Use one node voltage to connect several loops
You will be able to: Solve signed branch currents and verify charge and energy conservation.
How do junction and loop rules work together?
Two source-resistor branches feed a shared node, while a third resistor connects that node to the return. One source can supply current while the other absorbs it, depending on the node voltage.
A useful starting point: A loop returns to its starting potential →
Words and symbols before equations
- Node voltage V
- Potential of the shared top node relative to the bottom return.
- Reference branch current
- An assigned direction used consistently in equations.
- Steady junction balance
- No continuing charge accumulation: incoming current equals outgoing current.
What this picture assumes
Two source-resistor branches and one load share top and bottom nodes. Reference I₁ and I₂ enter the top node; I₃ leaves downward. Each ideal source has positive series resistance; unequal ideal sources are never directly placed in parallel.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Node V = 6 V. I₁ = 2, I₂ = 0, I₃ = 2 A. Incoming signed sum equals I₃. Net source power 24 W equals resistor power 24 W.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
The first source feeds the node through R₁, so I₁ = (ε₁−V)/R₁. The second feeds it through R₂, so I₂ = (ε₂−V)/R₂. The load current downward is I₃ = V/R₃.
Set I₁+I₂ = I₃. Solving gives V = (ε₁/R₁+ε₂/R₂)/(1/R₁+1/R₂+1/R₃). This uses loop voltage relations inside a junction balance.
If V exceeds ε₂, I₂ is negative: current leaves the node into that source branch. The two unequal ideal sources are not directly wired in parallel; each has its own positive series resistor. The complete power check is ε₁I₁+ε₂I₂ = ΣI_i²R_i.
A worked example, step by step
Take ε₁ = 12 V, ε₂ = 6 V and all three resistors 3 Ω. Find the node voltage and branch currents.
- Numerator is 12/3+6/3 = 6 A; denominator is 1 Ω⁻¹.
- V = 6 V.
- I₁ = 2 A, I₂ = 0 A and I₃ = 2 A.
- The junction balances; the 6 V source branch carries no current in this state.
Do not force every assumed branch current to be positive or combine resistors without checking their nodes.
Is charge continuously piling up at a steady junction?
Compare with an explanation
No. The total entering rate equals the leaving rate.
Predict. Change one thing. Explain.
Increase ε₁ while holding the other values fixed. Find when I₂ reverses and explain how the signed junction equation still balances.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Node V = 6 V. I₁ = 2, I₂ = 0, I₃ = 2 A. Incoming signed sum equals I₃. Net source power 24 W equals resistor power 24 W.
Two source-resistor branches and one load share top and bottom nodes. Reference I₁ and I₂ enter the top node; I₃ leaves downward. Each ideal source has positive series resistance; unequal ideal sources are never directly placed in parallel.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant current, circuit topology, charge or energy conservation, or RC relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionUse ε₁ = 12 V, ε₂ = 0 V and R₁ = R₂ = R₃ = 3 Ω. Find V and signed currents, and check power.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: V = (12/3)/(1/3+1/3+1/3) = 4 V.
- 1 point: I₁ = 8/3 A, I₂ = −4/3 A and I₃ = 4/3 A.
- 1 point: 8/3−4/3 = 4/3 verifies the junction.
- 1 point: Source power is 32 W and the three I²R terms sum to 32 W.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Which conservation law gives the junction rule?
Conservation of electric charge.
RECALL 2How many node voltages are needed here?
One unknown, after choosing the bottom node as zero.
RECALL 3Why are the source branches well-defined?
Each source has a positive series resistance.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Use one node voltage to connect several loops
- I₁+I₂ = I₃ for the stated reference directions.
- I₁ = (ε₁−V)/R₁; I₂ = (ε₂−V)/R₂.
- I₃ = V/R₃.
Remember: Do not force every assumed branch current to be positive or combine resistors without checking their nodes.
Conditions: Two source-resistor branches and one load share top and bottom nodes. Reference I₁ and I₂ enter the top node; I₃ leaves downward. Each ideal source has positive series resistance; unequal ideal sources are never directly placed in parallel.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 4 (official Unit 11) · Objectives 11.7.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 11.7, objectives 11.7.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 4: Electric Circuits, numbered Unit 11 in the official combined Physics C sequence. Topics 11.1–11.8 retain their official identifiers. Models include signed charge flow, prescribed current-density and resistivity integrals, DC resistor networks, nonideal batteries and meters, capacitor combinations and finite-resistance RC transients. Circuit schematics use conventional-current references and explicit node connectivity. Initial capacitor voltage and positive time constants are stated. Unequal ideal sources are never directly wired in parallel. The optional spatial wire view supplements a complete 2D current-density explanation. Checked with the Fall 2026 clarifications. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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