DC networks and Kirchhoff's laws
Complex direct-current networks are solved by combining charge conservation at junctions with energy conservation around closed loops. Measurement instruments then test the model without significantly disturbing the circuit.
1. Junction rule: conservation of charge
In steady state, charge does not accumulate at a junction. Therefore the algebraic sum of currents at every node is zero: ΣI = 0. Equivalently, total current entering a node equals total current leaving it. Choose current directions before calculating. A negative result means the real current flows opposite to the chosen arrow; it is not a failed solution.
2. Loop rule: conservation of energy
A charge returning to its starting point has zero net energy change, so ΣΔV = 0 around any closed loop. Traversing a source from its negative to positive terminal gives +ε; the reverse gives −ε. Across a resistor in the chosen current direction the change is −IR, and against the current it is +IR. Only independent junction and loop equations are needed.