Direct-Current Circuits in Physics

A circuit problem is a bookkeeping problem with two rules. Charge cannot pile up at a junction, so whatever flows in must flow out, and energy cannot change over a closed loop, so the rises and falls in potential must cancel exactly. Between them they are enough to solve any network, however tangled, without ever needing to know what the electrons are doing. The blocks add the one piece of realism that simple circuit diagrams leave out, the internal resistance of the source, which is why a battery's terminal voltage always sags while it works, and they finish with the timed behaviour of a capacitor, where the current starts large and fades away rather than being constant at all.