Kirchhoff's junction rule
/ KEER-khofs JUNK-shun rule /
Kirchhoff's junction rule is just the bookkeeping of charge at a fork in a circuit: whatever flows in must flow out, because charge cannot pile up or vanish at a point. Picture a road junction or a plumbing tee, the total flow arriving must equal the total flow leaving, or water would be appearing from nowhere. It answers the practical question at every branch point of a circuit: if I know some of the currents meeting here, what must the others be?
Precisely, at any junction (or node) where wires meet, the sum of currents flowing in equals the sum of currents flowing out; equivalently, the algebraic sum of all currents at the node is zero, sum of I = 0, if you count inflows as positive and outflows as negative. This is a direct consequence of the conservation of electric charge in a steady state: charge is neither created nor destroyed, and none accumulates at a point, so the flows must balance exactly.
The junction rule is one of the two tools (with the loop rule) for solving any circuit too tangled for simple series-parallel shortcuts. It is what tells you how current divides among parallel branches. One honest caveat: the rule holds for steady currents and for the terminals of a capacitor treated as a whole; strictly, at a plate of a charging capacitor charge does accumulate, which is handled at a deeper level by the idea of displacement current. For ordinary DC circuit analysis, though, currents in equal currents out at every node, full stop.
At a junction, two wires bring in currents of 3 A and 2 A, and one wire leaves. By the junction rule the outgoing current must be 3 + 2 = 5 A, so that inflow equals outflow.
Charge is conserved at a node: currents in equal currents out.
The junction rule is conservation of charge in action: no charge builds up at a node in steady DC. It, plus the loop rule, solves circuits that series-parallel tricks cannot.