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Various types of equilibrium processes involve electric fields. In somecases, the electrical energy appears to be negative (e.g. if the voltage isfixed by an external source). This paper explains how to derive the correctthermo-dynamic potential for electro-static phenomena, whether the voltage isfixed, or the charge is fixed, or some combination is fixed. In particular, weexplain, 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 alumped-parameter argument (i.e. a lumped-capacitor model). The secondexplanation uses a distributed parameter model (i.e. a partial differentialequation (PDE) model) of a dielectric medium, in this case, we allow fornon-linearity and external polarization effects. Connections with Legendre(duality) transforms are also discussed.

On The Correct Thermo-dynamic Potential for Electro-static Dielectric Energy