nLab Kirchhoff's laws

Idea

In the context of electromagnetism, Kirchhoff’s laws are a kind of coarse-grained form of Maxwell's equations. Where the latter deal with infinitesimal quantities, Kirchhoff’s law involve macroscopic current? and voltage in electrical circuits?.

One speaks of two laws,

1. the Kirchhoff voltage law (abbreviated KVL)

2. the Kirchhoff current law (abbreviated KCL)

The derivation of these two laws from Maxwell's equations is spelled out for instance in lecture 7 (here).

Nevertheless, Kirchhoff’s laws preserve the cohomological nature of Maxwell’s equations: where the latter involves de Rham cohomology and hence “infinitesimal cochains”, Kirchhoff’s laws can neatly be formulated in terms of cochains on finite cell complexes. (See for instance appendix B of (Frankel) or the section “Basic concepts” in (Baez)).

Related concepts

• Maxwell's equations

References

Reviews include

lecture 7 in

• (pdf)

appendix B in

• Theodore Frankel, The Geometry of Physics - An Introduction .

section Basic concepts in

• John Baez, Circuit theory

The cohomological nature of Kirchoff’s laws was maybe first made explicit in

• Hermann Weyl, Repartición de corriente en una red conductora, Rev. Mat. Hisp. Amer. 5 (1923), 153-164.

A textbook on electromagnetism amplifying this point of view is

• P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, Cambridge University Press, (2004)