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AP Physics C: E&M Unit 13 Review: Electromagnetic Induction

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.

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What is AP Physics C: E&M unit 13?

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.

Flux and Faraday's law

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.

Inductance and energy storage

An inductor opposes changes in current. Its inductance L_sol = μ_core N² A / ℓ depends on geometry and core material. Energy stored is U_L = (1/2)LI², and any change in current produces a back emf of -L dI/dt.

LR and LC circuit dynamics

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.

Why electromagnetic induction unifies the unit

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.

AP Physics C: E&M unit 13 topics

13.1

Magnetic Flux

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.

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13.2

Electromagnetic Induction

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.

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13.3

Induced Currents and Magnetic Forces

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.

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13.4

Inductance

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.

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13.5

Circuits with Resistors and Inductors (LR Circuits)

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.

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13.6

Circuits with Capacitors and Inductors (LC Circuits)

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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Hardest AP Physics C: E&M unit 13 topics

This snapshot uses Fiveable practice activity to show where students tend to miss questions and which review moves are worth prioritizing first.

54%average MCQ accuracy

Across 839 multiple-choice practice attempts for this unit.

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0%average FRQ score

Across 1 scored free-response attempts for this unit.

Hardest topics in unit 13

MCQ miss rate

13.4

Inductance

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54%104 tries

13.3

Induced Currents and Magnetic Forces

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49%177 tries

13.5

Circuits with Resistors and Inductors (LR Circuits)

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48%219 tries

13.1

Magnetic Flux

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38%157 tries

Unit 13 review notes

13.1

Magnetic Flux

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.

Φ_B = B · A: Flux for a uniform field over a flat surface; equals BA cos θ where θ is the angle between B and the area vector.
Φ_B = ∫ B · dA: General surface integral form used when the field varies across the surface or the surface is curved.
Area vector: A vector perpendicular to the surface with magnitude equal to the area; for a closed surface it points outward.
Sign of flux: Positive when B and the area vector have a component in the same direction; negative when they oppose; zero when perpendicular.

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

13.2

Electromagnetic Induction: Faraday's Law and Lenz's Law

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.

emf = -dΦ_B/dt: Faraday's law: the magnitude of the induced emf equals the rate of flux change; the negative sign indicates Lenz's law direction.
Lenz's law: The induced current flows in the direction that opposes the flux change causing it; determines the sign of the induced emf.
|emf_sol| = N|dΦ_B/dt|: For a solenoid, the total induced emf is N times the single-loop emf because flux links all N turns.
Motional emf: When a conductor of length ℓ moves at velocity v perpendicular to field B, the induced emf is emf = Bℓv.

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

13.3

Induced Currents and Magnetic Forces

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.

F_B = ∫ I(dℓ × B): Magnetic force on a current-carrying conductor segment; only segments inside the external field contribute to the net force on the loop.
Induced current I = emf/R: Ohm's law applied to the loop: the induced emf drives a current inversely proportional to the loop's resistance.
Magnetic braking: The opposing force on a loop moving through a field region; arises because the induced current experiences a retarding magnetic force.
Newton's second law application: The net magnetic force on the loop equals ma, allowing calculation of translational or rotational acceleration of the loop.

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

13.4

Inductance

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.

L_sol = μ_core N² A / ℓ: Solenoid inductance depends on core permeability, number of turns squared, cross-sectional area, and inverse of length.
emf_L = -L(dI/dt): Back emf produced by an inductor; opposes the change in current through it.
U_L = (1/2)LI²: Energy stored in the magnetic field of an inductor carrying current I.
dI/dt: Rate of change of current; determines the magnitude of the back emf across an inductor.

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

13.5

LR Circuits

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.

τ = L/R_eq: LR time constant; larger L or smaller R means slower approach to steady state.
I(t) = (emf/R)(1 - e^(-t/τ)): Current growth after switch closes; starts at zero and approaches emf/R asymptotically.
I(t) = I_0 e^(-t/τ): Current decay after battery is removed; starts at I_0 and decays to zero.
Steady state: After many time constants, dI/dt = 0, so the inductor has no voltage drop and current is constant at emf/R.
emf = L(dI/dt) + IR: Kirchhoff's loop rule for a series LR circuit; the first-order linear ODE whose solution gives the transient behavior.

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

13.6

LC Circuits

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.

d²q/dt² = -(1/LC)q: Governing equation for charge in an LC circuit; identical in form to the SHM equation d²x/dt² = -(k/m)x.
ω = 1/√(LC): Angular frequency of LC oscillation; depends only on inductance and capacitance, not on initial conditions.
I_max = Q_0/√(LC): Maximum current in the inductor, found by setting initial capacitor energy equal to maximum inductor energy.
Energy exchange: Total energy U = (1/2)CV² + (1/2)LI² is constant; energy shifts between capacitor and inductor each quarter cycle.
Simple harmonic motion analogy: Charge q maps to displacement x, current I maps to velocity v, L maps to mass m, and 1/C maps to spring constant k.

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)

Practice AP Physics C: E&M unit 13 questions

Try AP-style multiple-choice questions and written prompts after you review the notes.

Example AP-style MCQs

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MCQ

AP-style practice question

Question

An LC circuit contains a $50\ \text{mH}$ inductor. If the maximum current observed in the circuit is $4.0\ \text{A}$ , what is the maximum energy stored in the capacitor during the oscillation cycle?

$0.10\ \text{J}$

$0.20\ \text{J}$

$0.40\ \text{J}$

$0.80\ \text{J}$

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MCQ

AP-style practice question

Question

A series LR circuit with time constant $\tau$ is connected to a DC battery at $t=0$ . At what time $t$ is the magnitude of the potential difference across the resistor equal to the magnitude of the potential difference across the inductor?

The magnitude of the voltages is equal when time $t$ reaches $\tau \ln 2$

The magnitude of the voltages is equal when time $t$ reaches $\tau / 2$

The magnitude of the voltages is equal when time $t$ reaches $\tau$

The magnitude of the voltages is equal when time $t$ reaches $2\tau$

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Example FRQs

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FRQ

Motional EMF in moving conductor bars

4. A U-shaped conducting track lies in the plane of the page, as shown in Figure 1. The rails are separated by a distance $L = 0.30\ \text{m}$ and are connected by a resistor of resistance $R = 2.0\ \Omega$ at the left end. A conducting bar of negligible resistance makes sliding contact with the rails and is pulled to the right at constant speed. A uniform magnetic field of magnitude $B = 0.80\ \text{T}$ is directed into the page over the entire region shown. The bar is at a distance $x$ from the resistor, where $x$ increases as the bar moves to the right. Gravitational effects and friction are negligible.

Figure 1. Sliding conducting bar on parallel rails in a uniform magnetic field (into the page).

At the instant shown in Figure 1, the bar is pulled to the right with speed $v_1 = 1.5\ \text{m/s}$ . The induced potential difference across the resistor is $\Delta V_1$ . Later, the bar is pulled to the right with a greater speed $v_2 = 3.0\ \text{m/s}$ while all other quantities remain the same, and the induced potential difference across the resistor is $\Delta V_2$ .

Indicate whether $\Delta V_2$ is greater than, less than, or equal to $\Delta V_1$ by writing one of the following.

$\Delta V_2 > \Delta V_1$
$\Delta V_2 < \Delta V_1$
$\Delta V_2 = \Delta V_1$

Justify your answer.

Derive an expression for the magnitude of the induced potential difference $\Delta V$ across the resistor in Figure 1 in terms of $B$ , $L$ , and the bar speed $v$ . Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Your derivation must explicitly relate magnetic flux through the loop to the induced emf.

Figure 2. Same rails and sliding bar, with a series RL element (ideal inductor and resistor) at the left end; current direction shown.

Indicate whether the magnitude of the pulling force required to keep the bar moving at constant speed is greater at the instant immediately after the bar begins moving from rest or greater after a long time, by writing one of the following. The circuit is modified as shown in Figure 2 by placing an ideal inductor of inductance $L_{\text{ind}} = 0.50\ \text{H}$ in series with the resistor $R = 2.0\ \Omega$ . The bar continues to be pulled to the right at constant speed $v_1 = 1.5\ \text{m/s}$ in the same magnetic field. Immediately after the bar begins moving from rest, the current in the circuit is very small. After a long time, the circuit reaches a steady state with a constant current.

$F_{\text{initial}} > F_{\text{long}}$
$F_{\text{initial}} < F_{\text{long}}$
$F_{\text{initial}} = F_{\text{long}}$

Briefly justify your answer by referencing your derivation in part B and the physical/electrical properties of an inductor in an $RL$ circuit.

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FRQ

Electromagnetic induction in moving conductor loop

1. A rectangular conducting loop of total resistance $R = 2.0\ \Omega$ is formed by two vertical rails connected by a movable conducting bar of length $\ell = 0.30\ \text{m}$ , as shown in Figure 1. The rails and bar lie in the plane of the page. A uniform external magnetic field of magnitude $B = 0.80\ \text{T}$ is directed into the page in a region that begins at a vertical boundary and extends to the left. The bar is pulled to the right with a constant speed $v = 4.0\ \text{m/s}$ , so that the portion of the loop within the magnetic field decreases with time. The bar remains perpendicular to the rails and maintains electrical contact with them at all times.

Figure 1. Conducting-rail loop with a sliding bar leaving a uniform magnetic-field region (field into the page).

Figure 2. Axes for induced current magnitude I versus time t (0 to 0.15 s).

Using the definition $\Phi_B = \int \vec B ·d\vec A$ , derive an expression for the magnetic flux $\Phi_B(t)$ through the loop while part of the loop remains in the magnetic-field region. Let the portion of the bar that is still inside the magnetic-field region have width $x(t)$ , where $x(t) = x_0 - vt$ . Express your answer in terms of $B$ , $\ell$ , and $x(t)$ , and include the appropriate sign for the flux given that $\vec B$ is into the page.

ii.

While $x(t) > 0$ , derive an expression for the magnitude of the induced electric potential difference $|\mathcal{E}|$ around the loop. Begin your derivation by writing Faraday’s law. Express your answer in terms of $B$ , $\ell$ , and $v$ .

iii.

Assume $x_0 = 0.40\ \text{m}$ at $t = 0$ . The bar is pulled at constant speed $v = 4.0\ \text{m/s}$ . On the axes shown in Figure 2, sketch a graph of the induced current magnitude $I$ in the loop as a function of time $t$ from $t = 0$ until $t = 0.15\ \text{s}$ . Label the time at which the current first becomes zero and indicate the corresponding current values on the graph.

Figure 3. Same sliding-bar loop with an inductor L inserted in series on the top segment (RL circuit).

Derive an expression for the induced current $I(t)$ in the RL circuit for the interval $0 ≤ t ≤ t_1$ , where $t_1$ is the time when the bar just leaves the magnetic-field region. Begin your derivation by writing the loop equation that includes the inductor (Kirchhoff’s loop rule with the inductor term). Then calculate (i) the time constant $\tau$ and (ii) the current $I(t_1)$ . Express your final numerical answers with units. An ideal inductor of inductance $L = 0.060\ \text{H}$ is inserted in series with the loop, as shown in Figure 3. The total resistance of the circuit remains $R = 2.0\ \Omega$ . The bar continues to be pulled with constant speed $v = 4.0\ \text{m/s}$ in the same magnetic field $B = 0.80\ \text{T}$ , and $x_0 = 0.40\ \text{m}$ at $t=0$ .

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FRQ

Conducting loop exiting magnetic field region

2. A rectangular conducting loop of total resistance R = 0.80 \Omega is made of rigid wire and has height h = 0.20 m (vertical dimension) and width w = 0.30 m (horizontal dimension). The loop is initially entirely inside a region of uniform magnetic field of magnitude B = 0.60 T directed into the page. At time t = 0, the loop begins moving to the right at constant speed v = 1.5 m/s, as shown in Figure 1, so that it exits the magnetic-field region across a straight boundary. For 0 ≤ t ≤ w/v, the area of the loop that remains inside the magnetic field decreases at a constant rate. Ignore fringing fields and assume the magnetic field is zero outside the region.

Figure 1. Rectangular conducting loop exiting a uniform magnetic-field region at constant speed.

Figure 2. Bar chart for induced emf magnitude at four specified times; the bar at t = 3w/(4v) is the reference height ε₀.

In Figure 2, draw bars to represent $\varepsilon$ at times $t = 0$ , $w/(4v)$ , and $w/(2v)$ relative to $\varepsilon$ shown at $3w/(4v)$ . If $\varepsilon = 0$ , write a "0" in that column. For 0 ≤ t ≤ w/v, the induced emf magnitude is $\varepsilon = \left|\frac{d\Phi_B}{dt}\right|$ . The partially completed bar chart in Figure 2 shows a bar that represents $\varepsilon$ at $t = 3w/(4v)$ .

Derive an expression for the magnitude of the induced current $I$ in the loop during the interval $0 ≤ t ≤ w/v$ in terms of $B$ , $h$ , $v$ , and $R$ . Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Your derivation must include an explicit evaluation of $d\Phi_B/dt$ using the time rate of change of the area of the loop within the magnetic field.

Figure 3. Axes for magnetic force magnitude F versus time t during 0 ≤ t ≤ w/v.

On the axes shown in Figure 3, sketch a graph of the magnitude of the magnetic force $F$ exerted on the loop as a function of time $t$ during the time interval $0 ≤ t ≤ w/v$ . The magnetic force is due to the interaction between the external magnetic field and the induced current in the loop. Assume the only horizontal magnetic force is on the vertical side of the loop that remains in the field, and use the relationship $F = I h B$ for that side.

Indicate whether the current in Circuit 1 reaches $0.95\,\varepsilon_0/R_1$ before the source turns off at $t = w/v$ . Briefly justify your answer by comparing the relevant RL time constant to the on-time of the source and by stating the functional dependence of the current on time in an RL circuit. A student models the induced emf found in parts A and B as an ideal emf source of constant magnitude $\varepsilon_0$ that is turned on at $t = 0$ and turned off at $t = w/v$ , where $\varepsilon_0 = B h v$ . The student then connects this source (while it is on) to two different circuits.

Circuit 1 (RL): The source is connected in series with a resistor $R_1 = 1.2\ \Omega$ and a single inductor of inductance $L = 0.040\ \text{H}$ .

Circuit 2 (LC): The source is connected in series with the same inductor $L = 0.040\ \text{H}$ and a capacitor $C = 220\ \mu\text{F}$ . The capacitor is initially uncharged.

Use $B = 0.60\ \text{T}$ , $h = 0.20\ \text{m}$ , $v = 1.5\ \text{m/s}$ , and $w = 0.30\ \text{m}$ .

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Key terms

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.

Common unit 13 mistakes

Ignoring the angle in flux calculations

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.

Forgetting the negative sign in Faraday's law

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.

Treating the inductor as a resistor at t = 0

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.

Using the wrong N dependence for solenoid inductance

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.

Confusing the LC angular frequency with the LR time constant

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.

How this unit shows up on the AP exam

Deriving and solving differential equations

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.

Multi-step flux and force problems

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.

Energy conservation and graphical analysis

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.

Final unit 13 review checklist

Unit 13 final review checklistUse this list to confirm you can handle every major skill in the unit before exam day.
Calculate magnetic fluxCompute 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.
Apply Faraday's and Lenz's lawsUse 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.
Analyze forces on induced currentsFind 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.
Work with inductance formulasCalculate 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.
Solve LR circuit transientsWrite 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.
Describe LC circuit oscillationsIdentify 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).

How to study unit 13

Step 1: Build fluency with flux (Topic 13.1)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.

Step 2: Work through Faraday's and Lenz's laws (Topic 13.2)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.

Step 3: Connect flux changes to forces (Topic 13.3)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.

Step 4: Understand inductance and energy storage (Topic 13.4)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.

Step 5: Solve LR and LC circuit problems (Topics 13.5-13.6)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.

More ways to review

Topic study guides

Open the individual guides for Unit 13 when you want a closer review of one topic.

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FRQ practice

Practice free-response reasoning and compare your answer with scoring guidance.

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Cheatsheets

Use unit cheatsheets for a quick visual review after you work through the notes.

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Score calculator

Estimate your broader AP score goal after you review the course and exam format.

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Frequently Asked Questions

What topics are covered in AP Physics E&M Unit 13?

AP 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.

How much of the AP Physics E&M exam is Unit 13?

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.

What's on the AP Physics E&M Unit 13 progress check (MCQ and FRQ)?

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.

How do I practice AP Physics E&M Unit 13 FRQs?

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.

Where can I find AP Physics E&M Unit 13 practice questions?

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.

How should I study AP Physics E&M Unit 13?

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.

Ready to review Unit 13?Start with the notes, check the topic cards, and use the practice or resource links when they are available for this course.

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AP Physics C: E&M Unit 13 Review: Electromagnetic Induction

What is AP Physics C: E&M unit 13?

Flux and Faraday's law

Inductance and energy storage

LR and LC circuit dynamics

AP Physics C: E&M unit 13 topics

Magnetic Flux

Electromagnetic Induction

Induced Currents and Magnetic Forces

Inductance

Circuits with Resistors and Inductors (LR Circuits)

Circuits with Capacitors and Inductors (LC Circuits)

Hardest AP Physics C: E&M unit 13 topics

Hardest topics in unit 13

Unit 13 review notes

Magnetic Flux

Electroma­gnetic Induction: Faraday's Law and Lenz's Law

Induced Currents and Magnetic Forces

Inductance

LR Circuits

LC Circuits

Practice AP Physics C: E&M unit 13 questions

Example AP-style MCQs

AP-style practice question

AP-style practice question

Example FRQs

Motional EMF in moving conductor bars

Electromagnetic induction in moving conductor loop

Conducting loop exiting magnetic field region

Key terms

Common unit 13 mistakes

Ignoring the angle in flux calculations

Forgetting the negative sign in Faraday's law

Treating the inductor as a resistor at t = 0

Using the wrong N dependence for solenoid inductance

Confusing the LC angular frequency with the LR time constant

How this unit shows up on the AP exam

Deriving and solving differential equations

Multi-step flux and force problems

Energy conservation and graphical analysis

Final unit 13 review checklist

How to study unit 13

More ways to review

Topic study guides

FRQ practice

Cheatsheets

Score calculator

Frequently Asked Questions

What topics are covered in AP Physics E&M Unit 13?

How much of the AP Physics E&M exam is Unit 13?

What's on the AP Physics E&M Unit 13 progress check (MCQ and FRQ)?

How do I practice AP Physics E&M Unit 13 FRQs?

Where can I find AP Physics E&M Unit 13 practice questions?

How should I study AP Physics E&M Unit 13?

Electromagnetic Induction: Faraday's Law and Lenz's Law