VARIATIONAL THERMODYNAMIC DERIVATION OF THE FORMULA FOR PRESSURE DIFFERENCE ACROSS A CHARGED CONDUCTING LIQUID SURFACE AND ITS RELATION TO THE THERMODYNAMICS OF ELECTRICAL CAPACITANCE
Nn. Ljepojevic et Rg. Forbes, VARIATIONAL THERMODYNAMIC DERIVATION OF THE FORMULA FOR PRESSURE DIFFERENCE ACROSS A CHARGED CONDUCTING LIQUID SURFACE AND ITS RELATION TO THE THERMODYNAMICS OF ELECTRICAL CAPACITANCE, Proceedings - Royal Society. Mathematical and physical sciences, 450(1938), 1995, pp. 177-192
The formula for pressure difference across a charged conducting liquid
surface has conventionally been derived by adding a Maxwell stress te
rm to the pressure-difference formula for the field-free case. As far
as can be established, no derivation applying direct energy-based meth
ods to the charged-surface case has ever been clearly formulated. This
paper presents a first-principles variational derivation, starting fr
om the laws of thermodynamics and modelled on Gibbs's (1875) approach
to the field-free case. The derivation applies to the static equilibri
um situation. The method is to treat the charged liquid and its enviro
nment as a heterogeneous system in thermodynamic equilibrium, and cons
ider the effects of a small virtual. variation in the shape of the con
ducting-liquid surface. Expressions can be obtained for virtual change
s in the free energies of relevant system components and for the virtu
al electrical work done on the system. By converting the space integra
l of the variation in electrostatic field energy to an integral over t
he surface of the liquid electrode, the usual pressure-difference form
ula is retrieved. It is also shown how the problem can be formulated,
in various ways, as a free-energy problem in a situation involving ele
ctric stresses and capacitance. The most satisfactory approach involve
s the definition of an unfamiliar form of free energy, that can be see
n as the electrical analogue of the Gibbs free energy and may have use
in other contexts.