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Application of Gibbs variational principle and Lagrange multiplier method to problems of electrostatics. Part 2. General conditions of equilibrium

Keywords:

I.N. Aliev – Dr.Sc.(Phys.-Math.), Professor, Department «Physics», Bauman Moscow State Technical University
Z.A. Samedova – Post-graduate Student, Lecturer, Department «Physics», Bauman Moscow State Technical University


When considering the conditions of mechanical and thermomechanical conditions for the equilibrium of a liquid (gaseous) medium or solid, the principle of virtual works is widely used. This approach is based both on the principle of virtual mechanical displacements and on the Gibbs-Planck thermodynamic principle, which make it possible not only to write classical conditions for thermomechanical equilibrium, but also to extend the application of these principles to the case of external field influence. The variational principles by which the problem is solved allow us to calculate the energy of the electrostatic field, and then obtain a variation of this energy. The result obtained will allow us to derive not only all the laws of electrostatics, but also some additional relationships. This study demonstrates the application of the variational principles of electrostatics on a specific problem, which is very relevant given the insufficiently detailed exposition of the variational principles of magnetostatics and electrostatics in scientific and educational literature. In addition, the concept of surface charge and the physical meaning of a given quantity is expanding.
In this paper we consider a body placed in a thermostat and bounded from the outside by a conducting shell, which is maintained at constant potential and charge. First, we find the distribution of electrical potentials using the boundary conditions. We determine the formulation of boundary value problems, after which we calculate the energy of the electrostatic field, and then the variation of the energy of the electrostatic field.
The variational methods considered in this paper make it possible to obtain relations describing the general equilibrium conditions, the equilibrium of liquid and solid conductors, as well as dielectrics with frozen charges.

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