On equivalent resistance of electrical circuits

Abstract

While the standard (introductory physics) way of computing the equvalent resistance of non-trivial electrical ciruits is based on Kirchhoff's rules, there is a mathematically and conceptually simpler approach, called the method of nodal potentials, whose basic variables are the values of electric potential at the circuit's nodes. In this paper, we review the method of nodal potentials and illustrate it using the Wheatstone bridge as an example. At the end, we derive - in a closed form - the equivalent resistance of a generic circuit, which we apply to a few sample circuits. The final result unveils a curious interplay between electrical circuits, matrix algebra, and graph theory and its applications to computer science. The paper is written at a level accessible by undergraduate students who are familiar with matrix arithmetic. For the more inquisitive reader, additional proofs and technical details are provided in the appendix.