Kirchhoff's current law (KCL)

Kirchhoff's current law says that at any node in a circuit — any junction where two or more wires meet — the total current flowing in equals the total current flowing out. Written symbolically: ∑ I_in = ∑ I_out, or equivalently ∑ I_k = 0 (with signs indicating direction). It is conservation of charge applied to a point: charge cannot pile up at a wire junction, so whatever flows in must flow out on the same timescale.

The underlying physics is the continuity equation ∂ρ/∂t + ∇·J = 0. In DC and low-frequency circuits — where the charge density inside conductors is essentially constant — the continuity equation reduces to ∇·J = 0, and integrated over any closed surface surrounding a node, it gives ∮J·dA = 0, which is KCL. In high-frequency regimes where capacitors can charge and discharge appreciably during a single cycle, the displacement current term has to be added to keep the law exact; in practice this means treating capacitors as having a current I_C = C dV/dt flowing "through" them.

KCL is used in tandem with KVL to solve networks. For a circuit with N nodes and B branches, N−1 independent node equations (one fewer than nodes, because the Nth follows from the others) combine with B−(N−1) independent loop equations to uniquely determine all the branch currents. This is nodal analysis when node voltages are the unknowns, and mesh analysis when loop currents are the unknowns; both ultimately rest on the same two Kirchhoff sentences. SPICE simulators, hand calculations, and every textbook network problem use one or the other formulation.