Kirchhoff’s Rules

You’ll be able to

Apply the junction rule as a statement of charge conservation
Apply the loop rule as a statement of energy conservation
Assign consistent signs to EMFs and resistor voltage drops around a loop

The junction rule: charge is conserved

Where wires meet at a junction, charge cannot pile up or vanish, so whatever flows in must flow out: ΣI_in = ΣI_out. This is Kirchhoff’s junction (node) rule, and it is simply conservation of charge applied to a point in the circuit. If 5 A enters a junction and one branch carries off 2 A, the remaining branches must carry the other 3 A. The rule lets you track how current splits and recombines through a multi-branch network.

The loop rule: energy is conserved

Follow any closed loop around a circuit and add up every potential change — batteries raise the potential, resistors drop it (by IR). When you return to your starting point you must be back at the same potential, so the changes sum to zero: ΣΔV = 0. This loop rule is conservation of energy for a unit charge: whatever energy the battery gives a charge is exactly spent crossing the resistors before it comes home. Signs are the whole game — a consistent sign convention makes the algebra work out.

Kirchhoff’s rules

Junction: ΣI_in = ΣI_out · Loop: ΣΔV = 0

Going through a resistor in the direction of current is a voltage drop (−IR); passing from − to + inside a battery is a rise (+EMF). Reverse the sign if you traverse against that direction.

Worked example

At a junction, 5.0 A flows in along one wire. Two wires carry current away: one carries 2.0 A. Separately, a single 9.0 V battery drives a lone 3.0 Ω resistor loop. Find (a) the current in the second outgoing wire and (b) the current in the resistor loop.

1.(a) Junction rule: current in = current out, so 5.0 = 2.0 + I₂.
2.Solve: I₂ = 5.0 − 2.0 = 3.0 A.
3.(b) Loop rule on the single-resistor loop: +9.0 − I(3.0) = 0.
4.Solve: I = 9.0 / 3.0 = 3.0 A.

Answer: (a) The second wire carries 3.0 A; (b) the resistor loop carries 3.0 A. Each rule is just conservation — of charge at the junction, of energy around the loop.

Checkpoint

At a junction, 5.0 A flows in and then splits into two branches. If one branch carries 2.0 A, what does the other branch carry?

Tip

Before writing loop equations, draw an arrow for the assumed current direction and pick a direction to walk the loop. If a current comes out negative, it simply means it actually flows the other way — the magnitude is still correct.

Checkpoint

Kirchhoff’s loop rule — that the potential changes around any closed loop sum to zero — is fundamentally a statement of the conservation of:

On the exam

Pair the rules with their conservation law: junction rule = conservation of charge, loop rule = conservation of energy. On multi-loop problems, write one junction equation and one loop equation per unknown current, then solve the system.

Answer the 2 checkpoints as you read.