the solution. Although the methods described in this paper are given for right angled isosceles triangle, which is used in the time detector design, the method is in fact applicable to any triangles. The same problem has also been solved using the standard numerical method in which the electric potential is calculated on a set of grid points. The accuracy of the numerical solution depends on the size of the grid and the precision of iteration, which are generally limited by the computing resources. In comparison with the numerical method the analytical solution yields required information in a much more efficient and accurate way.