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Why must the voltage changes around any closed loop in a circuit add to zero?

Topic 11.6 Kirchhoff's Loop Rule: apply conservation of energy to the voltage changes around any closed loop of a circuit.

A focused answer to AP Physics 2 Topic 11.6, covering Kirchhoff's loop rule as conservation of energy, the sign conventions for emf sources and resistors, how to write loop equations, and its use in multi-loop circuits, with full worked examples.

Generated by Claude Opus 4.811 min answerUpdated 2026-06-04

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Quick answer

Kirchhoff's loop rule states that the sum of the voltage changes around any closed loop is zero: $\sum \Delta V = 0$ . It expresses conservation of energy, because a charge that travels around a loop and returns to its starting point has no net change in potential energy. To use it, pick a direction around the loop and add up the voltage changes: a source raises the potential by its emf when you traverse it from $-$ to $+$ (a rise, $+\mathcal{E}$ ), and a resistor drops the potential by $IR$ when you go in the direction of the current (a fall, $-IR$ ). Setting the total to zero gives one equation per loop. Combined with the junction rule (conservation of charge), the loop rule solves circuits too complex for series and parallel reduction. The key skill is consistent sign bookkeeping as you walk around the loop.

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What this topic is asking
What the loop rule says
Sign conventions
Using the loop rule
Try this

What this topic is asking

The College Board (Topic 11.6) wants you to apply Kirchhoff's loop rule: the sum of the voltage changes around any closed loop of a circuit is zero, a statement of conservation of energy. You must use sign conventions correctly and write loop equations.

What the loop rule says

The loop rule is energy conservation in circuit language. As a charge goes around a loop, it gains energy passing through sources and loses energy in resistors, and when it arrives back where it started, the gains and losses must exactly cancel, otherwise its potential energy would not return to its starting value. This is why the voltage drops across the resistors in a series loop add up to the source's emf.

Sign conventions

The sign rules are where most marks are won or lost. The recipe is mechanical: pick a loop direction, walk around it, and for each component write its voltage change with the correct sign, a battery is a rise when entered at its negative terminal, a resistor is a drop when crossed along the current. The signs are the whole difficulty; once the equation is written, it is simple algebra. If a solved current comes out negative, the real current simply flows opposite to your guess; the magnitude is still correct.

Using the loop rule

For a single-loop circuit, one loop equation gives the current directly. For multi-loop circuits, you assign a current to each branch, write a loop equation for each independent loop, add junction equations (Topic 11.7), and solve the simultaneous equations. The strategic role of the loop rule is that it, together with the junction rule, can analyze any circuit, including those that are not simple series-parallel combinations (such as bridge circuits or networks with multiple batteries). It also underlies the series-resistor result: in a series loop, the loop rule directly gives $\mathcal{E} = I R_1 + I R_2 + \dots$ , which is why series resistances add. Mastering the sign conventions here is what lets you handle the hardest circuit problems on the exam.

A two-resistor loop with sign bookkeeping

A loop has a $12$ V battery and resistors of $3.0$ ohms and $5.0$ ohms in series. Find the current using the loop rule.

step 1 Choose a loop direction and current

Let the current $I$ flow clockwise out of the battery's $+$ terminal, and traverse the loop clockwise.

step 2 Write the voltage changes

Crossing the battery from $-$ to $+$ : $+12$ V (rise). Crossing the $3.0$ -ohm resistor along the current: $-I(3.0)$ (fall). Crossing the $5.0$ -ohm resistor along the current: $-I(5.0)$ (fall).

step 3 Apply the loop rule

$12 - 3.0 I - 5.0 I = 0$ , so $12 = 8.0 I$ .

step 4 Solve and check

$I = \dfrac{12}{8.0} = 1.5$ A. The drops are $V_1 = (1.5)(3.0) = 4.5$ V and $V_2 = (1.5)(5.0) = 7.5$ V, summing to $12$ V, the emf, as energy conservation requires.

Try this

Q1. State the conservation law that Kirchhoff's loop rule expresses. [1 point]

Cue. Conservation of energy.

Q2. A loop has a $6.0$ V source and a single $4.0$ -ohm resistor. Write the loop equation and find the current. [2 points]

Cue. $6.0 - 4.0 I = 0$ , so $I = 1.5$ A.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2024 (style)6 marksSection II (short FRQ). A single loop contains a

9.0

V battery and two resistors in series,

2.0

ohms and

4.0

ohms. (a) State Kirchhoff's loop rule and the conservation law it expresses. (b) Write the loop equation and solve for the current. (c) Calculate the voltage across each resistor and verify they sum to the emf.

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A 6-point FRQ on the loop rule.

(a) Statement (2 points): the sum of the voltage changes around any closed loop is zero. It expresses conservation of energy: a charge returning to its start has no net change in potential energy.
(b) Loop equation (2 points): $9.0 - I(2.0) - I(4.0) = 0$ , so $9.0 = 6.0 I$ and $I = 1.5$ A.
(c) Voltages (2 points): $V_1 = (1.5)(2.0) = 3.0$ V; $V_2 = (1.5)(4.0) = 6.0$ V. Sum: $3.0 + 6.0 = 9.0$ V, equal to the emf.

Markers reward the zero-sum statement with energy conservation, the loop equation, and the voltages summing to the emf.

AP 2023 (style)1 marksSection I (multiple choice). Kirchhoff's loop rule is a direct consequence of which conservation law? (A) conservation of charge (B) conservation of energy (C) conservation of momentum (D) conservation of mass. Justify your reasoning.

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A 1-point MCQ on the basis of the loop rule. The answer is (B).

The loop rule says the voltage changes around a closed loop sum to zero, which means a charge that travels around and returns to its start has no net change in potential energy: that is conservation of energy. The junction rule, by contrast, expresses conservation of charge. The trap is (A): that is the junction rule, not the loop rule.

Related dot points

Sources & how we know this

AP Physics 2: Algebra-Based Course and Exam Description — College Board (2024)