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A review of tools and methods to calculate electrostatic potentials and fields inside high-purity germanium detectors in various configurations is given. The methods are illustrated concretely with a new educational program named GeFiCa - Germanium detector Field Calculator. Demonstrated in GeFiCa are generic numerical calculations based on the successive over-relaxation method as well as analytic ones whenever simplification is possible due to highly symmetric detector geometries. GeFiCa is written in C++, and provided as an extension to the CERN ROOT libraries widely used in the particle physics community. Calculation codes for individual detectors, provided as ROOT macros and python scripts, are distributed along with the GeFiCa core library, serving as both examples showing the usage of GeFiCa and starting points for customized calculations. They can be run without compilation in a ROOT interactive session or directly from a Linux shell. The numerical results are saved in a ROOT tree, making full use of the I/O optimization and plotting functionalities in ROOT. The speed and precision of the calculation are comparable to other commonly used packages, which qualifies GeFiCa as a scientific research tool. However, the main focus of GeFiCa is to clearly explain and demonstrate the analytic and numeric methods to solve Poisson's equation, practical coding considerations and visualization methods, with intensive documentation and example macros. It serves as a one-stop resource for people who want to understand the operating mechanism of such a package under the hood.