Some results on the cross-capacitances per unit length of cylindrical three-terminal capacitors with thin dielectric films on their electrodes

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The effect on the cross-capacitances per unit length of cylindrical 3-terminal capacitors of thin dielectric films on the capacitor electrodes is discussed. It is assumed that the cross-section of such dielectric films remains constant throughout the length of the capacitor. Some conformal transformations of basic cylindrical capacitor cross-sections are given. When these are applied to a symmetric cylindrical capacitor with a thin dielectric film on its electrodes, the results suggest that the mean capacitance per unit length may remain constant to the first order despite the presence of the dielectric film. The same methods also suggest that the individual cross-capacitances per unit length may remain constant to the first order provided that the dielectric film is disposed symmetrically with respect to the capacitor symmetry plane. Further support for these conjectures is given by the results of a detailed calculation of the cross-capacitances per unit length of a parallel-plate cylindrical capacitor with a thin uniform dielectric film on one electrode. In the last Section, a lemma concerning the existence of an equivalent dielectricless capacitor is given and this is followed by the proof of general results of the type suggested by the previous working.