3.6 Kirchhoff's Loop Rule

Kirchhoff's Loop Rule Basics

Kirchhoff's Loop Rule (voltage law) states that the sum of voltage changes around any closed loop in a circuit equals zero. This reflects conservation of energy: as charge moves around a complete loop and returns to its starting point, it gains and loses equal amounts of potential energy per unit charge.

Applying Sign Conventions

When tracing a loop, EMF sources add voltage (+ if moving from - to + terminal) while resistors cause voltage drops (- in the direction of assumed current flow). Choose a direction, sum all voltage changes, and set the total equal to zero to solve for unknown currents or voltages.

Multi-Loop Circuits

In circuits with multiple loops, the loop rule is applied to each independent loop, often combined with Kirchhoff's Junction Rule (conservation of charge) to create a system of equations. Solving these simultaneously gives the currents through each branch, essential for analyzing complex resistor networks.

Key terms

Kirchhoff's Loop Rule

The sum of voltage changes around any closed loop in a circuit equals zero, reflecting conservation of energy.

Voltage rise vs. drop

Crossing an EMF source from - to + is a rise (+); crossing a resistor in the direction of current is a drop (-).

Why does the loop rule work?

Because energy is conserved: a charge that returns to its starting point has zero net change in potential energy.

Kirchhoff's Junction Rule (companion law)

The sum of currents entering a junction equals the sum of currents leaving it, based on conservation of charge.

Using the Loop Rule in practice

Pick a loop direction, sum EMFs and resistor voltage drops with correct signs, and set the total equal to zero to solve for unknowns.

Try an AP-style question

A 12 V battery is connected to 2 Ω and 4 Ω resistors in series. What is the current?

1. 2 A
2. 6 A
3. 1 A
4. 3 A