  Various types of equilibrium processes involve electric fields. In some
cases, the electrical energy appears to be negative (e.g. if the voltage is
fixed by an external source). This paper explains how to derive the correct
thermo-dynamic potential for electro-static phenomena, whether the voltage is
fixed, or the charge is fixed, or some combination is fixed. In particular, we
explain, in complete detail, why fixing the voltage introduces "a minus sign"
in the electrical energy.
  Two explanations are given. The first explanation is based on a
lumped-parameter argument (i.e. a lumped-capacitor model). The second
explanation uses a distributed parameter model (i.e. a partial differential
equation (PDE) model) of a dielectric medium, in this case, we allow for
non-linearity and external polarization effects. Connections with Legendre
(duality) transforms are also discussed.
