Kirchhoff's Current Law and the Continuity Equation

Kirchhoff’s current law says that current does not accumulate. The continuity equation says that current does accumulate when currents and voltages vary. The continuity equation is the more general result, derivable from conservation of charge or the Maxwell equations. Kirchhoff’s current law is often a poor approximation when applied to the fast time varying signals of modern circuits. Engineers deal with this difficulty by supplementing circuits with parasitic (stray) capacitances that provide the displacement current ignored by Kirchhoff’s law. Maxwell used a different unfamiliar definition of true (or total) current that avoids this difficulty, because it includes the displacement current. The engineering approach works well as our electronic technology demonstrates. Maxwell’s approach may work better, after all he said “ … the time-variation of the electric displacement, must be considered in estimating the total movement of electricity.”