Complex electrical networks cannot always be reduced to combinations of series and parallel connections: they may contain several sources, branches that interweave, and nodes where three or more wires converge. To analyse them in full generality one uses the two Kirchhoff’s laws, which add no new physics but translate into equations two already familiar conservation principles: that of charge and that of energy.
First law — the junction rule
A node is a point where three or more conductors meet. In a steady-state regime, charge cannot accumulate at a node: everything that enters must, at the same instant, leave.
Principle — Junction rule (Kirchhoff's first law)
The sum of the currents entering a node equals the sum of the currents leaving it:
Physically this is conservation of charge: charges are neither created nor destroyed, and under steady-state conditions they do not pile up at nodes. If it were otherwise, a net charge would grow indefinitely at that point, with a growing electric field — a situation incompatible with the steady state. The current behaves like an incompressible fluid: the flow arriving at a junction is exactly the flow leaving it.
Second law — the loop rule
A loop is any closed path within the network. Travelling around it and summing all the potential changes (drops across resistors and electromotive forces of sources) brings us back to the starting point, where the potential necessarily has the same value as before.
Principle — Loop rule (Kirchhoff's second law)
The algebraic sum of the potential changes along any closed path is zero:
This is conservation of energy: the electric potential is a state function, that is, it takes a single, well-defined value at every point of the circuit. Returning to the starting point after a full loop, the net change in potential must vanish, exactly as altitude returns to its initial value after a circular route in the mountains. In energy terms: the energy supplied by the sources around the loop equals the energy dissipated in the resistors.
The two laws in one line
First law (nodes): conservation of charge. Second law (loops): conservation of energy.
Try it — circuit simulator
Build a multi-loop network with several sources, then check both laws: sum the currents at a node (should be zero) and the potential changes around a loop (likewise). Hover over components to read current and voltage.
Links
Topics: Circuiti elettrici Concepts: Leggi di Kirchhoff
Related exercises: Problema — due batterie con Kirchhoff · Esercizio svolto — due generatori, due maglie · Esercizio svolto — circuito a tre rami con due generatori