United StatesPhysics C: Electricity and MagnetismSyllabus dot point

How does conservation of energy constrain the voltages around a circuit loop?

Topic 11.6 Kirchhoff's Loop Rule: apply the loop rule (energy conservation) to write voltage equations for multi-loop circuits.

A calculus-based answer to AP Physics C E&M Topic 11.6, covering the loop rule as energy conservation, sign conventions for EMFs and resistors, writing loop equations, and solving multi-loop circuits.

Generated by Claude Opus 4.810 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

Kirchhoff's loop rule states that the algebraic sum of the potential changes around any closed loop is zero: $\sum_{\text{loop}} \Delta V = 0$ . This is conservation of energy: a charge taken once around a loop returns to its starting potential. To apply it, pick a loop direction, then account for each element's potential change with a consistent sign convention: across a resistor, the potential drops ( $-IR$ ) when you traverse it in the direction of the current and rises ( $+IR$ ) against the current; across a battery, the potential rises ( $+\varepsilon$ ) from the minus to the plus terminal and falls ( $-\varepsilon$ ) the other way. Writing one loop equation per independent loop, together with the junction rule, gives enough equations to solve for all unknown currents.

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What this topic is asking
The loop rule and energy conservation
Sign conventions
Solving multi-loop circuits
Try this

What this topic is asking

The College Board (Topic 11.6) wants you to apply Kirchhoff's loop rule, the statement that the potential changes around any closed loop sum to zero, to write and solve voltage equations for circuits too complex for simple series-parallel reduction. The loop rule is conservation of energy per charge.

The loop rule and energy conservation

Physically, a unit charge carried once around the loop gains energy at sources and loses it at resistors, and must end with the same potential energy it began with. The energy bookkeeping is exact, making the loop rule reliable for any circuit.

Sign conventions

The whole skill is consistent signs. Choose a direction to traverse the loop, then for each element:

Resistor: if you cross it with the assumed current, the potential drops: write $-IR$ . If against the current, it rises: write $+IR$ .
Battery (EMF): if you cross from the minus to the plus terminal, the potential rises: write $+\varepsilon$ . From plus to minus, it falls: write $-\varepsilon$ . (This is independent of the current direction.)

Set the signed sum to zero. If a solved current comes out negative, it simply flows opposite to your assumed direction; the magnitude is still correct.

Solving multi-loop circuits

For a circuit with several loops:

Label every branch current with an assumed direction.
Write the junction rule at enough nodes (Topic 11.7) and the loop rule for enough independent loops so that the number of equations equals the number of unknown currents.
Solve the simultaneous equations.

A loop with two opposing batteries

A single loop has a $10$ V battery and a $4.0$ V battery wired so their EMFs oppose, with resistors $R_1 = 3.0\,\Omega$ and $R_2 = 2.0\,\Omega$ in the loop. Find the current.

step 1 Choose a current direction and loop direction

Assume the current $I$ flows in the direction the $10$ V battery drives it, and traverse the loop that way.

step 2 Write the potential changes

Crossing the $10$ V battery minus-to-plus: $+10$ . Crossing each resistor with the current: $-IR_1$ and $-IR_2$ . Crossing the opposing $4.0$ V battery plus-to-minus (it opposes): $-4.0$ .

step 3 Apply the loop rule

$+10 - IR_1 - IR_2 - 4.0 = 0$ , so $10 - 4.0 = I(3.0 + 2.0)$ .

step 4 Solve

$6.0 = 5.0\,I$ , so $I = 1.2$ A in the assumed direction. The positive result confirms the $10$ V battery wins, driving current that charges the $4.0$ V battery.

Try this

Q1. State the physical principle the loop rule expresses. [1 point]

Cue. Conservation of energy (the potential returns to its value after a closed loop).

Q2. Crossing a resistor in the direction of the current, does the potential rise or fall? [1 point]

Cue. It falls: write $-IR$ in the loop equation.

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 2022 (style)1 marksSection I (multiple choice). The Kirchhoff loop rule is a statement of the conservation of (A) charge (B) current (C) energy (D) momentum. Justify your reasoning.

Show worked answer →

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

The loop rule says the sum of potential changes around any closed loop is zero. Potential energy per charge returns to its starting value after a full loop, which is conservation of energy. The junction rule, by contrast, expresses conservation of charge. The trap is confusing the two rules.

AP 2024 (style)5 marksSection II (FRQ, quantitative). A single loop contains a

12

V battery, a

9.0

V battery opposing it, and two resistors

R_1 = 2.0\,\Omega

and

R_2 = 1.0\,\Omega

in series. (a) Write the loop equation. (b) Solve for the current. (c) State which battery is being charged and explain.

Show worked answer →

A 5-point FRQ applying the loop rule with opposing EMFs.

(a) Loop equation (2 points): choosing a direction and summing potential changes: $+12 - IR_1 - IR_2 - 9.0 = 0$ , that is $12 - 9.0 = I(2.0 + 1.0)$ .
(b) Current (2 points): $3.0 = 3.0 I$ , so $I = 1.0$ A in the chosen (12 V driving) direction.
(c) Charging (1 point): the current is driven by the stronger $12$ V battery and flows into the $9.0$ V battery's positive terminal against its EMF, so the $9.0$ V battery is being charged.

Markers reward correct signs in the loop sum, the current, and identifying the charged battery.

Related dot points

Sources & how we know this

AP Physics C: Electricity and Magnetism Course and Exam Description — College Board (2024)