Review AP Physics 2 Unit 11 to build a complete picture of how electric circuits work, from current and resistance through Kirchhoff's rules and RC transients. This unit carries 15-18% of the exam and rewards students who can reason through circuit behavior qualitatively and quantitatively.
Use the topic guides, practice questions, FRQ practice, and AP score calculator available for this unit to focus your review.
Electric circuits build directly on the electric potential concepts from Unit 10. In Unit 11, potential difference drives current through resistors and capacitors arranged in series, parallel, or mixed networks. The unit progresses from defining current and resistance to analyzing multi-loop circuits and finally to the transient charging and discharging behavior of RC circuits.
Current I = Δq/Δt measures charge flow rate in amperes. Resistance depends on material resistivity and geometry via R = ρℓ/A. Ohm's law I = ΔV/R links these quantities for ohmic materials, where an I-V graph is a straight line with slope 1/R.
Resistors in series share the same current; resistors in parallel share the same voltage. Equivalent resistance lets you simplify networks. Power dissipated by any element is P = IΔV, also written as P = I²R or P = (ΔV)²/R, and bulb brightness scales directly with power.
The loop rule (ΣΔV = 0) applies conservation of energy around any closed loop. The junction rule (ΣI_in = ΣI_out) applies conservation of charge at any node. In RC circuits, the time constant τ = R_eq C_eq sets how quickly a capacitor charges to 63% or discharges to 37% of its value.
Every major tool in Unit 11 is an application of a conservation law. Kirchhoff's loop rule is conservation of energy: a charge returning to its starting point has zero net change in potential. Kirchhoff's junction rule is conservation of charge: current cannot accumulate at a node. Even the RC time constant reflects how charge redistributes over time while total charge is conserved. Recognizing which conservation law applies is the core reasoning skill the exam tests.
Current I = Δq/Δt measures charge flow rate. Conventional current direction is defined by positive charge motion, opposite to electron flow in metals. emf drives charge through a circuit.
Circuits are closed loops of elements including batteries, resistors, bulbs, and meters. Closed circuits allow current; open circuits do not. Schematics use standard symbols and element arrangement determines behavior.
Resistance R = ρℓ/A depends on material and geometry. Ohm's law I = ΔV/R applies to ohmic materials with constant resistance. An I-V graph for an ohmic resistor is a straight line with slope 1/R.
Power P = IΔV = I²R = (ΔV)²/R gives the rate of energy transfer. Bulb brightness increases with power, allowing qualitative ranking of bulbs in series or parallel circuits.
Series resistors add directly; parallel resistors combine by reciprocal sums. Real batteries have internal resistance, reducing terminal voltage below emf. Mixed networks are simplified by finding equivalent resistance step by step.
ΣΔV = 0 around any closed loop, reflecting conservation of energy. Track potential rises across batteries and drops across resistors to write loop equations and solve for unknown quantities.
ΣI_in = ΣI_out at every junction, reflecting conservation of charge. Apply at nodes where current branches split or recombine to write current equations for multi-loop circuits.
Capacitors combine in series (reciprocal sum) or parallel (direct sum). The time constant τ = R_eq C_eq sets the charging and discharging rate. At t = 0 an uncharged capacitor acts as a wire; at steady state it acts as an open circuit.
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Review Resistance, Resistivity, and Ohm's Law with attention to how the concept appears in AP-style source and evidence questions.
Review Electric Current with attention to how the concept appears in AP-style source and evidence questions.
Review Kirchhoff's Junction Rule with attention to how the concept appears in AP-style source and evidence questions.
Review Electric Power with attention to how the concept appears in AP-style source and evidence questions.
Current is the rate at which charge passes through a cross-sectional area of a wire, defined as I = Δq/Δt and measured in amperes. Charge moves in response to an electric potential difference, also called emf (ε). Conventional current direction is defined as the direction positive charge would move, which is opposite to the actual motion of electrons in a metal. Current is not a vector, but it does have a direction that is independent of any coordinate system.
A circuit is one or more electrical loops containing elements such as batteries, resistors, bulbs, capacitors, switches, ammeters, and voltmeters. Charge can only flow in a closed loop. Circuit schematics use standard symbols for each element, and the physical arrangement of elements determines circuit behavior. Ammeters are placed in series; voltmeters are placed in parallel with the element being measured.
| Circuit type | Current flows? | Potential difference across elements? |
|---|---|---|
| Closed circuit | Yes | Yes, distributed across elements |
| Open circuit | No | Full source voltage appears across the gap |
| Short circuit | Yes, very large | Zero across the shorted element |
Resistance measures how strongly an object opposes charge flow. For a uniform conductor, R = ρℓ/A, where ρ is the material's resistivity, ℓ is length, and A is cross-sectional area. Resistivity is a material property that typically increases with temperature for conductors. Ohm's law, I = ΔV/R, applies to ohmic materials, which have constant resistance regardless of current. On an I-V graph, an ohmic resistor produces a straight line; the slope equals 1/R. Resistors also convert electrical energy to thermal energy (Joule heating), which can raise the temperature of the resistor and its surroundings.
Power is the rate at which energy is transferred or dissipated by a circuit element. The fundamental equation is P = IΔV. Combining with Ohm's law gives two derived forms: P = I²R (useful when you know current and resistance) and P = (ΔV)²/R (useful when you know voltage and resistance). Bulb brightness increases with power, so you can rank bulbs in a circuit by comparing their power values without calculating exact brightness.
| Formula | Best used when | Variable held constant |
|---|---|---|
| P = IΔV | Both I and ΔV are known | Neither |
| P = I²R | Current I is known | Current same in series |
| P = (ΔV)²/R | Voltage ΔV is known | Voltage same in parallel |
Resistors in series carry the same current; their equivalent resistance is R_eq = R1 + R2 + ... Resistors in parallel share the same potential difference; their equivalent resistance satisfies 1/R_eq = 1/R1 + 1/R2 + ... Real batteries have internal resistance r, so the terminal voltage is ΔV_terminal = ε - Ir, which is less than the emf when current flows. Simplifying a mixed network means repeatedly combining series and parallel groups until one equivalent resistance remains.
| Property | Series connection | Parallel connection |
|---|---|---|
| Current | Same through all resistors | Splits among branches |
| Voltage | Divides across resistors | Same across all branches |
| Equivalent resistance | R_eq = ΣR_i (larger than any single R) | 1/R_eq = Σ(1/R_i) (smaller than any single R) |
Kirchhoff's loop rule states that the sum of all potential differences around any single closed loop equals zero: ΣΔV = 0. This is a direct consequence of conservation of energy: a charge that travels around a complete loop returns to the same electric potential it started at. When applying the rule, track potential rises (across a battery from - to +) and potential drops (across a resistor in the direction of current) consistently. A graph of electric potential versus position around a loop shows these rises and drops as a visual check.
Kirchhoff's junction rule states that the total current entering any junction equals the total current leaving it: ΣI_in = ΣI_out. This follows from conservation of charge: charge cannot accumulate at a node in a steady-state circuit. The junction rule is the key tool for writing equations at nodes in multi-branch circuits, and it works together with the loop rule to solve for all unknown currents.
| Rule | Conservation law | Equation | Where to apply |
|---|---|---|---|
| Loop rule | Conservation of energy | ΣΔV = 0 | Around any closed loop |
| Junction rule | Conservation of charge | ΣI_in = ΣI_out | At any node where currents split or merge |
Capacitors in series satisfy 1/C_eq = Σ(1/C_i), giving an equivalent capacitance smaller than the smallest individual capacitor. Capacitors in parallel satisfy C_eq = ΣC_i. In an RC circuit, the time constant τ = R_eq C_eq determines how quickly the capacitor charges or discharges. After one time constant, a charging capacitor reaches about 63% of its final charge; a discharging capacitor retains about 37% of its initial charge. Immediately after a switch closes, an uncharged capacitor behaves like a short circuit (wire). After a long time, it behaves like an open circuit and no current flows through its branch.
| Capacitor configuration | Equivalent capacitance formula | Compared to individual values |
|---|---|---|
| Series | 1/C_eq = Σ(1/C_i) | Smaller than the smallest C |
| Parallel | C_eq = ΣC_i | Larger than the largest C |
Try stimulus-based AP practice questions and written prompts after you review the notes.
The circuit shown contains an ideal battery and four identical lightbulbs labeled A, B, C, and D. Bulb A is in series with the battery, while bulbs B, C, and D are connected in a separate parallel combination that is in series with bulb A.
The figure shows two cylindrical resistors, P and Q, made of the same material. Resistor P has length L and cross-sectional area A. Resistor Q has length 2L and cross-sectional area A/2. A student applies the resistance equation to compare the two resistors.
3. In Experiment 1, a student is given a battery of unknown emf and unknown internal resistance. The student is asked to predict how the battery’s terminal potential difference changes as the current supplied by the battery changes when different external resistors are connected.
Describe a procedure for collecting data that would allow the student to determine the battery’s emf and internal resistance. In your description, include the measurements to be made. Include any steps necessary to reduce experimental uncertainty.
Describe how the collected data could be analyzed to determine the battery’s emf and internal resistance. Include references to appropriate equations and to relationships between measured and known quantities.
Figure 1. Circuit used to measure the battery terminal potential difference and current for different external resistors.
Figure 2. Graph of battery terminal potential difference versus current for determining emf and internal resistance.
External resistance R (Ω) | Current I (A) | Battery terminal potential difference V_terminal (V) |
|---|---|---|
4.0 | 0.30 | 1.70 |
6.0 | 0.20 | 1.80 |
8.0 | 0.15 | 1.85 |
12.0 | 0.10 | 1.90 |
24.0 | 0.05 | 1.95 |
In Experiment 2, the student uses the setup in Figure 1 with five different external resistors. For each trial, the student records the current I in the circuit and the battery terminal potential difference V_terminal. The data are shown in Table 1.
Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine the battery’s emf and internal resistance.
Vertical axis: Horizontal axis:
On Figure 2, create a graph of the quantities indicated in part C(i) that can be used to determine the battery’s emf and internal resistance.
Use Table 2 to record the data points or calculated quantities that you will plot.
Clearly label the axes, including units as appropriate.
Plot the points you recorded in Table 2.
Draw a best-fit line for the data graphed in part C(ii).
Using the best-fit line in part D and appropriate circuit relationships, calculate the power dissipated in the external resistor (i) immediately after the switch is closed and (ii) after a long time has passed. Express each answer in watts. The student analyzes the graph from Experiment 2 and finds that the best-fit line for as a function of is
The student then connects an external resistor in series with the battery. The student also connects a capacitor of capacitance in parallel with the external resistor. Immediately after the switch is closed, the capacitor is uncharged. After a long time, the capacitor is fully charged.
4. A circuit contains a battery of emf with internal resistance , a resistor , and two capacitors and , as shown in Figure 1. The two capacitors are connected in parallel with each other, and that parallel combination is connected in series with the resistor and the battery. A switch is initially open, so no current flows and both capacitors are uncharged. At time the switch is closed and the circuit begins charging. The circuit is allowed to reach steady state.
Figure 1. Series circuit containing a 12.0 V battery with internal resistance 1.0 Ω, a switch S, a 5.0 Ω resistor, and a parallel combination of capacitors C1 = 6.0 µF and C2 = 3.0 µF.
A student claims that immediately after the switch is closed at , the current in the circuit is determined only by the resistors and because the uncharged capacitors initially behave like a wire, and that at very long times the current becomes zero because charges stop moving through the circuit.
Indicate whether the student's claim is correct or incorrect. Without manipulating equations, justify your answer by describing (i) the movement of electric charges in the circuit at and as and (ii) how the potential difference across the capacitor combination changes during the charging process.
Derive an expression for the initial current in the circuit at and an expression for the steady-state potential difference across the parallel capacitor combination as . Express your answers in terms of , , and only. Begin your derivation by writing Kirchhoff's loop rule for the circuit at the appropriate time.
Indicate whether your expressions from part B are or are not consistent with your answer from part A. Briefly justify your answer by referencing how the expressions depend on and and what happens to the current and potential differences at and as .
2. A student builds the circuit shown in Figure 1 using an ideal battery, resistors, and capacitors. At time the switch is closed. Before the switch is closed, both capacitors are uncharged. The battery has emf and negligible internal resistance. Resistor and resistor . Capacitor and capacitor . All connecting wires have negligible resistance.
Figure 1. Parallel RC branches connected to a 12.0 V ideal battery through switch S, with junctions A and B labeled.
Figure dot. Dot for charge-motion representation (wire segment just to the right of switch S).
Immediately after the switch is closed at , the capacitors behave as uncharged capacitors.
On the dot shown in Figure dot, representing a small segment of the metal wire just to the right of the switch, draw and label
Each arrow must start on, and point away from, the dot.
Derive an expression for the total current delivered by the battery immediately after the switch is closed in terms of , , and . Begin your derivation by writing a fundamental physics principle or an equation from the reference information. In your derivation, clearly indicate how Kirchhoff's junction rule applies at junction A.
Figure 2. Axes for sketching current through resistor R1 as a function of time.
On the axes provided in Figure 2, sketch the expected relationship between the current through resistor and time from to long after the switch is closed. Draw an arrow on your sketch to indicate the direction of time. Your sketch should be consistent with the behavior of a resistor-capacitor series branch and with the fact that the two branches are in parallel. Long after the switch is closed, the circuit reaches steady state. At that time, the capacitors are fully charged and no longer changing charge.
Indicate whether the total energy stored in the capacitors in the modified circuit at steady state is greater than, less than, or equal to the total energy stored in the capacitors in the original circuit at steady state. The student now modifies the circuit by removing capacitor and replacing it with a new capacitor while keeping all other components the same. The switch is again closed at , and long after the switch is closed the circuit reaches steady state.
Given values: ; ; ; ; .
Greater than
Less than
Equal to
Briefly justify your answer by referencing at least one feature of your answers to parts B or C and by using an expression for energy stored in a capacitor. If you make a calculation, show enough work to support your claim.
| Term | Definition |
|---|---|
| emf | The energy per unit charge provided by a battery or voltage source, measured in volts, that drives current through a circuit. |
| R = ρℓ/A | The formula relating resistance to resistivity, length, and cross-sectional area of a uniform conductor; longer or thinner wires have higher resistance. |
| I-V graph | A graph of current versus potential difference; an ohmic resistor produces a straight line with slope equal to 1/R. |
| Joule heating | Thermal energy dissipated in a resistor due to current flow, given by P = I²R or P = (ΔV)²/R. |
| P = IV | The fundamental formula for electrical power; the rate at which energy is transferred or dissipated equals current times potential difference. |
| series resistors | Resistors connected end-to-end so the same current flows through each; equivalent resistance is the direct sum R_eq = ΣR_i. |
| parallel resistors | Resistors sharing the same potential difference across each branch; equivalent resistance satisfies 1/R_eq = Σ(1/R_i). |
| Voltage divider | A series resistor configuration where each resistor's voltage is proportional to its fraction of the total resistance. |
| open circuit | A circuit with a broken path where charge cannot flow; current is zero throughout the loop. |
| short circuit | A low-resistance path that allows charge to flow with no potential difference across the bypassed elements. |
| conservation of charge | The principle underlying Kirchhoff's junction rule: current entering a node must equal current leaving it because charge cannot accumulate there. |
| equivalent capacitance | A single capacitance representing multiple capacitors; series combinations give 1/C_eq = Σ(1/C_i), parallel combinations give C_eq = ΣC_i. |
| time constant | τ = R_eq C_eq; the time for a charging capacitor to reach 63% of final charge, or a discharging capacitor to fall to 37% of initial charge. |
| Transient response | The time-dependent circuit behavior after a switch changes state; an uncharged capacitor initially acts as a short circuit and approaches open-circuit behavior at steady state. |
| steady state | The condition after many time constants in an RC circuit where capacitor voltage and branch currents no longer change; no current flows through the capacitor branch. |
P = I²R is most useful in series circuits where current is the same through all elements. P = (ΔV)²/R is most useful in parallel circuits where voltage is the same across all branches. Mixing these up leads to incorrect brightness rankings and power calculations.
Students often apply the series resistor rule (add directly) to capacitors. For capacitors in series, 1/C_eq = Σ(1/C_i), so the equivalent capacitance is less than the smallest individual capacitor, which is the opposite of what happens with resistors.
The emf of a battery is not the same as the voltage it delivers when current flows. Terminal voltage = ε - Ir, so a battery under load always delivers less than its rated emf. Treating ε as the terminal voltage overestimates current and power in the circuit.
A potential drop across a resistor is only negative if you traverse it in the direction of current. If you traverse it opposite to current, it is a potential rise. Inconsistent sign choices produce wrong loop equations even when the algebra is correct.
Immediately after a switch closes, an uncharged capacitor acts as a short circuit (zero voltage across it, maximum current through it). Only after a long time (many time constants) does it act as an open circuit. Reversing these initial and final conditions is a frequent error in RC circuit analysis.
AP Physics 2 frequently asks you to predict what happens to current, voltage, or bulb brightness when a circuit element is added, removed, or changed. Practice explaining these changes using Ohm's law, power formulas, and the series/parallel rules rather than just calculating numbers. Justify your reasoning in terms of how equivalent resistance or voltage distribution shifts.
Free-response questions often present a circuit with multiple loops and ask you to write loop and junction equations, solve for unknown currents, and then calculate power or potential difference at a specific element. Show your sign convention explicitly and verify that your junction equations are consistent with your loop equations.
Exam questions on RC circuits commonly ask you to describe or sketch how current or capacitor voltage changes from the moment a switch closes to steady state, identify the time constant, and explain the physical meaning of the initial and final conditions. Be prepared to connect the 63%/37% benchmarks to the definition of τ and to explain why a fully charged capacitor carries no steady-state current.
Open the individual guides for Unit 11 when you want a closer review of one topic.
browse guidesPractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Physics 2 Unit 11 covers 8 topics built around capacitors and circuit analysis: Current and Resistance, Electric Power, Resistance/Resistivity/Ohm's Law, Series and Parallel Circuits, Analysis of Circuits, Capacitors in Circuits, RC Circuits, and Electrical Power in Circuits. Together they connect conservation of energy to real circuit behavior. See the full topic breakdown at AP Physics 2 Unit 11.
Unit 11 makes up 15-18% of the AP Physics 2 exam, making it one of the heavier-weighted units. It covers electric circuits topics including capacitors, series and parallel circuits, Ohm's Law, RC circuits, and electrical power. Expect multiple MCQ questions and at least one FRQ that draws from this material.
The AP Physics 2 Unit 11 progress check includes both MCQ and FRQ parts drawn from all 8 unit topics. MCQ questions test current and resistance, Ohm's Law, and series and parallel circuits. The FRQ portion typically asks you to analyze a circuit, work with capacitors, or explain RC circuit behavior quantitatively and conceptually. Practice with questions matched to every progress check topic at AP Physics 2 Unit 11.
The best way to practice AP Physics 2 Unit 11 FRQs is to focus on the three highest-yield topics: Analysis of Circuits, Capacitors in Circuits, and RC Circuits. FRQs from this unit typically ask you to derive current or voltage using Kirchhoff's rules, explain how adding a capacitor changes circuit behavior, or sketch and interpret RC charging/discharging graphs. Start by writing out full solutions, not just plugging numbers. Show your reasoning for each step, since College Board awards points for justification. Find FRQ-style practice sets at AP Physics 2 Unit 11.
You can find AP Physics 2 Unit 11 practice questions, including MCQ and practice test sets, at AP Physics 2 Unit 11. The page organizes practice by topic, so you can drill series and parallel circuits separately from capacitors or RC circuits before taking a full unit practice test. For the best results, mix MCQ drills with at least one timed FRQ attempt per study session.
Start AP Physics 2 Unit 11 by building a solid foundation in current and resistance and Ohm's Law before moving to series and parallel circuits, since later topics like RC circuits and capacitors stack directly on those ideas. Here's a practical study sequence: 1. **Topics 11.1-11.3** (Current, Resistance, Ohm's Law): Practice drawing V-I graphs and using R = V/I in multiple forms. 2. **Topic 11.4** (Series and Parallel Circuits): Redraw every circuit you see. Label current paths and voltage drops before calculating anything. 3. **Topics 11.5 and 11.8** (Analysis and Electrical Power): Apply Kirchhoff's rules to multi-loop circuits. Connect power dissipation to real components like light bulbs. 4. **Topics 11.6 and 11.7** (Capacitors and RC Circuits): Understand charging and discharging curves conceptually first, then work through the math. After each topic, do a short MCQ check, then finish with a timed FRQ. Find topic-by-topic practice at AP Physics 2 Unit 11.