A review of the variational methods for solving DC circuits

Abstract

Direct current (DC) circuits are usually taught in upper-level physics curricula, and Kirchhoff's laws are stated and used to solve the steady-state currents. However, students are not often introduced to alternative techniques, such as variational principles, for solving circuits. Many authors have tried to derive the steady-state distribution of currents in circuits from variational principles, and the initial attempts were carried out by great physicists, such as Kirchhoff, Maxwell and Feynman. In this article, we shall review such variational principles in physics and illustrate how they can be used to solve DC circuits. We will also explore how they are related, in a fundamental way, to the entropy production principles of irreversible thermodynamics. We believe that the introduction of variational alternatives for solving circuits can be a good opportunity for the instructor to present the techniques of the calculus of variations and irreversible thermodynamics to students.

Beyond the model of conductor with fixed cross-section and length: Exactly and numerically solvable problems

“…According to this equation the contact resistance increases sharply with decreasing the contact spot size. It is interesting that Equations ( 3), ( 4) can be derived by using Equation (8). Equations ( 3), ( 4), ( 8) is often applied to estimate leakage resistance [5] of spherical, cylindrical and other types of capacitors.…”

“…7 we plot dependences R(a/b) calculated with the help of Equations ( 1), ( 16)-(20) at different values of b/l and n. It is seen that in all cases the real (true) value of the resistance is larger than its value obtained in supposition of the homogeneous electric field. This fact can be explained on the basis of the minimum electric power principle [8]. According to this principle the currents distribute themselves inside the piece of conductor so that the electric power is as little as possible.…”