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Electrostatic shape energy differences of one-dimensional line charges

We investigate the electrostatic energy of one-dimensional line charges, focusing on the energy difference between lines of different shapes. The self-energy of a strictly one-dimensional charge is infinite, but one can quantify the energy by considering geometries that approach a one-dimensional curve, for example, thin wires, thin strips, or chains of close point charges. In each model, the energy diverges logarithmically as the geometry approaches a perfect one-dimensional curve, but the energy also contains a finite term depending on the shape of the line—the “shape energy.” The difference in shape energy between a straight line and a circle is checked to be the same using a range of models. To calculate the shape energy of more complex shapes numerically, we propose a line integral where the singularity in the integrand is canceled. This integral is used to calculate the shape energy of a helix.