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:
. Maxwell’s ‘true’ current is defined as
. The universal displacement current found everywhere is
. The conduction current
is carried by any charge with mass, no matter how small, brief, or transient, driven by any source, e.g., diffusion. The second term
is the usual approximation to the polarization currents of ideal dielectrics. The dielectric constant
is a dimensionless real number. Real dielectrics can be very complicated. They require a complete theory of polarization to replace the
term. The Maxwell current law
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
is
not solenoidal. Conduction current
accumulates significantly in many chemical and biological applications. Total current
does not accumulate in any time interval or in any circumstance where the Maxwell equations are valid.
does not accumulate during the transitions of electrons from orbital to orbital within a chemical reaction, for example.
should be included in chemical reaction kinetics. The classical Kirchhoff current law
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
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.
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