Circuits are found by the billions throughout our technology. Kirchhoff’s laws are widely used to design circuits, but the words ‘circuit’ and ‘Kirchhoff’ are not found in the index of some widely used textbooks of electrodynamics. Electrodynamics and circuit theory could benefit if Kirchhoff’s current law and circuit theory were taught as an integral part of electrodynamics. Circuits cannot be analyzed by Coulomb’s or Gauss’ law applied to charges. Circuits contain too many charges to allow interactions to be calculated by Coulomb’s law, charge by charge. Circuits can be analyzed by the currents that flow in them, traditionally by Kirchhoff’s current law applied to conduction current. Maxwell insisted that a generalization of Kirchhoff’s current law is needed that includes displacement current as well as the conduction current. Circuit designers provide the displacement currents intuitively, using imagined circuit elements called ‘parasitic capacitors’. Computer packages like LT-Spice could use Maxwell’s generalization, thereby using mathematics instead of imagination to deal with displacement current. Using Maxwell’s generalization would be a continuous reminder of the importance of displacement current in circuit theory and electrodynamics, uniting both disciplines. Using Maxwell’s generalization would show respect for historical precedent while increasing the realism of circuit simulations, although not to perfection: details of layout and device properties would be needed as well to design computer circuits that work at computer speeds. This paper shows how circuit analysis can be taught and used as part of electrodynamics.