Review AP Physics C: E&M Unit 13 to build fluency with magnetic flux, Faraday's law, Lenz's law, inductance, and the transient and oscillatory behavior of LR and LC circuits. This unit connects the magnetic field concepts from Unit 12 to real circuit behavior and carries up to 20% of the exam.
Use the topic guides, practice questions, FRQ practice, and score calculator available for this unit to work through every concept before exam day.
Electromagnetic induction is the mechanism behind electric generators, transformers, wireless charging, and Wi-Fi. Unit 13 gives you the mathematical tools to quantify how changing magnetic flux produces electric potential differences and how inductors store and release energy in circuits.
Unit 13 covers how a changing magnetic flux induces an emf (Faraday's law), how the direction of that emf is determined (Lenz's law), how inductors behave as circuit elements, and how LR and LC circuits evolve over time.
Magnetic flux Φ_B = ∫ B · dA is the foundation. Faraday's law states that any change in flux through a loop induces an emf equal to -dΦ_B/dt. The negative sign encodes Lenz's law: the induced current opposes the flux change that caused it.
In an LR circuit, current grows or decays exponentially with time constant τ = L/R. In an LC circuit with no resistance, energy oscillates between the capacitor and inductor at angular frequency ω = 1/√(LC), analogous to simple harmonic motion.
Every topic in Unit 13 follows from one principle: a changing magnetic flux drives an electric effect. Flux (13.1) defines what changes. Faraday's and Lenz's laws (13.2) quantify the induced emf. Forces on induced currents (13.3) show the mechanical consequences. Inductance (13.4) describes how a conductor's own changing current creates flux. LR circuits (13.5) and LC circuits (13.6) show how that inductance shapes time-dependent circuit behavior.
Define and calculate Φ_B = ∫ B · dA. Understand the area vector orientation, the role of the dot product in determining sign, and when to use the integral form versus the simple BA cos θ formula.
Apply Faraday's law emf = -dΦ_B/dt to find induced emf when B, A, or the angle between them changes. Use Lenz's law to determine the direction of the induced current. Handle the solenoid case with N turns.
Calculate the induced current using I = emf/R, then find the magnetic force on current-carrying segments via F_B = ∫ I(dℓ × B). Apply Newton's second law to determine translational or rotational acceleration of the loop.
Calculate solenoid inductance with L_sol = μ_core N² A / ℓ. Find the back emf using emf_L = -L(dI/dt) and the stored energy using U_L = (1/2)LI². Understand how each physical parameter of the solenoid affects L.
Derive and solve the LR differential equation emf = L(dI/dt) + IR. Interpret the time constant τ = L/R, sketch current vs. time for growth and decay, and identify steady-state behavior.
Recognize the SHM structure of d²q/dt² = -(1/LC)q. Calculate ω = 1/√(LC), find I_max from energy conservation, and describe the phase relationship between charge and current throughout the oscillation cycle.
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Review Inductance with attention to how the concept appears in AP-style source and evidence questions.
Review Induced Currents and Magnetic Forces with attention to how the concept appears in AP-style source and evidence questions.
Review Circuits with Resistors and Inductors (LR Circuits) with attention to how the concept appears in AP-style source and evidence questions.
Review Magnetic Flux with attention to how the concept appears in AP-style source and evidence questions.
Magnetic flux Φ_B measures how much magnetic field passes through a surface. For a uniform field, Φ_B = B · A = BA cos θ, where θ is the angle between the field vector and the area vector. For a nonuniform field or curved surface, you must use the surface integral Φ_B = ∫ B · dA. The area vector points perpendicular to the surface and outward for a closed surface. The sign of flux is determined by the dot product: flux is positive when B and the area vector point in the same general direction and negative when they oppose.
A flat circular loop of area 0.05 m² sits in a uniform 2 T field at 30° to the plane of the loop. What is the magnetic flux through the loop?
| Situation | Formula to use | Key variable |
|---|---|---|
| Uniform B, flat surface | Φ_B = BA cos θ | Angle θ between B and area vector |
| Nonuniform B or curved surface | Φ_B = ∫ B · dA | Differential area element dA |
Faraday's law states that the induced emf in a loop equals the negative rate of change of magnetic flux: emf = -dΦ_B/dt. If the area is constant, emf = -A(dB_perp/dt). If the field is constant but the area changes, emf = -B(dA_perp/dt). For a solenoid with N turns, the total induced emf is |emf_sol| = N|dΦ_B/dt|. Lenz's law gives the direction: the induced current flows in the direction that opposes the change in flux. A loop entering a region of increasing flux will have an induced current that creates a field opposing the increase.
A square loop of side 0.1 m is in a region where B increases at 3 T/s perpendicular to the loop. What is the magnitude of the induced emf, and in which direction does the induced current flow?
| What changes | Induced emf expression | Example |
|---|---|---|
| Magnetic field B (constant area) | emf = -A(dB/dt) | Loop in a time-varying solenoid field |
| Area A (constant field) | emf = -B(dA/dt) | Conducting rod sliding on rails |
| Both B and A change | emf = -d(BA cos θ)/dt | Rotating loop in a uniform field |
When flux through a conducting loop changes, an induced current I = emf/R flows. That current-carrying loop sits in the external magnetic field, so the field exerts a force on it via F_B = ∫ I(dℓ × B). Only the segments of the loop actually inside the external field experience a net force. The result can be translational acceleration (a loop entering or exiting a field region) or rotational acceleration (a loop in a torque-producing geometry). By Lenz's law, the net magnetic force always opposes the motion that caused the flux change, which is the basis of magnetic braking and eddy current damping.
A rectangular loop of resistance R moves at constant velocity into a uniform field B. Identify which segments carry induced current, which experience a magnetic force, and in what direction the net force acts.
| Motion type | Relevant equation | Physical effect |
|---|---|---|
| Translational (loop entering field) | F = BIℓ = B²ℓ²v/R | Retarding force opposing entry |
| Rotational (loop in torque geometry) | τ = μ × B = IAB sin θ | Angular acceleration of the loop |
Inductance L is the property of a conductor that opposes changes in current by generating a back emf: emf_L = -L(dI/dt). For a solenoid, L_sol = μ_core N² A / ℓ, so inductance increases with more turns, larger cross-sectional area, shorter length, and higher core permeability. Straight wires are modeled as having zero inductance. An inductor stores energy in its magnetic field: U_L = (1/2)LI². That stored energy can later be dissipated through a resistor or transferred to a capacitor.
A solenoid has 500 turns, length 0.2 m, cross-sectional area 4 × 10⁻⁴ m², and an air core. Calculate its inductance and the energy stored when 2 A flows through it.
| Physical change to solenoid | Effect on L |
|---|---|
| Double the number of turns N | L increases by factor of 4 (L ∝ N²) |
| Double the length ℓ | L decreases by factor of 2 (L ∝ 1/ℓ) |
| Double the cross-sectional area A | L increases by factor of 2 (L ∝ A) |
| Replace air core with ferrite core | L increases proportional to μ_core |
Applying Kirchhoff's loop rule to a series LR circuit with battery emf gives the differential equation emf = L(dI/dt) + IR. The solution is an exponential function with time constant τ = L/R_eq. When the switch closes, current grows as I(t) = (emf/R)(1 - e^(-t/τ)). When the battery is removed, current decays as I(t) = I_0 e^(-t/τ). At steady state (t >> τ), the inductor acts like a short circuit (zero voltage drop across an ideal inductor) and current equals emf/R. At t = τ, the current has reached about 63% of its final value during growth or dropped to about 37% during decay.
An LR circuit has emf = 12 V, R = 4 Ω, and L = 0.2 H. Find τ, the steady-state current, and the current at t = τ after the switch closes.
| Quantity | At t = 0 | At t = τ | At t >> τ (steady state) |
|---|---|---|---|
| Current I | 0 (switch just closed) | 0.63 × (emf/R) | emf/R |
| Voltage across L | emf | 0.37 × emf | 0 |
| Voltage across R | 0 | 0.63 × emf | emf |
An ideal LC circuit with no resistance allows energy to oscillate indefinitely between the electric field of the capacitor and the magnetic field of the inductor. Applying Kirchhoff's loop rule gives the differential equation d²q/dt² = -(1/LC)q, which has the same form as simple harmonic motion. The angular frequency of oscillation is ω = 1/√(LC). The maximum current occurs when the capacitor is fully discharged, and conservation of energy gives I_max = Q_0/√(LC). Charge and current are 90° out of phase: when charge on the capacitor is maximum, current is zero, and vice versa.
An LC circuit has L = 0.1 H and C = 10 μF. Find ω, the period T, and the maximum current if the capacitor is initially charged to Q_0 = 2 × 10⁻³ C.
| LC circuit quantity | SHM analogy | Formula |
|---|---|---|
| Charge q | Displacement x | q(t) = Q_0 cos(ωt) |
| Current I = dq/dt | Velocity v = dx/dt | I(t) = -Q_0 ω sin(ωt) |
| Inductance L | Mass m | Opposes change |
| 1/C (inverse capacitance) | Spring constant k | Restoring tendency |
| Angular frequency ω | ω = √(k/m) | ω = 1/√(LC) |
Try AP-style multiple-choice questions and written prompts after you review the notes.
A timed, AP-style test built only from unit 13 questions — multiple choice plus free response, scored with AI feedback. Use the standard format or build your own.
| Term | Definition |
|---|---|
| area vector | A vector perpendicular to the plane of a surface with magnitude equal to the area; for a closed surface it points outward. Used in Φ_B = B · A to determine both the magnitude and sign of magnetic flux. |
| dot product | The scalar product B · A = BA cos θ that determines the component of B perpendicular to the surface; sets both the magnitude and sign of magnetic flux. |
| dI/dt | Rate of change of current with respect to time; the back emf across an inductor is emf_L = -L(dI/dt), so a rapidly changing current produces a large opposing voltage. |
| energy stored in an inductor | U_L = (1/2)LI²; the energy held in the magnetic field of an inductor carrying current I. This energy can be dissipated through a resistor or transferred to a capacitor. |
| ideal inductor | An inductor with negligible resistance. At steady state in an LR circuit, an ideal inductor has zero potential difference across it and acts like a short circuit. |
| steady state | The condition in an LR circuit after many time constants when dI/dt = 0, the inductor voltage is zero, and current equals emf/R. |
| RL circuit | A series circuit containing a resistor and inductor governed by emf = L(dI/dt) + IR. Current changes exponentially with time constant τ = L/R. |
| LC circuit | A circuit containing only a capacitor and an inductor. Energy oscillates between electric energy in the capacitor and magnetic energy in the inductor at angular frequency ω = 1/√(LC). |
| angular frequency | ω = 1/√(LC) for an LC circuit; the rate of charge oscillation in radians per second, determined entirely by the values of L and C. |
| simple harmonic motion | The oscillatory behavior described by d²q/dt² = -(1/LC)q in an LC circuit. Charge q plays the role of displacement, L plays the role of mass, and 1/C plays the role of spring constant. |
| Maximum emf | The peak induced emf in a rotating loop, emf_max = NBAω, occurring when the plane of the loop is parallel to the field and the rate of flux change is greatest. |
| Sinusoidal emf | The time-varying emf produced by a loop rotating at constant angular speed in a uniform field: emf(t) = emf_max sin(ωt), a direct consequence of Faraday's law. |
| F_B = ∫I(dℓ × B) | The magnetic force on a current-carrying conductor; in Topic 13.3, this force acts on the segments of an induced-current loop that lie within the external magnetic field. |
Φ_B = BA cos θ uses the angle between B and the area vector, not the angle between B and the plane of the loop. If B is parallel to the plane, θ = 90° and flux is zero. Confusing these two angles is a frequent error.
The negative sign is Lenz's law in mathematical form. Dropping it means you get the magnitude of the induced emf but assign the wrong direction to the induced current, which affects force and torque calculations in Topic 13.3.
At the instant a switch closes, an inductor with zero initial current acts like an open circuit (infinite opposition to current change), not a short circuit. At steady state it acts like a wire. Mixing these two limits is a common LR circuit error.
L_sol scales as N², not N. Doubling the number of turns quadruples the inductance. Students often write L ∝ N and underestimate the effect of adding turns.
ω = 1/√(LC) is an angular frequency in rad/s for oscillation; τ = L/R is a time constant in seconds for exponential decay. They are different quantities with different units and should not be substituted for each other.
AP Physics C: E&M free-response questions frequently ask you to apply Kirchhoff's loop rule to an LR or LC circuit, write the resulting differential equation, and either solve it or verify that a given exponential or sinusoidal function is a solution. You should be comfortable showing that I(t) = (emf/R)(1 - e^(-t/τ)) satisfies emf = L(dI/dt) + IR and that q(t) = Q_0 cos(ωt) satisfies d²q/dt² = -(1/LC)q.
A common exam task chains together several Unit 13 skills: calculate the flux through a moving or rotating loop, apply Faraday's law to find the induced emf, use Ohm's law to find the induced current, then apply F_B = ∫ I(dℓ × B) to find the retarding force and Newton's second law to find acceleration. Each step must be shown clearly with correct sign conventions.
Exam questions on LC circuits often ask you to use conservation of energy to find I_max or the voltage across each element at a given moment, then sketch graphs of q(t), I(t), U_C(t), and U_L(t) over one full cycle. For LR circuits, you may need to read or sketch I vs. t graphs and identify τ from the graph. Connecting the mathematical solution to a correctly labeled quantitative graph is a tested skill.
Use this list to confirm you can handle every major skill in the unit before exam day.
Compute Φ_B = BA cos θ for uniform fields and set up Φ_B = ∫ B · dA for nonuniform or curved-surface cases. Correctly orient the area vector and determine the sign of flux.
Use emf = -dΦ_B/dt to find induced emf when B, A, or θ changes. Determine the direction of the induced current using Lenz's law and the right-hand rule.
Find the induced current with I = emf/R, identify which loop segments are in the field, compute the magnetic force via F_B = ∫ I(dℓ × B), and apply Newton's second law to find acceleration.
Calculate L_sol = μ_core N² A / ℓ, find the back emf emf_L = -L(dI/dt), and compute stored energy U_L = (1/2)LI². Predict how changing N, A, ℓ, or core material affects L.
Write and solve the Kirchhoff loop equation for an LR circuit. Use τ = L/R to find the time constant, write I(t) for growth and decay, and identify voltages across L and R at t = 0, t = τ, and steady state.
Identify the SHM analogy, calculate ω = 1/√(LC) and period T = 2π√(LC), use energy conservation to find I_max = Q_0/√(LC), and describe the phase relationship between q(t) and I(t).
Read the Topic 13.1 guide and practice computing Φ_B = BA cos θ for loops at various angles. Then set up the integral form for a nonuniform field. Sketch diagrams showing the area vector and field vector to confirm sign conventions before moving on.
Read the Topic 13.2 guide. Practice differentiating Φ_B with respect to time for cases where B changes, A changes, and the loop rotates. For each scenario, use Lenz's law to determine current direction before checking with the formula.
Read the Topic 13.3 guide. Work problems involving a rectangular loop entering or exiting a field region: find the induced emf, the induced current, the force on each segment, and the net force. Apply Newton's second law to find acceleration.
Read the Topic 13.4 guide. Practice calculating L_sol for solenoids with different parameters and finding U_L = (1/2)LI². Make sure you can derive the back emf emf_L = -L(dI/dt) from Faraday's law applied to the solenoid's own flux.
Read the Topic 13.5 and 13.6 guides back to back. For LR circuits, practice writing I(t) for growth and decay and reading values off exponential graphs. For LC circuits, identify the SHM analogy, calculate ω and T, and use energy conservation to find I_max. Use available practice questions and FRQ practice to test both circuit types under timed conditions.
Open the individual guides for Unit 13 when you want a closer review of one topic.
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get printablesAP Physics E&M Unit 13 covers electromagnetic induction across 6 topics: Magnetic Flux (13.1), Electromagnetic Induction (13.2), Induced Currents and Magnetic Forces (13.3), Inductance (13.4), Circuits with Resistors and Inductors or LR Circuits (13.5), and Circuits with Capacitors and Inductors or LC Circuits (13.6). These topics build on each other, so magnetic flux and Faraday's law come first, then you apply those ideas to real circuit behavior with inductors. See AP Physics E&M Unit 13 for matched practice on each topic.
Electromagnetic induction makes up 10-20% of the AP Physics E&M exam, making it one of the heavier-weighted units. That range covers everything from magnetic flux and Faraday's law to inductance and the behavior of LR and LC circuits. Expect both multiple-choice and free-response questions that test these concepts. Because the weight is that significant, it's worth spending real time on the calculus-based derivations in this unit, not just the conceptual ideas.
The AP Physics E&M Unit 13 progress check in AP Classroom includes both MCQ and FRQ parts drawn from all six topics in the unit. The MCQ section tests conceptual and quantitative understanding of magnetic flux, electromagnetic induction, induced currents, and inductance. The FRQ part typically asks you to analyze LR or LC circuit behavior, derive expressions using Faraday's law, or calculate induced EMF. The progress check is one of the best low-stakes ways to find gaps before the real exam. Head to AP Physics E&M Unit 13 for practice that mirrors the same topic breakdown.
The best way to practice AP Physics E&M Unit 13 FRQs is to focus on the topics that show up most often: electromagnetic induction, inductance, and LR and LC circuit analysis. FRQs in this unit typically ask you to derive an expression for induced EMF using Faraday's law, sketch or interpret current vs. time graphs for LR circuits, or analyze energy storage in an LC circuit. For each problem, write out your setup equation before plugging in numbers, and show the calculus steps clearly since AP graders award method points. Visit AP Physics E&M Unit 13 to find FRQ practice sets organized by topic.
For AP Physics E&M Unit 13 practice questions, including multiple-choice and practice test sets, go to AP Physics E&M Unit 13. That page has MCQ and FRQ practice covering all six topics: magnetic flux, electromagnetic induction, induced currents, inductance, LR circuits, and LC circuits with capacitors and inductors. When you work through MCQs, flag any question involving magnetic flux or inductance that trips you up, then review the matching topic before moving on to a full practice test.
To study AP Physics E&M Unit 13 on electromagnetic induction, start with magnetic flux (13.1) and make sure you can set up and evaluate the flux integral before moving on. Faraday's law and Lenz's law in topic 13.2 are the backbone of the whole unit, so spend extra time there. Here's a concrete plan: - **Topics 13.1-13.3:** Practice drawing field diagrams, calculating magnetic flux, and applying Lenz's law to find induced current direction. - **Topic 13.4:** Work through inductance derivations and understand the energy stored in an inductor. - **Topics 13.5-13.6:** Solve LR and LC circuit differential equations step by step. Connecting LC circuits to capacitors and oscillatory behavior is a common FRQ angle. After each topic, do a short timed problem set, then check your work before moving forward. Visit AP Physics E&M Unit 13 for topic-by-topic practice to structure your review.