AP Physics C: Electricity & Magnetism Unit 13: Electromagnetic Induction 2027
Dedicated Unit 13 chapter hub for 6 current topics. Use it to move through the lesson sequence, connect formulas and labs, and hand off cleanly to the previous or next unit without mixing topic and unit search intent.
Unit 13 map: flux change, induced emf, magnetic forces, inductors, LR transients, and LC oscillations
Unit 13 is the synthesis of the E&M course. Magnetic flux links field geometry to induced emf; Lenz’s law supplies direction; induced currents create magnetic forces and energy transfers; inductance makes current itself a dynamical state variable; LR circuits introduce a first-order transient analogous to RC circuits; and ideal LC circuits produce a second-order oscillator mathematically parallel to simple harmonic motion.
All six Unit 13 topics
Magnetic Flux
Define signed flux through area vectors and surface integrals, including nonuniform fields and orientation changes.
Current lesson · 13.1 →Electromagnetic Induction
Use Faraday’s law and Lenz’s law for changes in field, area, orientation, or flux linkage.
Open 13.2 →Induced Currents and Magnetic Forces
Connect motional emf, induced current, magnetic force, mechanical work, and resistive dissipation.
Open 13.3 →Inductance
Model flux linkage, self-induced emf, and magnetic-field energy stored by an inductor.
Open 13.4 →LR Circuits
Analyze current growth/decay, inductor voltage, time constant L/R, energy storage, and resistor dissipation.
Open 13.5 →LC Circuits
Derive charge/current oscillations and track energy exchange between capacitor electric energy and inductor magnetic energy.
Open 13.6 →Induction model-selection table
| Question feature | Best starting relation | Direction/limit check |
|---|---|---|
| Changing magnetic flux through a loop | ε=−dΦ_B/dt, with Φ_B=∫B·dA | Define the area-vector sign before interpreting the minus sign. |
| Conducting bar moving through uniform B | |ε|=Bℓv for the standard perpendicular geometry | Use magnetic force on charge or flux change to determine polarity/current direction. |
| Induced current creates mechanical resistance | I=ε/R then F=IℓB in the simple perpendicular bar model | Force must oppose the flux-changing motion consistently with energy conservation. |
| Self-inductance | ε_L=−L dI/dt and U_L=½LI² | Inductor current cannot jump instantaneously in the ideal finite-voltage model. |
| Series LR transient | τ=L/R and the appropriate growth/decay exponential | Check I at t=0 and t→∞; distinguish inductor voltage from resistor voltage. |
| Ideal LC oscillator | q/C+L dI/dt=0 with I=dq/dt | Energy alternates between capacitor and inductor; total is constant only in the ideal lossless model. |
Calculus and differential-equation ladder
Flux as an integral
Magnetic flux is a surface integral. For a uniform field through a flat surface it reduces to BA cosθ, but nonuniform fields or curved surfaces require the integral and a defined area-vector orientation.
Faraday as a time derivative
Induced emf depends on the time rate of change of flux, not merely on the amount of flux. A large constant flux produces no Faraday emf if it is not changing.
LR first-order dynamics
For a series source–R–L circuit, ε−IR−L dI/dt=0. The current approaches ε/R with time constant L/R; during source-free decay it falls exponentially.
LC second-order dynamics
With ideal L and C, q/C+L dI/dt=0 and I=dq/dt give d²q/dt²=−q/(LC). Thus ω=1/√(LC), exactly the mathematical SHM form.
Energy identity for magnetic braking
In the simple sliding-bar model, external mechanical power Fv is converted into electrical power I²R while the bar moves at constant speed. This is a strong cross-check on direction and magnitude.
Lenz as conservation reasoning
Do not answer only “by Lenz’s law.” State what flux is changing, what induced field would oppose that change, then infer the current direction from the right-hand rule.
Three Unit 13 synthesis problems
Loop entering a magnetic-field region
As the loop enters, the area inside the field grows, so |Φ_B| changes. Determine the sign of dΦ_B/dt from the chosen area vector, use Faraday–Lenz for emf polarity, then find current and the magnetic force that opposes the entering motion. At constant speed, compare mechanical input power with I²R.
LR current-growth graph
From an I(t) graph, read the final current I∞=ε/R and the time at which the current reaches 1−e^(−1) of its final value to estimate τ=L/R. Use the initial slope dI/dt=ε/L to obtain an independent inductance check.
LC energy exchange
Start with a charged capacitor and zero current. Use U_C=q²/(2C)+½LI²=constant and the differential equation to connect Q_max, I_max, period, and phase. At maximum |q| the current is zero; at q=0 the current magnitude is maximum.
Laboratory and data-analysis tasks
Faraday-law slope test
Vary the rate of magnetic-flux change through a coil and measure induced emf. Plot |ε| versus |dΦ/dt| or versus N|dΦ_single/dt| and test linearity.
Motional-emf scaling
Move a conducting bar at controlled speeds through a field and test ε∝v for fixed B and active length. Identify contact resistance and field nonuniformity as limitations.
LR time constant
Record current or resistor voltage during growth/decay, fit an exponential, and test τ∝L and τ∝1/R by changing one element at a time.
LC period test
Measure oscillation period for several L or C values in a low-loss simulation/experiment. Test T²∝L at fixed C or T²∝C at fixed L and identify damping as a departure from the ideal model.
Unit 13 traps that cost points
Flux exists, so emf exists
Faraday emf requires changing magnetic flux. Constant flux gives zero induced emf even if B itself is large.
Lenz without a mechanism
State the direction of the original flux change, the induced field that opposes it, and then the induced-current direction.
Inductor current jumps
Ideal inductor current is continuous unless an unphysical infinite voltage is allowed; inductor voltage can change abruptly.
Capacitor and inductor endpoint confusion
Capacitor voltage cannot jump instantaneously in the ideal RC model; inductor current cannot jump in the ideal LR model. Keep the state variables distinct.
Wrong time constant
RC uses RC; LR uses L/R. Their units and physical state variables differ.
LC with resistive decay hidden
The ideal LC model conserves electromagnetic energy. Significant resistance produces damping and is outside that ideal conservation model.
Where Unit 13 completes the course
Unit 12 · Magnetic fields
Flux and induction depend on magnetic-field direction, geometry, and source models developed in Unit 12.
Unit 11 · Circuits
Kirchhoff rules, resistance, power, and RC-transient habits prepare the differential-equation reasoning for LR and LC circuits.
Mechanics Unit 7 · Oscillations
Ideal LC charge obeys the same second-order mathematical form as SHM, providing a direct cross-course comparison of phase and energy exchange.
Course synthesis
Across Units 8–13, fields, potential, circuits, magnetism, and induction become one connected model system rather than isolated formula sets.
Unit 13 mastery checklist
Unit 13 lesson sequence
This dedicated unit hub owns the chapter intent. Every lesson keeps its own canonical topic URL and returns here for unit context.
Topic 13.1 · Magnetic Flux
Magnetic flux Φ_B is the signed surface integral of B⃗ through an oriented area. For a uniform field across a flat surface it reduces to Φ_B=B⃗·A⃗=BA cosθ, where θ is measured from the area normal—not from the plane. Flux is the quantity whose change drives electromagnetic induction, so sign, area-vector orientation, and the difference between open-surface flux and closed-surface net flux must be controlled before using Faraday’s law.
Open Topic 13.1 →Topic 13.2 · Electromagnetic Induction
Electromagnetic induction occurs when magnetic flux through a circuit changes. Faraday’s law gives the induced emf as the negative time derivative of flux, while Lenz’s law interprets the minus sign: the induced current produces a magnetic effect that opposes the change in flux. Flux can change because B changes, the threaded area changes, the loop rotates, or the overlap between a conductor and a magnetic-field region changes.
Open Topic 13.2 →Topic 13.3 · Induced Currents and Magnetic Forces
A changing magnetic flux can induce a current, and any segment of that induced-current loop that lies in an external magnetic field experiences magnetic force. The complete AP Physics C model therefore links Faraday–Lenz induction to I=ε/R, the current-element force dF⃗=I(dℓ⃗×B⃗), and Newton’s second law. In moving-loop problems the resulting force typically opposes the motion that produced the flux change, converting mechanical energy into thermal energy unless an external agent supplies power.
Open Topic 13.3 →Topic 13.4 · Inductance
Inductance measures how strongly a conductor’s own changing current changes linked magnetic flux and therefore generates an emf that opposes that current change. For a linear inductor, flux linkage λ=NΦ_B is proportional to current, λ=LI, giving ε_L=−L dI/dt. A long solenoid provides the central geometry model L=μ_core N²A/ℓ, and an inductor carrying current stores magnetic-field energy U_L=½LI².
Open Topic 13.4 →Topic 13.5 · Circuits with Resistors and Inductors (LR Circuits)
A series LR circuit is governed by a first-order differential equation because the resistor’s voltage drop depends on current while the inductor’s voltage depends on dI/dt. With a battery, ε=IR+L dI/dt and the current approaches ε/R exponentially with time constant τ=L/R. During decay through resistance, I=I₀e^(−t/τ). At one time constant, growth reaches about 63% of its final value and decay falls to about 37% of its initial value.
Open Topic 13.5 →Topic 13.6 · Circuits with Capacitors and Inductors (LC Circuits)
An ideal LC circuit contains a capacitor network, represented by its equivalent capacitance C, and a single inductor but no resistance. Energy therefore oscillates between the capacitor electric field and the inductor magnetic field instead of being dissipated. Kirchhoff’s loop rule gives a second-order differential equation, d²q/dt²=−q/(LC), mathematically identical to simple harmonic motion. The natural angular frequency is ω=1/√(LC), so larger L or C produces slower oscillation. At maximum capacitor charge the current is zero; when the capacitor is momentarily uncharged, the current and magnetic energy are maximum.
Open Topic 13.6 →Unit 13 resource network
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