Kirchhoff’s Current Law: A Derivation from Maxwell’s Equations

Kirchhoff’s current law was originally derived for systems such as telegraphs that switch in 0.1 s. It is used widely today to design circuits in computers that switch in ~0.1 nanoseconds, one billion times faster. Current behaves differently in one second and one-tenth of a nanosecond. A derivation of a current law from the fundamental equations of electrodynamics—the Maxwell equations—is needed. Here is a derivation in one line: div curl B/µ0 = 0 = div(J + (εr – 1) ε0∂E/∂t + ε0∂E/∂t = div Jtotal. Maxwell’s ‘true’ current is defined as Jtotal. The universal displacement current found everywhere is ε0∂E/∂t.The conduction current J is carried by any charge with mass, no matter how small, brief, or transient, driven by any source, e.g., diffusion. The second term (εr – 1) ε0∂E/∂t is the usual approximation to the polarization currents of ideal dielectrics. The dielectric constant εr is a dimensionless real number. Real dielectrics can be very complicated. They require a complete theory of polarization to replace the (εr – 1) ε0∂E/∂t term. The Maxwell current law div Jtotal= 0 defines the solenoidal field of total current that has zero divergence, typically characterized in two dimensions by streamlines that end where they begin, flowing in loops that form circuits. Note that the conduction current J is not solenoidal. Conduction current J accumulates significantly in many chemical and biological applications. Total current Jtotal does not accumulate in any time interval or in any circumstance where the Maxwell equations are valid. Jtotal does not accumulate during the transitions of electrons from orbital to orbital within a chemical reaction, for example. Jtotal should be included in chemical reaction kinetics. The classical Kirchhoff current law div J = 0 is an approximation used to analyze idealized topological circuits found in textbooks. The classical Kirchhoff current law is shown here by mathematics to be valid only when J >> ε0∂E/∂t, typically in the steady state. The Kirchhoff current law is often extended to much shorter times to help topological circuits approximate some of the displacement currents not found in the classical Kirchhoff current law. The original circuit is modified. Circuit elements—invented or redefined—are added to the topological circuit for that purpose.