Kirchhoff’s Junction Rule
Track signed branch currents with a steady-junction charge account.
AP Physics 2 Topic 11.7 · College Board CED printed p. 71 / PDF p. 78 · See this topic in the course outline
A junction cannot keep swallowing charge in a steady circuit. What enters per second must leave per second.
For a supporting original reading, see Kirchhoff’s Laws. Follow charge flow at a node and interpret negative answers relative to an assumed current arrow.
Before you start
- Current as charge rate
- Charge conservation and signed branch arrows
Conservation of charge gives the junction rule
For a small region around a steady junction, charge is not accumulating. Incoming charge per time therefore equals outgoing charge per time: . This is Kirchhoff’s junction rule. Multiplying by a common time interval gives equal total incoming and outgoing charge.
At a split, a source current may divide unequally among branches. It does not have to choose one branch, and it is not consumed by the first component. If 3.0 A enters and 1.0 A leaves through one path, 2.0 A leaves through the other. At a recombination node the branch currents add back to the source current.
The rule concerns signed flow rates along branches, not vector components. Wire angles do not enter. If you assume an unknown arrow outward and calculate a negative current, that branch is actually incoming. In an unbranched series path, the rule at each connecting region requires the same steady current throughout. If measurable charge were accumulating in a junction region, the simple equality would need the accumulation term; our steady DC approximation excludes that.
Use charge and energy conservation together
A junction equation relates branch rates, while a loop equation relates energy per charge. They supply different constraints. In the 12 V source, 1 Ω feed, and parallel 6 Ω/3 Ω network, I₀ = I₁ + I₂ is needed along with the two branch-loop equations. Loop relations show I₂ = 2I₁; charge conservation then gives I₀ = 3I₁.
At the return junction, I₁ + I₂ = I₀ states the same information for this network, so it checks the solution but does not add an independent unknown constraint. For a larger network, select enough independent junctions and loops, solve the linear equations, and test every remaining conservation relation.
A negative answer reveals an incoming branch
At a node, 0.80 A enters in branch A and 0.25 A leaves in B. Branch C is known to carry 0.70 A outward. Assume unknown branch D is outward and solve for its current.
With the assumed arrows, 0.80 = 0.25 + 0.70 + ID.
Why this step? Incoming and outgoing charge rates must balance.
ID = −0.15 A. Actual branch D brings 0.15 A into the junction.
Why this step? A negative result reverses only the assumed D arrow.
Actual incoming total is 0.80 + 0.15 = 0.95 A, equal to outgoing 0.25 + 0.70 = 0.95 A.
Why this step? A reconstructed physical-flow check prevents treating a negative magnitude as a lost charge.
Answer and check
ID = −0.15 A relative to the assumed outward direction, so 0.15 A actually enters.
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
At a junction 6 mC enters each millisecond, while 2 A leaves one branch. Find the other outward current.
Worked answer and reason
The incoming current is 6 A, so the other outward branch carries 4 A.
Convert charge/time to amperes, then balance charge rates.
Question 2
Two currents of 1 A and 2 A merge with wires at right angles. Is the outgoing current √5 A or 3 A? Explain.
Worked answer and reason
3 A. They are rates of charge arrival that add; the branch currents are not spatial vectors.
Junction conservation is scalar signed bookkeeping along wire paths.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- The current divides equally at every junction.
- Branch currents depend on resistances and shared voltage; equality needs symmetry.
- A negative solved current means the circuit is impossible.
- It generally means the actual direction opposes your assumed arrow.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 13 · junction charge-rate ledger
- Problem 14 · signed directions and network continuity
- Problem 18 · nonreducible bridge and independent equations
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
- Ask for a region boundary around the junction and charge amounts over one second.
- Require a final arrow redraw when a solution is negative.
Source objectives and scope
- 11.7.A — Describe a circuit or elements of a circuit by applying Kirchhoff’s junction 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 · 4 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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Circuit node
A circuit node is a connected conducting region treated as having one potential in the ideal-wire model. Branches meet at nodes, so topology rather than the drawing’s shape determines how elements are connected. For example, moving a wire on a schematic without changing its connections leaves the circuit nodes unchanged.
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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 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.
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Vector component
A vector component describes the part of a vector along a chosen axis. Its signed value changes when the axes change, even though the physical vector remains the same. For example, a diagonal launch velocity can be resolved into horizontal and vertical components that share the same elapsed time.