Review AP Physics C: E&M Unit 11 to build fluency with electric current, resistance, power, Kirchhoff's Rules, and RC circuit behavior. This unit carries 15-25% of the exam weight and connects directly to capacitors from Unit 10 and electromagnetic induction in Unit 13.
Use the topic guides, key terms, and FRQ practice available for this unit to work through every concept from drift velocity to RC time constants.
Electric circuits describe how charge flows through connected components to transfer and dissipate energy. Unit 11 builds from the microscopic picture of drifting charge carriers all the way to multi-loop DC networks and time-dependent RC behavior.
Unit 11 is about analyzing electric circuits: how current flows, how resistance and power are related, how to apply Kirchhoff's Rules to find unknown currents and voltages, and how capacitors charge and discharge through resistors over time.
Current is defined as I = dq/dt and arises from charge carriers drifting with average velocity v_d. Resistance depends on material resistivity and geometry through R = rho*l/A. Ohm's law I = delta-V/R connects these to potential difference.
The loop rule (sum of delta-V = 0) enforces conservation of energy around any closed path. The junction rule (sum I_in = sum I_out) enforces conservation of charge at every node. Together they let you solve any DC resistor network.
When a capacitor charges or discharges through a resistor, charge and current follow exponential functions governed by the time constant tau = R_eq * C_eq. At t = 0 the capacitor acts like a short circuit; at t >> tau it acts like an open circuit.
Every major tool in Unit 11 is a conservation law in disguise. Kirchhoff's loop rule is conservation of energy; the junction rule is conservation of charge. Even the exponential behavior of RC circuits comes from applying the loop rule to a circuit where charge accumulates on a capacitor. Recognizing which conservation principle applies in a given situation is the core skill of this unit.
Defines current as I = dq/dt and connects it to microscopic drift velocity through I = nqv_d*A and current density J = nqv_d. Introduces emf as the driving potential difference.
Covers closed, open, and short circuits; circuit schematic symbols; series and parallel connections; and how a single element can belong to multiple loops.
Derives R = rho*l/A for uniform conductors and extends to variable resistivity with integration. Applies Ohm's law I = delta-V/R and distinguishes ohmic from non-ohmic materials using I-V graphs.
Introduces P = Idelta-V and derived forms P = I^2R and P = delta-V^2/R. Uses power to predict lightbulb brightness and analyze energy dissipation as Joule heating.
Reduces series-parallel resistor networks to equivalent resistance. Models real batteries with internal resistance r and terminal voltage emf - I*r. Covers correct placement and ideal properties of ammeters and voltmeters.
States sum delta-V = 0 around any closed loop as a consequence of energy conservation. Establishes sign conventions for voltage rises and drops and uses potential-versus-position graphs.
States sum I_in = sum I_out at any node as a consequence of charge conservation. Combines with the loop rule to solve multi-loop DC networks with multiple unknown branch currents.
Combines capacitor combination rules with Kirchhoff's loop rule to derive the RC differential equation. Solves charging and discharging behavior using exponential functions with time constant tau = R_eq*C_eq.
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Review Electric Current with attention to how the concept appears in AP-style source and evidence questions.
Review Resistor-Capacitor (RC) Circuits with attention to how the concept appears in AP-style source and evidence questions.
Review Compound Direct Current Circuits with attention to how the concept appears in AP-style source and evidence questions.
Review Resistance, Resistivity, and Ohm's Law with attention to how the concept appears in AP-style source and evidence questions.
Current is the rate of charge flow through a cross-sectional area: I = dq/dt. Inside a conductor, positive charge carriers drift with average velocity v_d, giving I = nqv_dA. Current density J = nqv_d is a vector, and the total current through a surface is I = integral of J dot dA. Microscopic Ohm's law connects field and density: J = sigmaE, where sigma is conductivity. Even when net current is zero, individual carriers still move randomly; only the net drift stops.
If a wire carries 3 A and has 5 x 10^28 free electrons per cubic meter with cross-section 2 x 10^-6 m^2, what is the drift velocity? Use v_d = I / (nqA).
| Quantity | Symbol | Equation | Units |
|---|---|---|---|
| Current | I | dq/dt | A |
| Current density | J | nqv_d | A/m^2 |
| Drift velocity | v_d | I / (nqA) | m/s |
A circuit is a set of closed loops containing elements such as batteries, resistors, capacitors, inductors, and switches. A closed circuit allows charge to flow; an open circuit has a break that stops flow; a short circuit provides a zero-resistance path with no potential difference change. Circuit schematics use standard symbols for each element. Series and parallel are the two fundamental connection types, and a single element can belong to multiple loops simultaneously.
Draw a schematic with a battery, two resistors in series, and a voltmeter placed correctly across the second resistor. Identify which loop the voltmeter belongs to.
| Circuit type | Charge flow? | Potential difference across break |
|---|---|---|
| Closed | Yes | N/A |
| Open | No | Nonzero |
| Short | Yes (bypassed) | Zero |
Resistance quantifies how strongly an object opposes current. For a uniform conductor, R = rho*l/A, where rho is resistivity, l is length, and A is cross-sectional area. If resistivity varies along the length, integrate: R = integral of rho(l) dl / A. Ohm's law, I = delta-V / R, applies to ohmic materials that maintain constant resistance regardless of current. On an I-V graph, ohmic materials produce a straight line through the origin; the slope equals conductance 1/R. Resistivity in metals typically increases with temperature.
A wire of resistivity 1.7 x 10^-8 ohm-m, length 2 m, and diameter 1 mm carries a current. Calculate its resistance using R = rho*l/A.
| Property | Series resistors | Parallel resistors |
|---|---|---|
| Current | Same through each | Splits among branches |
| Voltage | Divides proportionally | Same across each |
| Equivalent R | R_eq = sum R_i | 1/R_eq = sum 1/R_i |
Power is the rate of energy transfer by a circuit element: P = I * delta-V. Substituting Ohm's law gives two derived forms: P = I^2 * R (useful when current is known) and P = delta-V^2 / R (useful when voltage is known). Energy dissipated in a resistor appears as thermal energy, called Joule heating. Lightbulb brightness increases with power, so comparing P values lets you rank brightness without calculating exact temperatures. In series circuits, the largest resistor dissipates the most power; in parallel circuits, the smallest resistor dissipates the most power.
Two resistors, 4 ohms and 8 ohms, are connected in parallel across a 12 V battery. Which dissipates more power, and by how much? Use P = delta-V^2 / R for each.
| Configuration | Which R has highest P? | Formula to use |
|---|---|---|
| Series | Largest R (same I) | P = I^2 R |
| Parallel | Smallest R (same V) | P = V^2 / R |
Compound circuits mix series and parallel connections. Simplify by finding equivalent resistances step by step: R_eq,series = sum R_i and 1/R_eq,parallel = sum 1/R_i. A real battery has internal resistance r modeled as a resistor in series with an ideal emf source. Terminal voltage is delta-V_terminal = emf - I*r, so it drops below emf when current flows. Ammeters (zero resistance, placed in series) measure current; voltmeters (infinite resistance, placed in parallel) measure potential difference. Nonideal meters alter the circuit they measure.
A battery with emf 9 V and internal resistance 1 ohm drives a 4-ohm external resistor. Find the current, terminal voltage, and power lost to internal resistance.
| Meter | Connection | Ideal resistance | Measures |
|---|---|---|---|
| Ammeter | Series | Zero | Current at a point |
| Voltmeter | Parallel | Infinite | Potential difference across element |
Kirchhoff's loop rule states that the sum of all potential differences around any closed loop equals zero: sum delta-V = 0. This is conservation of energy applied to circuits. To use it, choose a loop direction, assign a voltage rise to each battery traversed from minus to plus terminal, and assign a voltage drop of I*R to each resistor traversed in the direction of current. Write one equation per independent loop. You can also graph electric potential versus position around the loop to visualize where energy is gained and lost.
Write the loop equation for a single loop containing a 12 V battery, a 3-ohm resistor, and a 5-ohm resistor all in series. Solve for the current.
Kirchhoff's junction rule states that the total current entering any junction equals the total current leaving it: sum I_in = sum I_out. This is conservation of electric charge applied to circuit nodes. In a multi-loop circuit, label each branch with its own current variable, apply the junction rule at each independent node, and combine with loop equations to solve the system. The junction rule is especially important when parallel branches carry different currents.
At a junction, currents of 2 A and 3 A enter. Two branches leave. If one carries 1 A, what does the other carry? Apply sum I_in = sum I_out.
| Rule | Conservation law | Equation | Applied at |
|---|---|---|---|
| Loop rule | Conservation of energy | sum delta-V = 0 | Any closed loop |
| Junction rule | Conservation of charge | sum I_in = sum I_out | Any circuit node |
Capacitors in series combine as 1/C_eq = sum 1/C_i; capacitors in parallel combine as C_eq = sum C_i (opposite rules from resistors). In an RC circuit, applying Kirchhoff's loop rule gives the differential equation emf = R(dq/dt) + q/C. The solution for charging is q(t) = Cemf(1 - e^(-t/RC)) and for discharging q(t) = Q_0 * e^(-t/RC). The time constant tau = R_eq * C_eq sets the timescale: after one tau, a charging capacitor reaches about 63% of its final charge; a discharging capacitor retains about 37%. At t = 0, a capacitor acts like a wire (short circuit); at t >> tau, it acts like an open circuit with no current flowing through its branch.
A 10-microfarad capacitor charges through a 5-kilohm resistor from a 20 V source. Find tau, the charge at t = tau, and the current at t = 0.
| Combination | Capacitors | Resistors |
|---|---|---|
| Series | 1/C_eq = sum 1/C_i | R_eq = sum R_i |
| Parallel | C_eq = sum C_i | 1/R_eq = sum 1/R_i |
Try AP-style multiple-choice questions and written prompts after you review the notes.
A timed, AP-style test built only from unit 11 questions — multiple choice plus free response, scored with AI feedback. Use the standard format or build your own.
| Term | Definition |
|---|---|
| drift velocity | The average velocity at which charge carriers move through a conductor in response to an applied electric field; appears in I = nqv_d*A. |
| conventional current | The direction of current defined as the direction positive charges would move, opposite to electron flow in metal conductors. |
| electromotive force | The energy per unit charge provided by a source (such as a battery) to drive charge around a circuit; symbol emf or epsilon. |
| Joule heating | Thermal energy generated in a resistor by current flow at a rate equal to I^2*R; represents conversion of electrical energy to heat. |
| series connection | A circuit configuration where charge must pass through all elements in sequence; current is the same through each element. |
| parallel connection | A circuit configuration where charge can take multiple paths; the same potential difference appears across each branch. |
| internal resistance | Resistance r inside a real battery modeled as a series resistor; causes terminal voltage to equal emf - I*r when current flows. |
| terminal voltage | The potential difference measured across a battery's terminals; equals emf minus the voltage drop across internal resistance. |
| conservation of electric charge | The principle that charge is neither created nor destroyed at a circuit junction; the basis of Kirchhoff's junction rule. |
| equivalent capacitance | A single capacitance representing multiple capacitors: sum C_i for parallel, reciprocal sum for series (opposite of resistor rules). |
| steady state | The condition in an RC circuit after t >> tau when the capacitor is fully charged or discharged and no current flows through its branch. |
| transient response | The time-dependent exponential behavior of charge and current in an RC circuit between t = 0 and steady state. |
| open circuit | A circuit with a break in the loop so no charge flows; a fully charged capacitor in an RC circuit behaves as an open circuit at steady state. |
| short circuit | A path with negligible resistance and no potential difference change; an uncharged capacitor behaves as a short circuit at t = 0. |
Capacitor combination rules are the reverse of resistor rules. Capacitors in series use the reciprocal formula; capacitors in parallel add directly. Applying resistor rules to capacitors is one of the most frequent errors on RC circuit problems.
P = I^2R and P = delta-V^2/R give different answers if you substitute inconsistent values. In parallel circuits, voltage is the same across branches, so P = delta-V^2/R is the cleaner choice. In series circuits, current is the same, so P = I^2R avoids extra steps.
The terminal voltage of a real battery is emf - Ir, not just emf. Forgetting the Ir drop leads to incorrect currents and power calculations throughout the circuit.
You must choose a loop direction and apply it consistently. A resistor traversed against the current direction gives a voltage rise of +I*R, not a drop. Switching conventions mid-loop produces wrong equations.
At the instant a switch closes, an uncharged capacitor has zero voltage across it and acts like a short circuit, allowing maximum current. The open-circuit behavior only applies at steady state (t >> tau).
Free-response questions in AP Physics C: E&M frequently ask you to apply Kirchhoff's loop rule to an RC circuit, write the resulting differential equation emf = R(dq/dt) + q/C, and solve it to obtain q(t) or I(t). You may also be asked to sketch the exponential graph, identify tau, and describe initial and steady-state conditions in words.
Expect problems that require setting up a system of equations from loop and junction rules simultaneously. Skills tested include assigning current directions, writing consistent sign-convention loop equations, and solving for multiple unknown branch currents or voltages. Graphing potential versus position around a loop is another common task.
Questions often ask you to predict how adding or removing a resistor changes current, voltage, and power throughout a circuit, or to rank lightbulb brightness before and after a switch opens or closes. These tasks require you to reason about equivalent resistance changes and apply power formulas without necessarily solving the full circuit numerically.
Use this list to confirm you can handle every major skill in Unit 11 before exam day.
Apply I = dq/dt and I = nqv_d*A to find current or drift velocity. Integrate J over a cross-section when current density is non-uniform.
Compute resistance from material and geometry. Apply I = delta-V/R and identify ohmic behavior from a linear I-V graph.
Use P = Idelta-V, P = I^2R, and P = delta-V^2/R. Rank lightbulb brightness by comparing power dissipated in each branch.
Combine series and parallel resistors step by step to find equivalent resistance. Account for internal resistance and compute terminal voltage.
Write independent loop equations using sum delta-V = 0 and junction equations using sum I_in = sum I_out. Solve the resulting system for all unknown currents and voltages.
Find tau = R_eq*C_eq. Write and interpret charging and discharging equations. Identify initial (short-circuit) and steady-state (open-circuit) capacitor behavior.
Read the topic guides for 11.1 and 11.2. Practice converting between I = dq/dt and I = nqv_d*A. Draw and label circuit schematics with correct symbols. Identify closed, open, and short circuits in diagrams.
Work through R = rho*l/A problems including variable-resistivity integrals. Sketch I-V graphs for ohmic and non-ohmic elements. Practice all three power formulas and rank bulb brightness in series and parallel examples.
Reduce multi-resistor networks step by step. Include internal resistance in battery problems and compute terminal voltage. Practice placing ammeters and voltmeters correctly in circuit diagrams.
Write loop equations using sum delta-V = 0 with consistent sign conventions. Write junction equations using sum I_in = sum I_out. Set up and solve systems of equations for two- and three-loop circuits. Sketch potential-versus-position graphs for a loop.
Combine capacitors in series and parallel. Derive the RC differential equation from Kirchhoff's loop rule. Practice writing q(t) and I(t) for charging and discharging. Identify initial and steady-state conditions and calculate tau for given circuits. Use the AP score calculator to estimate your overall exam score as you complete practice.
Open the individual guides for Unit 11 when you want a closer review of one topic.
Use AP-style practice after you review the notes so you can check what you understand.
Practice free-response reasoning and compare your answer with scoring guidance.
Open the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
Estimate your broader AP score goal after you review the course and exam format.
Print the unit cheatsheets ($3) or unit exams ($5) for Unit 11. Yours to keep, no subscription needed.
get printablesAP Physics E&M Unit 11 covers 8 topics centered on resistance and electric circuits: Electric Current, Simple Circuits, Resistance/Resistivity and Ohm's Law, Electric Power, Compound Direct Current Circuits, Kirchhoff's Loop Rule, Kirchhoff's Junction Rule, and Resistor-Capacitor (RC) Circuits. The unit builds from basic current and resistance up to analyzing multi-loop circuits with Kirchhoff's Rules, then finishes with RC circuits where capacitors charge and discharge over time. See AP Physics E&M Unit 11 for matched practice on each topic.
Unit 11 makes up 15-25% of the AP Physics E&M exam, making it one of the most heavily weighted units. That weight reflects how central resistance and electric circuits are to the course. Topics like Ohm's Law, Kirchhoff's Rules, and RC circuits all appear regularly on both the multiple-choice and free-response sections.
The AP Physics E&M Unit 11 progress check includes both MCQ and FRQ parts drawn from all 8 topics in the unit. The MCQ section tests resistance, Ohm's Law, electric power, and circuit analysis with Kirchhoff's Rules. The FRQ part typically asks you to analyze a compound DC circuit or an RC circuit, set up equations, and interpret results. College Board designs the progress check to mirror real exam difficulty, so it's one of the best checkpoints before test day. Head to AP Physics E&M Unit 11 for practice questions matched to each progress check topic.
The best way to practice AP Physics E&M Unit 11 FRQs is to focus on the topics that generate the most free-response questions: Compound DC Circuits, Kirchhoff's Loop and Junction Rules, and RC Circuits. FRQs in this unit typically ask you to derive an expression for resistance or current, apply Kirchhoff's Rules to a multi-loop circuit, or sketch and interpret a charge-vs-time graph for an RC circuit. For each practice problem, write out every step explicitly, because College Board awards points for correct setup even if your final answer is off. You can find FRQ-style problems organized by topic at AP Physics E&M Unit 11.
For AP Physics E&M Unit 11 practice questions, including multiple-choice and practice test sets, go to AP Physics E&M Unit 11. That page organizes MCQ and FRQ practice by topic, covering resistance and Ohm's Law, Kirchhoff's Rules, electric power, and RC circuits. For the best results, mix MCQ drills with full practice test sections so you get comfortable with both the calculation-heavy multiple-choice questions and the longer free-response format this unit is known for.
Start with resistance and Ohm's Law (Topic 11.3) before anything else, since that relationship underpins every other topic in the unit. From there, build up to series and parallel circuits (11.5), then tackle Kirchhoff's Loop Rule (11.6) and Junction Rule (11.7) together since they work as a system for solving multi-loop problems. Finish with RC circuits (11.8), which add a time-dependent layer on top of everything before. A few concrete steps that help: - Draw every circuit diagram yourself rather than just reading the textbook's version. - Practice setting up Kirchhoff's equations before solving them, since setup is where most points are lost on FRQs. - For RC circuits, make sure you can sketch charge and current vs. time graphs and explain what the time constant means physically. Check AP Physics E&M Unit 11 for topic-by-topic practice to test your understanding as you go.