Kirchhoff's voltage law (KVL)

Kirchhoff's voltage law says that if you walk around any closed loop in a circuit, adding up the voltage drops across each component you pass through, the total is zero. Stated symbolically: ∑ V_k = 0 around any loop. It is conservation of energy for the circuit: the work done per unit charge to move once around a closed path must return the charge to the same energy it started with. There is no free energy in a loop — what the source supplies, the components absorb.

The deeper origin is that the electrostatic field is conservative: ∇×E = 0 in electrostatics, so ∮E·dℓ = 0 around any closed path. In quasi-static circuit analysis — where frequencies are low enough that ∂B/∂t is negligible compared to the resistive voltage drops — Faraday's law reduces to exactly this condition, and Kirchhoff's voltage law follows as a direct corollary. In truly high-frequency regimes (wavelengths comparable to circuit size), induced EMFs from time-varying magnetic flux must be included explicitly, and KVL is replaced by its Faraday-generalised version.

Practically, KVL turns any schematic into a system of linear equations. For each independent loop in the circuit, write ∑V_k = 0, expressing each V_k via Ohm's law (V_R = IR), the capacitor equation (V_C = Q/C), or an EMF source (V_ε = ±ε). The result is one equation per loop, and combined with Kirchhoff's current law at each node, the whole network reduces to linear algebra. Every SPICE simulator, every hand-solved homework, every DC circuit analysis runs through this reduction. The two sentences Kirchhoff wrote at twenty-one in 1845 are the whole foundation of network analysis.