Kirchhoff’s Loop Rule
Write independent signed voltage ledgers around physical loops and check potential graphs.
AP Physics 2 Topic 11.6 · College Board CED printed p. 70 / PDF p. 77 · See this topic in the course outline
A charge returning to its starting point cannot gain net potential energy each trip. That is the energy basis of the loop rule.
For a supporting original reading, see Kirchhoff’s Laws. Trace actual closed paths and keep signed voltage changes. Use independent equations rather than combining disconnected branches into one loop.
Before you start
- Potential difference and energy per charge
- Ohm’s law and branch-current labels
A signed voltage ledger around a loop
Moving a charge q through a potential difference changes its electric potential energy by . A complete trip around a static DC loop returns to the same potential; conservation of energy requires . Batteries transfer energy from a chemical source into the electrical process; resistors convert electrical energy to thermal energy.
Choose a loop traversal and keep it fixed. Cross an ideal source from its negative to positive terminal: +ε; reverse traversal: −ε. Cross a resistor in the assumed conventional-current direction: −IR; go against it: +IR. Ideal wire changes are zero. Reversing the whole loop reverses every term and gives the same physical equation.
A negative solved current means the actual flow is opposite the chosen arrow. It does not invalidate the method. When multiple loops share a resistor, that resistor’s voltage is based on its actual branch current. Do not insert two different unrelated currents for one physical branch.
Draw the potential as you walk around the loop
A potential-versus-position graph is an energy-per-charge ledger. Starting from any reference, ideal wires are horizontal stretches, crossing a battery from − to + raises the graph by ε, and traveling through a uniform resistor with current lowers it by IR. The graph returns to its starting height after one full loop. A different zero shifts the entire graph vertically without changing drops or current.
| Position | Potential using negative terminal as zero | Change from previous position |
|---|---|---|
| Before source | 0 V | Reference |
| After source | 9 V | +9 V |
| After 3 Ω resistor | 6 V | −3 V |
| After 6 Ω resistor and return wire | 0 V | −6 V |
For each coulomb, the source provides 9 J, the first resistor converts 3 J, and the second 6 J. The sum +9 − 3 − 6 = 0 is both a loop equation and a graph-height check.
Write independent loops in a branching network
Consider a 12 V battery followed by a 1 Ω resistor carrying I₀ to a split. The branches are 6 Ω carrying I₁ and 3 Ω carrying I₂, both returning to the battery. Use one independent loop through each branch:
and . The junction relation is I₀ = I₁ + I₂.
Subtracting loops gives 6I₁ = 3I₂, so I₂ = 2I₁. Then I₀ = 3I₁ and 12 − 3I₁ − 6I₁ = 0. The results are I₁ = 4/3 A, I₂ = 8/3 A, and I₀ = 4 A. The shared 1 Ω drop is 4 V; either branch drops the remaining 8 V.
A third path going down one branch and up the other gives −6I₁ + 3I₂ = 0, already obtained by subtracting the two equations. It is a valid check but not an independent new equation. Label branches first so the algebra matches actual connections.
Two opposing sources in a single loop
In a single loop, traversal clockwise crosses a 12 V source from − to +, a 3 Ω resistor with assumed clockwise current, a 3 V source from + to −, and a 6 Ω resistor with current. Find I and describe the potential walk.
Write +12 − 3I − 3 − 6I = 0.
Why this step? Each sign follows the specified traversal, source polarity, and resistor-current direction.
9 = 9I gives I = 1 A clockwise.
Why this step? A positive solution confirms the assumed direction.
Choose 0 V before the 12 V source: the successive values are 0 → 12 → 9 → 6 → 0 V.
Why this step? Voltage rise and drops must return to the same starting potential.
Answer and check
I = 1.0 A clockwise; source energy supply 12 J/C, opposing source absorbs 3 J/C, and resistors convert a total 9 J/C.
A bridge needs branch equations
A 10 V ideal source fixes A at 10 V and D at 0 V. Use the five-resistor schematic below. Assume the five labeled currents in the drawn directions. Find the two node potentials and all branch currents using independent junction and loop equations.
At B, . At C, . Do not say R1 and R2 have equal current: the bridge provides another path.
Why this step? Conservation of charge gives two independent junction equations. The branch directions are chosen bookkeeping conventions.
Trace source–left path D→A→B→D: . Trace source–right D→A→C→D: . Trace the top bridge A→B→C→A: .
B and C are genuine three-way junctions. R1 and R2 cannot be reduced as a series pair, and neither can R3 and R4. Translucent traces mark selected closed walks through components, not new wires: orange D→A→B→D; blue D→A→C→D; green A→B→C→A. Why this step? Each equation follows one actual closed path. The last resistor in the top bridge loop is traversed opposite I3, producing a rise. Three independent loops and two junctions determine five unknown currents.
To solve cleanly, express each branch current with its own node voltage drop: I1 = 10 − VB, I2 = VB/2, I3 = (10 − VC)/2, I4 = VC, and I5 = (VB − VC)/2, all in amperes when volts and ohms are used. Substitute into the two junction equations to obtain and .
Why this step? Writing resistor voltage drops from node potentials already enforces the loop constraints. This is algebraic charge-and-energy accounting, with no special series reduction.
Eliminate VC: VB = 6 V and VC = 4 V. Then (I1,I2,I3,I4,I5) = (4,3,3,4,1) A. Check B: 4 = 3 + 1; check C: 3 + 1 = 4. Check the unused lower loop B→D→C→B: −2(3) + 1(4) + 2(1) = 0 V.
Why this step? The solution satisfies both junctions and an additional loop. Every positive answer agrees with its assumed arrow.
The source current is 4 + 3 = 7 A and source power is 10(7) = 70 W. Resistor powers sum to 4²(1) + 3²(2) + 3²(2) + 4²(1) + 1²(2) = 70 W.
Why this step? An independent energy-rate check catches branch or sign errors. This ideal resistor network has no stored-energy change at steady state.
Answer and check
VB = 6 V; VC = 4 V; currents are (4,3,3,4,1) A in the drawn directions. The source supplies 70 W, exactly the total resistor power. The bridge current is nonzero, so naive series-pair reduction would give the wrong circuit.
Make the model respond
Focused Physics 2 model
Two-resistor DC circuit
Predict first. Predict how total current changes when the same resistors are rewired in parallel.
Rₑq = R₁ + R₂; I₁ = I₂ = I; Vbattery = V₁ + V₂; P = I²R
On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.
Series: I₁ = 0.2 A, I₂ = 0.2 A, and total current = 0.2 A.
Read the model data and comparisons
| Quantity | Value | Unit | Meaning or limit |
|---|---|---|---|
| Battery voltage | 12 | V | |
| Equivalent resistance | 60 | Ω | |
| Battery current | 0.2 | A | |
| R₁ current | 0.2 | A | |
| R₂ current | 0.2 | A | |
| R₁ voltage drop | 4 | V | |
| R₂ voltage drop | 8 | V | |
| R₁ power | 0.8 | W | |
| R₂ power | 1.6 | W | |
| Battery power | 2.4 | W | |
| Power residual Pbattery − P₁ − P₂ | -2.220 × 10^−16 | W |
In series, every resistor carries the same current, and the voltage drops sum to the battery voltage. In parallel, every branch has the full battery voltage, and the branch currents sum at the junction. In either topology, the ideal battery power equals the sum of resistor powers. The battery moves charge around a closed circuit; resistors transfer electric energy to thermal energy.
Assumptions and limits. Ideal constant-voltage battery, ideal wires, two ohmic resistors with fixed resistance and no temperature dependence. Steady DC state; conventional-current direction from the positive terminal. No internal battery resistance or transient behavior.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
While traversing a loop you cross a resistor opposite its assumed current and a battery from + to −. Give the two signs. What happens to the equation if you reverse the traversal?
Worked answer and reason
Resistor: +IR; battery: −ε. Reversing traversal reverses all voltage-change signs and produces an equivalent equation.
Signs are local energy-per-charge changes along the chosen walk.
Question 2
A graph around a supposedly closed static DC loop starts at 2 V and ends at 7 V. Can that graph represent a consistent potential assignment? How could an arbitrary zero change the graph?
Worked answer and reason
No; one point must have one potential, so a full closed walk must end at 2 V. Changing the reference shifts every graph value by the same amount, without changing differences.
Loop closure is a conservation condition, not a choice of reference.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- Every resistor is a negative term in every loop equation.
- It is negative when traversal follows current and positive when traversal goes against current.
- Every drawable loop adds an independent equation.
- Some loop equations follow from others; use independent loops to solve, additional loops to check.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 11 · potential graph reconstruction
- Problem 12 · independent branch-loop equations and error diagnosis
- Problem 18 · nonreducible bridge and independent equations
- Problem 19 · branching RC endpoints and empirical time scale
Scientific calculator workspace
Keep intermediate precision and report units. This calculator supports arithmetic and elementary functions.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
For the teacher
- Require students to point to a physical element for each algebraic term.
- Ask them to redraw their potential graph after changing only the reference.
- Use explicit branch currents rather than ambiguous I labels.
Source objectives and scope
- 11.6.A — Describe a circuit or elements of a circuit by applying Kirchhoff’s loop rule.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 Course and Exam Description, Unit 11 (PDF pages 68–81) — Official topic order, learning objectives, all essential-knowledge clauses and boundaries. Equation pages 68,71,73,75,77–80 and relevant continuation pages visually inspected; independent reviewed-source-equation overlay consulted.
- OpenStax University Physics Volume 2, 9.3 Resistivity and Resistance — Independent verification of real material temperature dependence and the fixed-temperature qualification for linear metallic response. Teaching and assessment prose are original.
- OpenStax University Physics Volume 2, 10.3 Kirchhoff’s Rules — Independent verification of conservation-based branch and loop analysis; no calculus is needed in these authored tasks.
- Corrected original reading: Current and Resistance — Follow charge-flow rate and conventional-current direction; keep electron drift opposite the arrows.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Introduction to Circuits — Practice node connectivity, switches and common symbols. Check that any short-circuit idealization retains a finite current-limiting resistance.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Resistors in Circuits — Connect resistor voltage and current to energy transfer per second. Brightness comparisons require a stated bulb model.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Kirchhoff’s Laws — Trace actual closed paths and keep signed voltage changes. Use independent equations rather than combining disconnected branches into one loop.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Capacitors in Circuits — Compare equal-charge series and equal-voltage parallel capacitor arrangements.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: RC Circuits — Use initial and long-time states, τ and qualitative curves. Source exponential and logarithmic transient calculations extend beyond required AP Physics 2 modeling.Supporting context; original AP teaching and practice above remain independently authored.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Potential energy
Potential energy describes energy associated with the configuration of interacting parts of a system. Its numerical zero is a chosen reference, while differences in potential energy determine physical energy changes. For example, separating Earth and a ball changes their gravitational potential energy even though the energy belongs to their interaction.
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Potential difference
Potential difference is the change in electric potential between two locations. It represents electric potential-energy change per unit charge and is measured in volts. For example, the potential difference across a resistor describes the energy transferred per coulomb passing through it.
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Ohm’s law
Ohm’s law describes an element whose current is proportional to its potential difference under the stated conditions. Such an element has constant resistance over the modeled range, giving a straight-line current–voltage relationship through the origin. For example, a fixed-temperature ideal resistor follows this model, whereas a warming filament bulb need not.
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Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
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Electric potential energy
Electric potential energy is energy associated with the configuration of interacting charges. Its value depends on the chosen reference, and a charge pair is counted once when calculating a system’s total pair energy. For example, bringing two positive point charges closer requires positive external work in a slow controlled approach.
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Electric current
Electric current is the net rate of charge flow through a specified cross section. It is measured in amperes, or coulombs per second, with direction defined by positive-charge motion. For example, the same steady current passes through each element of a simple unbranched series loop.