TheoremProved

Kirchhoff circuit laws

Statement

At every node: total current in equals total current out (Kirchhoff current law — KCL). Around every closed loop: total emf equals total voltage drop (Kirchhoff voltage law — KVL).

Why is it true?

KCL is charge conservation: charge cannot pile up at a node. KVL is energy conservation: going around a loop must return to the same potential.

Together they solve any linear DC circuit — the foundation of modern circuit analysis (SPICE).

Place sources and resistors in two coupled loops, measure each current — verify ΣI_in = ΣI_out at the junction.

Applications

Circuit analysis (SPICE, LTspice), power-supply design, national power grids, smartphone/PC boards, traction networks.

Topics that use this theorem

Ohm’s law for a complete circuit and EMF

Related theorems

Ohm's law

Step-by-step proofs

Deriving Kirchhoff's two laws from charge and energy conservation

PhysicLabs, 2026

References

Gustav Kirchhoff (1845). Ueber den Durchgang eines elektrischen Stromes