Kirchhoff's loop rule

Kirchhoff's loop rule is the energy bookkeeping of a circuit: if you walk all the way around any closed loop and return to where you started, the voltage must add back up to zero, because you are back at the same electrical height. Picture hiking a trail that loops back to the trailhead, every metre you climb must be matched by metres you descend, so your net change in altitude is zero. It answers the question: how are the voltage gains from sources and the voltage drops across resistors balanced around a loop?

Precisely, around any closed loop in a circuit the algebraic sum of all the potential differences is zero, sum of V = 0. Going around, you add the EMF of each source you pass (a rise or a fall depending on direction) and subtract the I times R drop across each resistor. This is a direct expression of conservation of energy: a unit of charge carried all the way around a loop and back to its start must have the same potential energy it began with, so gains and losses cancel exactly.

The loop rule is the partner of the junction rule, and together they crack any DC network, however knotted. It is why the voltages across series resistors add up to the supply voltage, and it underlies every voltage divider. The honest fine print is bookkeeping discipline: you must choose a consistent direction around the loop and keep the signs of EMFs and I R drops straight, or the accounting fails. Note too that the loop rule assumes a conservative electric field; when a changing magnetic field threads the loop (electromagnetic induction), an extra induced EMF term must be added.