Bounds for the electrical resistance for homogeneous conducting body of rotation

Abstract

A mathematical model is developed for the steady-state electric current flow through in a homogeneous isotropic conductor whose shape is a body of rotation. The body of rotation considered is bounded by the coordinate surfaces of an orthogonal curvilinear coordinate system. The equations of the Maxwell’s theory of electric current flow in a homogeneous solid conductor body are used to formulate the corresponding electric boundary value problem. The studied steady-state conduction problem is axisymmetric. The determination of the steady motion of charges is based on the concept of the electrical conductance of the conductors the inverse of which is the electrical resistance. The exact (strict) value of the electrical resistance is known only for bodies with very simple shapes, therefore, the principles and the methods that can be used for creating lower and upper bounds to the numerical value of electrical resistance (electrical conductance) are important. The derivation of the upper and lower bound formulae for the electrical conductance of axisymmetric ring-like conductor is based on the two types of Cauchy–Schwarz inequality. The condition of equality of the derived lower and upper bounds is examined. Several examples illustrate the applications of the derived upper and lower bound formulae.