Capacitance and charging of metallic objects

The capacitance of arbitrarily shaped objects is reformulated in terms of the Neumann-Poincar\'{e} operator. Capacitance is simply the dielectric permittivity of the surrounding medium multiplied by the area of the object and divided by the squared norm of the Neumann-Poincar\'{e} eigenfunction that corresponds to its largest eigenvalue. The norm of this eigenfunction varies slowly with shape changes and allows perturbative calculations. This result is also extended to capacitors. For axisymmetric geometries a numerical method provides excellent results against finite element method results. Two scale-invariant shape factors and the capacitance of nanowires and of membrane in biological cells are discussed.