Mathematical expression for the capacitance of coplanar strips

Knowledge of the capacitance for the coplanar arrangement of long conducting strips is relevant for transmission lines and micro-antenna design for communications and medical applications. More generally, the evaluation of capacitances for non-trivial geometries is of continuous interest in electrostatics since the dawn of the discipline. The problem of finding an exact mathematical expression for the capacitance of coplanar conducting strips has not been solved. The purpose of this article is to show the corresponding derivation and its solution. The method used here starts from the charge density at the electrodes from the solution of the electrostatic integral equation, and subsequently computes the total charge and bias voltage at each electrode, from which the expression for the capacitance follows. The main result is an exact analytical expression for the capacitance as a function of the strips width and their separation. Moreover, the capacitance is found to depend non-trivially only on a single lumped dimensionless parameter that can be written in terms of the width and the separation of the strips. To gain insight into the dependence of the capacitance on geometrical parameters, we evaluate the analytical expression and also compare it with limiting standard geometries.