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AP Course

AP Physics 2

Study AP Physics 2 Units 9–15, with the Fall 2026 radioactive-decay clarification applied and official exam updates linked separately.

Study Units 9–15 through original models, derivations, experiments, quizzes, and practice sets.

Lessons
Particle Motion Behind Temperature and PressureConnecting Gas Variables with the Ideal-Gas ModelHeat Flow and the Approach to Thermal BalanceEnergy Accounting for Thermodynamic ProcessesMaterial Response to Heating and Heat ConductionEntropy, Probability, and the Direction of Thermal ChangeCharge Interactions and Coulomb-Force ModelsTracking Charge During Contact and InductionMapping Electric Fields from Source ChargesEnergy of Configurations of ChargesElectric Potential and Equipotential ReasoningCapacitance and Energy Stored in Electric FieldsConserving Energy in Charged-Particle MotionCharge Flow and Conventional CurrentModeling Sources, Wires, and Loads in Simple NetworksHow Geometry and Material Set ResistanceElectrical Energy Transfer and PowerReducing Series-Parallel DC NetworksLoop Equations from Energy ConservationJunction Equations from Charge ConservationTransient Charging and Discharging in RC NetworksSources, Direction, and Strength of Magnetic FieldsMagnetic Forces and Curved Paths of ChargesForces Between Fields and Current-Carrying ConductorsChanging Magnetic Flux and Induced EMFReflection and Absorption at Material BoundariesLocating Images with Mirror Ray ModelsRefraction, Index, and Total Internal ReflectionLocating Images with Thin-Lens ModelsPulses, Wave Types, and Propagation SpeedPeriodic-Wave Measures and GraphsBoundary Changes and PolarizationElectromagnetic-Wave BehaviorFrequency Shifts from Relative MotionSuperposition, Interference, and Standing WavesDiffraction from Openings and EdgesTwo-Slit and Grating Pattern GeometryPhase Change and Thin-Film ColorQuantum Models and Dual Wave-Particle EvidenceEnergy Levels in a Hydrogen-Like AtomConnecting Spectral Lines to Energy TransitionsThermal Spectra and Blackbody CurvesPhoton Thresholds in the Photoelectric EffectPhoton-Electron Scattering and MomentumMass-Energy Accounting in Fission and FusionRandom Nuclear Decay, Activity, and Half-Life
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics 2 · Unit 11 · Topic 11.5

Reducing Series-Parallel DC Networks

A resistor network can be reduced only after its topology is identified: series elements lie on one unbranched current path, while parallel elements connect to the same two nodes. Replace one valid group at a time with an equivalent resistance that draws the same terminal current for the same terminal voltage.

Learning Goals

  • Identify series and parallel connections from nodes and current paths rather than visual appearance.
  • Derive the series and parallel equivalent-resistance rules from conservation laws and Ohm's law.
  • Reduce a mixed network, then reconstruct branch currents, voltage drops, and power.
  • Recognize a bridge network that cannot be simplified by series-parallel reduction alone.

1. What Equivalent Resistance Means

Viewed from two chosen terminals, an equivalent resistor reproduces the original network's current-voltage relation. For a resistive DC network,

\[R_{\mathrm{eq}}=\frac{V_{\mathrm{terminal}}}{I_{\mathrm{terminal}}}\]

Equivalence does not mean that every internal current or voltage is preserved. It means the source sees the same total load at the selected terminals.

2. Conditions of the Model

The reduction rules below assume steady DC behavior, ideal connecting wires, and resistors whose stated resistance is constant. Source internal resistance, temperature change, capacitors during a transient, and non-ohmic devices require additional modeling.

Resistance is measured in ohms: \(1\,\Omega=1\,\mathrm{V/A}\).

3. Decide from Nodes, Not from the Drawing

ConnectionReliable testShared quantityEquivalent resistance
SeriesThe junction between the elements has no other branch, so the same charge flow must pass through both.Current\(R_s=\sum_i R_i\)
ParallelBoth ends of every element attach to the same two nodes.Voltage\(\displaystyle \frac1{R_p}=\sum_i\frac1{R_i}\)

Two resistors drawn side by side are not automatically parallel. Trace the wires and label electrically identical points with the same node name before choosing a formula.

A mixed series-parallel resistor network A four-ohm resistor is in series with a parallel pair of six and three ohms, followed by a two-ohm resistor. 4 Ω6 Ω3 Ω2 Ω AB
Topology firstThe 6 Ω and 3 Ω resistors share nodes A and B, so they are parallel. That parallel equivalent is in series with the 4 Ω and 2 Ω resistors.

4. Why Series Resistances Add

A series path carries one current \(I\). Energy conservation around the loop makes the total potential drop the sum of the individual drops:

\[V=V_1+V_2+\cdots=IR_1+IR_2+\cdots\]
\[\boxed{R_s=R_1+R_2+\cdots}\]

Therefore \(R_s\) must be greater than any individual series resistance.

5. Why Parallel Conductances Add

Parallel branches share one voltage \(V\). Charge conservation at a junction gives \(I=I_1+I_2+\cdots\). Using \(I_i=V/R_i\),

\[\frac{V}{R_p}=\frac{V}{R_1}+\frac{V}{R_2}+\cdots\]
\[\boxed{\frac1{R_p}=\frac1{R_1}+\frac1{R_2}+\cdots}\]

Adding a parallel path increases total conductance, so \(R_p\) must be smaller than the smallest branch resistance.

6. Reduction and Reconstruction Workflow

  1. Mark the two source terminals and label every distinct node.
  2. Find one unmistakable series or parallel group.
  3. Replace only that group with its equivalent and redraw the simpler topology.
  4. Repeat until a single \(R_{\mathrm{eq}}\) remains.
  5. Use the source voltage to find total current.
  6. Work backward through the reductions: series sections share current; parallel branches share voltage.
  7. Check junction current, loop voltage, resistance bounds, and total power.
Worked Example · Complete Mixed-Network Solution

Find Every Current, Voltage, and Power Check

A \(12.0\,\mathrm V\) ideal source drives the network shown above: \(4.0\,\Omega\), then \(6.0\,\Omega\parallel3.0\,\Omega\), then \(2.0\,\Omega\).

Step 1 — reduce the parallel pair.

\[R_p=\left(\frac1{6.0}+\frac1{3.0}\right)^{-1}=2.0\,\Omega\]

Step 2 — add the three series sections.

\[R_{\mathrm{eq}}=4.0+2.0+2.0=8.0\,\Omega\]

Step 3 — find source current and series voltage drops.

\[I_{\mathrm{total}}=\frac{12.0}{8.0}=1.50\,\mathrm A\]
\[V_{4\Omega}=6.0\,\mathrm V,\quad V_{AB}=3.0\,\mathrm V,\quad V_{2\Omega}=3.0\,\mathrm V\]

Step 4 — reconstruct the branch currents.

\[I_{6\Omega}=\frac{3.0}{6.0}=0.50\,\mathrm A,\qquad I_{3\Omega}=\frac{3.0}{3.0}=1.00\,\mathrm A\]

Checks. The branch currents recombine to \(1.50\,\mathrm A\), and the drops add to \(12.0\,\mathrm V\). Source power is \(VI=18.0\,\mathrm W\); resistor powers \(9.0+1.5+3.0+4.5\) also total \(18.0\,\mathrm W\).

7. When Reduction Must Stop

In a bridge network, a resistor may connect the midpoints of two branches. Then adjacent elements do not necessarily carry the same current, and candidate branches may not share the same two nodes. No legal series or parallel group may remain.

Do not force a reduction. Keep the original circuit and use junction and loop equations, or another valid network method. A balanced bridge can sometimes have zero current in its bridge branch, but that must be justified from symmetry or equations.

8. Common Mistakes

  • Calling elements parallel because they look parallel on the page.
  • Calling elements series even though their shared node has another branch.
  • Adding parallel resistances directly instead of adding reciprocals.
  • Assigning the source voltage to each series resistor.
  • Assigning total current to each parallel branch.
  • Stopping after \(R_{\mathrm{eq}}\) when branch quantities are requested.
  • Accepting a parallel equivalent larger than its smallest branch.
Mastery Check

1. Find the equivalent resistance of \(12\,\Omega\parallel6\,\Omega\).

Show reasoning and answer

\(1/R_p=1/12+1/6=3/12\), so \(R_p=4\,\Omega\). It is correctly smaller than \(6\,\Omega\), the smaller branch.

2. A \(5\,\Omega\) resistor is in series with that parallel pair. What is the total resistance?

Show reasoning and answer

The reduced \(4\,\Omega\) block is in series with \(5\,\Omega\), giving \(R_{\mathrm{eq}}=9\,\Omega\).

3. Why do two components connected to the same single node not necessarily form a series pair?

Show reasoning and answer

If another branch also meets that node, charge can split or recombine there. The two components need not carry the same current, so the series condition fails.

Key Takeaways

  • Series and parallel are topological relationships defined by paths and nodes.
  • Series elements share current and add resistance; parallel elements share voltage and add conductance.
  • Reduce inward, then reconstruct outward to recover branch quantities.
  • Use bounds, Kirchhoff checks, and power conservation to catch errors.
  • A nonreducible bridge requires circuit equations rather than an invented series-parallel shortcut.

Official curriculum reference: College Board AP Physics 2 course page. The explanation and worked example are independently written for this study site.