Circuits, Currents, Kirchhoff, and Maxwell

Abstract

Electricity flows in circuits that bring us power and information. The current flow in circuits is defined by the Maxwell equations that are as exact and universal as any in science. The Maxwell-Ampere law defines the source of the magnetic field as a current. In a vacuum, like that between stars, there are no charges to carry that current. In a vacuum, the source of the magnetic field is the displacement current, \(\varepsilon_0\ \partial\mathbf{E}/\partial t\). Inside matter, the source of the magnetic field is the flux of charge added to the displacement current. This total current obeys a version of Kirchhoff’s current law that is implied by the mathematics of the Maxwell equations, and therefore is as universal and exact as they are. Kirchhoff's laws provide a useful coarse graining of the Maxwell equations that avoids calculating the Coulombic interactions of \({10}^{23}\) charges yet provide sufficient information to design the integrated circuits of our computers. Kirchhoff's laws are exact, as well as coarse grained because they are a mathematical consequence of the Maxwell equations, without assumption or further physical content. In a series circuit, the coupling in Kirchhoff’s law makes the total current exactly equal everywhere at any time. The Maxwell equations provide just the forces needed to move atomic charges so the total currents in Kirchhoff’s law are equal for any mechanism of charge movement. Those movements couple processes for any physical mechanism of charge movement. In biology, Kirchhoff coupling is an important part of membrane transport and enzyme function. For example, it helps the membrane enzymes cytochrome c oxidase and ATP-synthase produce ATP, the biological store of chemical energy.