Kirchhoff’s Loop Rule
Kirchhoff’s loop rule is the circuit form of energy conservation: when you traverse any closed loop and add the signed potential changes across every element, the algebraic sum is zero. The physics is simple; the challenge is keeping one traversal direction and one sign convention consistent from start to finish.
Topic sequence · 6 of 8
Learning objectives
Objective 1
Explain Kirchhoff’s loop rule as conservation of energy per unit charge and write ΣΔV=0 for a chosen closed circuit path.
Objective 2
Assign signs consistently to batteries, resistors, and other modeled circuit elements while solving symbolic or numerical loop equations.
Objective 3
Create and interpret electric-potential-versus-position graphs around a loop and use the graph as an independent check on the algebra.
- Topic 9.2 electric potential and potential difference as energy change per unit charge.
- Topics 11.2–11.5: circuit loops/nodes, Ohm’s law, power, equivalent resistance, emf and internal resistance.
- Algebra for simultaneous equations; Topic 11.7 supplies the charge-conservation equations needed for general multi-loop networks.
Definitions first
Representations you must read and create
Representation 1
A circuit schematic with an explicit clockwise traversal arrow and assumed branch-current arrows. The loop direction and current direction are separate choices.
Representation 2
Potential-versus-position graph: ideal wires are flat, an ideal battery produces a vertical rise or drop, and an ohmic resistor produces a drop or rise according to traversal relative to current.
Representation 3
Energy-per-charge ledger listing each element in traversal order and the corresponding signed ΔV before summing to zero.
Representation 4
Two different traversal directions around the same loop; every term changes sign but the solved current and physical potential differences remain unchanged.
Representation 5
Multi-loop diagram in which only independent loop equations are retained; a loop equation formed by adding two others supplies no new information.
Core relationships and calculus
Variables: ΔV V
Use: Kirchhoff’s loop rule for any chosen closed loop in the lumped-circuit model
Variables: ΔU J; q C; ΔV V
Use: energy change of charge q crossing a circuit element
Variables: ℰ V
Use: ideal source sign rule for the chosen traversal direction
Variables: I A; R Ω
Use: ohmic resistor sign rule for a chosen traversal direction
Variables: ℰ V; I A; R Ω
Use: single-loop source with series resistive drops under a consistent direction choice
Variables: V; I A; r Ω
Use: discharging nonideal source modeled as ideal emf plus internal resistance
The loop rule follows from energy bookkeeping. Each circuit element changes the electric potential energy of a charge by ΔU=qΔV. After a charge completes a closed circuit path and returns to the same circuit point, the net change in its state-dependent electric potential energy is zero, so qΣΔV=0 and therefore ΣΔV=0. In a resistor the circuit model uses ΔV=±IR; in a source the change is ±ℰ. The calculus-level field relation ΔV=−∫E⃗·dℓ⃗ explains potential differences in electrostatic regions, while Unit 11 treats sources and circuit elements with lumped potential changes.
Choose a closed loop and a traversal direction. For every element j, write the charge-energy change ΔU_j=qΔV_j. Summing over the complete loop gives ΣΔU_j=qΣΔV_j. Because the charge returns to the same circuit point, its net electric potential-energy change around the loop is zero after including source energy gains and element losses: 0=qΣΔV_j. For nonzero test charge q, divide by q to obtain ΣΔV_j=0. Substituting the model for each element produces the familiar circuit equation, such as ℰ−IR₁−IR₂=0.
Topic 11.6 uses the lumped electric-circuit model and potential differences across circuit elements. Do not confuse “ΣΔV=0 around the loop” with the statement that every point has the same potential. A battery can raise potential and a resistor can lower it while the total change around the closed path is zero. Time-varying magnetic-flux effects and induced emf are developed later in Unit 13; do not import induction formulas into an ordinary Unit 11 DC loop unless the problem explicitly introduces them.
Physical examples
Example 1
Crossing a 9 V ideal battery from its negative terminal to its positive terminal contributes +9 V to the loop sum; crossing it the opposite way contributes −9 V.
Example 2
Traversing a 4 Ω resistor in the same direction as a 2 A conventional current contributes −8 V; traversing against the current contributes +8 V.
Example 3
If your assumed current direction is wrong, the algebra normally returns a negative current. Do not change signs halfway through the solution; interpret the negative result at the end.
Example 4
Two opposing sources in one loop are not automatically added. Their signed emfs depend on how their polarities appear along the chosen traversal path.
Example 5
A potential graph must return to the starting potential after one full loop. Failure to close the graph exposes a missing or incorrectly signed circuit element.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Mark the traversal direction before writing any terms. Do not decide the sign of a voltage term from memory without asking how you cross the element.
Reasoning pattern 2
For a resistor, compare traversal with the assumed current arrow. With current: −IR. Against current: +IR.
Reasoning pattern 3
For an ideal source, inspect terminal polarity. Crossing − to + is +ℰ; crossing + to − is −ℰ.
Reasoning pattern 4
Use a potential-versus-position graph as a second representation. The graph must return exactly to its starting potential after one full loop.
Reasoning pattern 5
A negative solved current is not an algebra failure. It means the actual current is opposite the initially assumed arrow.
Reasoning pattern 6
In a multi-loop circuit, combine loop rule with Topic 11.7 junction equations and avoid counting algebraically redundant loops as independent information.
Deeper calculus and limiting-case check
A useful linear-algebra view is that a circuit can contain many geometrically possible loops but fewer independent loop equations. If one closed path is the algebraic sum of two previously used paths, its loop equation is also the sum of those equations and adds no new constraint. This is why equation selection matters in large networks. Another limiting check is R→0 for an ideal wire: its IR drop tends to zero, so an ideal wire segment is represented as constant potential on the loop graph.
Model-selection workflow: decide what the problem is before choosing an equation
For Kirchhoff’s Loop Rule, the fastest reliable route is not equation hunting. Start by identifying the nodes, branches, and circuit state, encode it with the circuit schematic/graph, choose the Kirchhoff/constitutive or transient equation, and then test the ideal-element and topology assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat closed loop and Kirchhoff’s loop rule as physical quantities first and symbols second.
Choose a representation
Build A circuit schematic with an explicit clockwise traversal arrow and assumed branch-current arrows. The loop direction and current direction are separate choices.. If the prompt supplies Potential-versus-position graph: ideal wires are flat, an ideal battery produces a vertical rise or drop, and an ohmic resistor produces a drop or rise according to traversal relative to current., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is Σ ΔV_loop = 0. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Topic 11.6 uses the lumped electric-circuit model and potential differences across circuit elements. Do not confuse “ΣΔV=0 around the loop” with the statement that every point has the same potential. A battery can raise potential and a resistor can lower it while the total change around the closed path is zero. Time-varying magnetic-flux effects and induced emf are developed later in Unit 13; do not import induction formulas into an ordinary Unit 11 DC loop unless the problem explicitly introduces them. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | A circuit schematic with an explicit clockwise traversal arrow and assumed branch-current arrows. The loop direction and current direction are separate choices. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | Σ ΔV_loop = 0 | Kirchhoff’s loop rule for any chosen closed loop in the lumped-circuit model. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | ΔU = qΔV | energy change of charge q crossing a circuit element. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | The loop rule follows from energy bookkeeping. Each circuit element changes the electric potential energy of a charge by ΔU=qΔV. After a charge completes a closed circuit path and returns to the same circuit point, the net change in its state-dependent electric potential energy is zero, so qΣΔV=0 and therefore ΣΔV=0. In a resistor the circuit model uses ΔV=±IR; in a source the change is ±ℰ. The calculus-level field relation ΔV=−∫E⃗·dℓ⃗ explains potential differences in electrostatic regions, while Unit 11 treats sources and circuit elements with lumped potential changes. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A single loop contains an ideal 12.0 V source, an opposing ideal 5.0 V source, and resistors R₁=2.0 Ω and R₂=1.0 Ω. Assume clockwise current I and traverse clockwise. Starting just before the 12 V source: +12.0 − I(2.0) − 5.0 − I(1.0)=0. Therefore 7.0−3.0I=0 and I=2.33 A. A potential graph can use V=0 as the arbitrary start: crossing the 12 V source raises V to 12 V; R₁ lowers it by 4.67 V to 7.33 V; the opposing 5 V source lowers it to 2.33 V; R₂ lowers it by 2.33 V back to 0. The exact closure is both an energy-conservation check and a sign-convention check. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 11.6
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship Σ ΔV_loop = 0 should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between A circuit schematic with an explicit clockwise traversal arrow and assumed branch-current arrows. The loop direction and current direction are separate choices. and Potential-versus-position graph: ideal wires are flat, an ideal battery produces a vertical rise or drop, and an ohmic resistor produces a drop or rise according to traversal relative to current.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Construct a low-voltage series loop with a battery and two resistors. Use a high-resistance voltmeter to measure the potential at successive nodes relative to one reference node, then plot measured electric potential versus traversal position. Verify that the measured rises and drops approximately close to zero around the loop. Repeat after reversing one battery or changing one resistance. Compare the fitted resistor drops with IR, include meter loading and resistor tolerance, and state how measurement uncertainty affects the closure sum.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Treating every battery as +ℰ regardless of traversal direction. The sign follows the terminals you cross. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Treating every resistor as −IR regardless of whether you traverse with or against the assumed current. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Changing the assumed current direction midway through the algebra after seeing an inconvenient sign. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Adding voltage magnitudes without signs, which destroys the energy-conservation meaning of the loop equation. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Forgetting that ideal wire segments are at the same potential in the lumped model and therefore contribute zero potential change. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using all possible loop equations as though they were independent, creating redundant equations rather than new physical constraints. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 9.2 defines potential difference and ΔU=qΔV, which gives the energy-per-charge interpretation of every loop-rule term. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 11.5 uses a loop-energy equation to derive the terminal voltage of a nonideal battery; Topic 11.6 formalizes that reasoning. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 11.7 supplies current-conservation equations at branch points so general multi-loop networks can be solved with the loop rule. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write Σ ΔV_loop = 0 and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 11.6 mastery ladder
Interactive Kirchhoff loop ledger
Assume clockwise current. Traverse a loop containing source ℰ₁, an opposing source ℰ₂, and series resistors R₁ and R₂. The tool solves the signed loop equation and lists the potential levels around the loop.
Experimental connection
Construct a low-voltage series loop with a battery and two resistors. Use a high-resistance voltmeter to measure the potential at successive nodes relative to one reference node, then plot measured electric potential versus traversal position. Verify that the measured rises and drops approximately close to zero around the loop. Repeat after reversing one battery or changing one resistance. Compare the fitted resistor drops with IR, include meter loading and resistor tolerance, and state how measurement uncertainty affects the closure sum.
Connections across Physics C
Connection 1
Topic 9.2 defines potential difference and ΔU=qΔV, which gives the energy-per-charge interpretation of every loop-rule term.
Connection 2
Topic 11.5 uses a loop-energy equation to derive the terminal voltage of a nonideal battery; Topic 11.6 formalizes that reasoning.
Connection 3
Topic 11.7 supplies current-conservation equations at branch points so general multi-loop networks can be solved with the loop rule.
Connection 4
Topic 11.8 uses the loop rule with I=dq/dt and ΔV_C=q/C to produce the RC differential equation.
Connection 5
Unit 13 later adds induced emf from changing magnetic flux, connecting circuit-loop equations to electromagnetic induction.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Treating every battery as +ℰ regardless of traversal direction. The sign follows the terminals you cross.
Avoid this
Treating every resistor as −IR regardless of whether you traverse with or against the assumed current.
Avoid this
Changing the assumed current direction midway through the algebra after seeing an inconvenient sign.
Avoid this
Adding voltage magnitudes without signs, which destroys the energy-conservation meaning of the loop equation.
Avoid this
Forgetting that ideal wire segments are at the same potential in the lumped model and therefore contribute zero potential change.
Avoid this
Using all possible loop equations as though they were independent, creating redundant equations rather than new physical constraints.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What does Kirchhoff’s loop rule state?
The algebraic sum of all potential differences around a chosen closed circuit loop is zero.
Why is the loop rule conservation of energy?
Potential difference is energy change per unit charge. A charge completing the loop returns to its starting circuit point, so its net energy-per-charge change around the full path must balance to zero.
Does clockwise traversal mean current is clockwise?
No. Traversal direction is only bookkeeping. Current arrows are separate assumptions that the algebra can later confirm or reverse.
What if I solve for a negative current?
The current magnitude is the absolute value; the negative sign means the actual direction is opposite your assumed arrow.
Why draw a potential-versus-position graph?
It makes battery rises and resistor drops visible and must close back to the starting potential, so it catches sign mistakes.
Are the loop and junction rules the same law?
No. The loop rule expresses energy conservation through potential differences; the junction rule expresses charge conservation through currents.