While the standard (introductory physics) way of computing the equivalent resistance of nontrivial electrical circuits 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 the 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. We then derive a closed-form expression for the equivalent resistance of a generic circuit, which we apply to a few sample circuits. The 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. Additional proofs and technical details are provided in appendices.
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Here, we assume that all the relevant nodes are electrically connected to the battery and that no circuit node is connected to the battery terminal via an ideal wire, i.e., via an edge of zero resistance.
When I offered this to my students as an optional homework problem with given numerical edge resistances (e.g., 1, 2, 3, 4, and 5 Ω), a typical solution would take a couple of pages.
It could be the case that all conductances but, say, σ1 were equal zero. Then on physical grounds, that would fix . Mathematically, this can be seen from Eq. (15) by giving σ3 (which would not affect the circuit connectivity or V2 anyway) a small nonzero value. The latter would result in cancellations and .
Interestingly, the matrix Σ is obtained from the conductance matrix σij via the same procedure as obtaining the Laplacian matrix from the adjacency matrix. Namely, the off-diagonal elements of Σ are simply −σij, whereas each diagonal entry is the sum of the elements of the corresponding row of σij. Such a matrix is sometimes referred to as the weighted Laplacian. Note also that the sum of all elements of Σ equals zero, hence at most n − 1 equations in Eq. (19) are independent.
Along the lines of Ref. 7, this could help to reinforce students' understanding and appreciation of the concept of electric potential.
