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AP Course

AP Physics 2

Study AP Physics 2 Units 9–15, with the Fall 2026 radioactive-decay clarification applied and official exam updates linked separately.

Study Units 9–15 through original models, derivations, experiments, quizzes, and practice sets.

Lessons
Particle Motion Behind Temperature and PressureConnecting Gas Variables with the Ideal-Gas ModelHeat Flow and the Approach to Thermal BalanceEnergy Accounting for Thermodynamic ProcessesMaterial Response to Heating and Heat ConductionEntropy, Probability, and the Direction of Thermal ChangeCharge Interactions and Coulomb-Force ModelsTracking Charge During Contact and InductionMapping Electric Fields from Source ChargesEnergy of Configurations of ChargesElectric Potential and Equipotential ReasoningCapacitance and Energy Stored in Electric FieldsConserving Energy in Charged-Particle MotionCharge Flow and Conventional CurrentModeling Sources, Wires, and Loads in Simple NetworksHow Geometry and Material Set ResistanceElectrical Energy Transfer and PowerReducing Series-Parallel DC NetworksLoop Equations from Energy ConservationJunction Equations from Charge ConservationTransient Charging and Discharging in RC NetworksSources, Direction, and Strength of Magnetic FieldsMagnetic Forces and Curved Paths of ChargesForces Between Fields and Current-Carrying ConductorsChanging Magnetic Flux and Induced EMFReflection and Absorption at Material BoundariesLocating Images with Mirror Ray ModelsRefraction, Index, and Total Internal ReflectionLocating Images with Thin-Lens ModelsPulses, Wave Types, and Propagation SpeedPeriodic-Wave Measures and GraphsBoundary Changes and PolarizationElectromagnetic-Wave BehaviorFrequency Shifts from Relative MotionSuperposition, Interference, and Standing WavesDiffraction from Openings and EdgesTwo-Slit and Grating Pattern GeometryPhase Change and Thin-Film ColorQuantum Models and Dual Wave-Particle EvidenceEnergy Levels in a Hydrogen-Like AtomConnecting Spectral Lines to Energy TransitionsThermal Spectra and Blackbody CurvesPhoton Thresholds in the Photoelectric EffectPhoton-Electron Scattering and MomentumMass-Energy Accounting in Fission and FusionRandom Nuclear Decay, Activity, and Half-Life
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics 2 · Unit 12 · Topic 12.4

Changing Magnetic Flux and Induced EMF

Connect magnetic flux to loop geometry, predict the direction of an induced current with Lenz's law, and calculate emf for changing fields, moving conductors, and rotating coils. The central idea is not simply “a magnetic field makes current,” but that a changing flux linkage produces an emf.

Learning Goals

  • Define signed magnetic flux through a surface and use \(\Phi_B=BA\cos\theta\) when the field is uniform.
  • Apply Faraday's law to average and instantaneous changes in flux linkage.
  • Use Lenz's law and the right-hand rule to determine induced-current direction.
  • Distinguish induced emf from induced current and identify when each exists.
  • Analyze motional emf, a loop crossing a field boundary, and a rotating-coil generator.
  • Explain why electromagnetic induction is consistent with energy conservation.
\(\Phi_B\)signed magnetic flux through one turn
\(\mathcal E\)induced emf around a circuit
\(N\Phi_B\)flux linkage for \(N\) identical turns
\(\hat n\)chosen area-normal direction

1. Magnetic Flux Measures Field Through a Surface

Magnetic flux is a signed scalar that measures how much magnetic field passes through an oriented surface. Divide a surface into tiny area vectors \(d\vec A=\hat n\,dA\), take the component of \(\vec B\) normal to each piece, and add:

\(\Phi_B=\int_S\vec B\cdot d\vec A\)

For a flat loop of area \(A\) in a uniform field, the integral reduces to

\(\Phi_B=BA\cos\theta\)

Here \(\theta\) is the angle between \(\vec B\) and the area normal, not the angle between the field and the plane. The SI unit is the weber: \(1\,\mathrm{Wb}=1\,\mathrm{T\,m^2}\).

The area normal sets the flux angleFlux is maximum when the normal is parallel to the field, zero when the field lies in the loop's plane, and negative when the chosen normal points more than \(90^\circ\) from the field.
Worked Example 1 · Flux and Orientation

Calculate Flux Through One Turn

A flat loop has area \(0.030\,\mathrm{m^2}\). A uniform \(0.40\,\mathrm T\) field makes a \(60^\circ\) angle with the loop's area normal.

\(\Phi_B=BA\cos\theta=(0.40)(0.030)\cos60^\circ=6.0\times10^{-3}\,\mathrm{Wb}\)

If \(60^\circ\) had instead been measured from the plane, the normal angle would be \(30^\circ\). Always identify the normal before substituting.

2. Signed Flux and the Choice of Normal

Either side of an open surface can be chosen as positive. Reversing \(\hat n\) reverses the sign of \(\Phi_B\), but a consistent sign convention also reverses the positive traversal direction around the loop. Physical predictions are unchanged.

  • \(\theta=0^\circ\): \(\Phi_B=+BA\), maximum positive flux.
  • \(\theta=90^\circ\): \(\Phi_B=0\), because the field is tangent to the surface.
  • \(\theta=180^\circ\): \(\Phi_B=-BA\), maximum negative flux.

3. Nonuniform Fields Require an Integral

The shortcut \(BA\cos\theta\) works only when the relevant field component is uniform over the surface. If the field or angle varies, use

\(\Phi_B=\int_S B_\perp\,dA\)

Only the component perpendicular to each small area contributes. A strong field over a small part of a loop can therefore produce the same flux as a weaker field spread across the whole loop.

4. Faraday's Law: Emf Comes from Changing Flux Linkage

For \(N\) tightly wound turns that share the same flux per turn, Faraday's law is

\(\mathcal E=-N\frac{d\Phi_B}{dt}\)

Over a finite interval, the average induced emf is

\(\mathcal E_{\mathrm{avg}}=-N\frac{\Delta\Phi_B}{\Delta t}\)

The magnitude depends on how quickly the flux linkage \(N\Phi_B\) changes. A large constant flux produces no emf. The minus sign is not an instruction to report a negative magnitude; it encodes Lenz's law relative to the chosen positive loop direction.

Worked Example 2 · Average Induced Emf

Track a Signed Flux Change

In a \(200\)-turn coil, the flux per turn changes from \(+8.0\times10^{-3}\,\mathrm{Wb}\) to \(+2.0\times10^{-3}\,\mathrm{Wb}\) in \(0.050\,\mathrm s\).

\(\Delta\Phi_B=(2.0-8.0)\times10^{-3}=-6.0\times10^{-3}\,\mathrm{Wb}\)
\(\mathcal E_{\mathrm{avg}}=-200\frac{-6.0\times10^{-3}}{0.050}=+24\,\mathrm V\)

The positive sign means the emf follows the loop direction paired with the chosen positive normal. Its magnitude is \(24\,\mathrm V\).

5. Three Ways to Change Magnetic Flux

Because \(\Phi_B=BA\cos\theta\) in the uniform-field case, flux can change by changing any factor:

  1. Field: change \(B\) or move the loop into a region with a different field.
  2. Effective area: change the loop area within the field, as when a loop crosses a field boundary.
  3. Orientation: rotate the loop so the angle between \(\vec B\) and \(\hat n\) changes.

Translating a rigid loop entirely inside an ideal, uniform, steady field does not change \(B\), \(A\), or \(\theta\), so it produces no net loop emf. “The loop is moving” is not enough; the flux linkage must change.

Worked Example 3 · Direction with Lenz's Law

Increasing Field Into the Page

A circular conducting loop lies in the page while an external magnetic field into the page increases. The increasing inward flux is the change to oppose, so the induced current produces a field out of the page. The right-hand rule then gives a counterclockwise induced current.

If the inward field were decreasing, the loop would try to maintain inward flux and the induced current would be clockwise.

6. A Reliable Lenz-Law Workflow

  1. Identify the external field through the loop.
  2. Decide whether the external flux is increasing, decreasing, or unchanged.
  3. Choose the induced field that opposes that change.
  4. Use the current-loop right-hand rule to obtain clockwise or counterclockwise current.
  5. Check whether the predicted force resists the motion or process causing the change.

7. Emf Is Not Automatically Current

An emf is energy transferred per unit charge around a complete path. It can be present even when the loop is open. A sustained current requires a closed conducting path.

\(I=\frac{\lvert\mathcal E\rvert}{R_{\mathrm{total}}}\)

This simple relation applies when the circuit can be modeled as resistive. The direction still comes from Lenz's law, not from the positive magnitude above.

Worked Example 4 · Rotation Through a Quarter Turn

Change Orientation at Fixed Field and Area

A \(50\)-turn coil with area \(0.010\,\mathrm{m^2}\) rotates in a \(0.20\,\mathrm T\) field. Its normal changes from parallel to the field to perpendicular in \(0.10\,\mathrm s\).

\(\Delta\Phi_B=BA(\cos90^\circ-\cos0^\circ)=-(0.20)(0.010)=-2.0\times10^{-3}\,\mathrm{Wb}\)
\(\lvert\mathcal E_{\mathrm{avg}}\rvert=N\frac{\lvert\Delta\Phi_B\rvert}{\Delta t}=50\frac{2.0\times10^{-3}}{0.10}=1.0\,\mathrm V\)

The average emf is \(1.0\,\mathrm V\). Its instantaneous value need not be constant during the rotation.

Worked Example 5 · Changing Field and Circuit Current

Separate the Induction Step from the Circuit Step

A \(100\)-turn coil of area \(5.0\times10^{-3}\,\mathrm{m^2}\) has its normal parallel to a field changing uniformly at \(0.30\,\mathrm{T/s}\). The total circuit resistance is \(6.0\,\Omega\).

\(\lvert\mathcal E\rvert=NA\left\lvert\frac{dB}{dt}\right\rvert=(100)(5.0\times10^{-3})(0.30)=0.15\,\mathrm V\)
\(I=\frac{0.15}{6.0}=2.5\times10^{-2}\,\mathrm A\)

First calculate emf from the flux change; then calculate current from the complete circuit. Lenz's law determines its direction.

8. Motional Emf in a Moving Conductor

A charge in a conductor moving with velocity \(\vec v\) through a magnetic field experiences \(q\vec v\times\vec B\). Charge separation along a rod can create a potential difference. For a rod of length \(\ell\) moving perpendicular to both its length and a uniform field,

\(\lvert\mathcal E\rvert=B\ell v\)

The same result follows from Faraday's law for a sliding rod on rails: \(dA/dt=\ell v\), so \(\lvert d\Phi_B/dt\rvert=B\ell v\). A more general moving-conductor contribution is

\(\mathcal E_{\mathrm{motional}}=\oint(\vec v\times\vec B)\cdot d\vec\ell\)
Worked Example 6 · Sliding Rod and Energy

Match Mechanical Input to Electrical Output

A \(0.30\,\mathrm m\) rod moves at \(4.0\,\mathrm{m/s}\) perpendicular to a \(0.50\,\mathrm T\) field on conducting rails. The circuit resistance is \(1.5\,\Omega\).

\(\mathcal E=B\ell v=(0.50)(0.30)(4.0)=0.60\,\mathrm V\)
\(I=\frac{0.60}{1.5}=0.40\,\mathrm A\)
\(F_B=I\ell B=(0.40)(0.30)(0.50)=0.060\,\mathrm N\)
\(P_{\mathrm{mech}}=F_Bv=0.24\,\mathrm W\qquad P_{\mathrm{thermal}}=I^2R=0.24\,\mathrm W\)

The magnetic force opposes the rod's motion. An external agent must supply power, which becomes thermal energy in the resistor in this idealized model.

9. A Loop Entering or Leaving a Field Region

Suppose a rectangular loop crosses a sharp boundary into a uniform field. While entering, the area inside the field grows at rate \(dA/dt=\ell v\), where \(\ell\) is the side crossing the boundary. The emf magnitude is \(B\ell v\). Once the whole loop is inside a uniform field, its flux becomes constant and the net emf falls to zero. While leaving, the flux decreases and the current direction reverses.

This analysis prevents a common mistake: using \(B\ell v\) for every moving loop even when equal motional contributions on opposite sides cancel.

Worked Example 7 · Loop Crossing a Boundary

Calculate Emf and Use Lenz's Law

A rectangular loop enters a region where \(B=0.80\,\mathrm T\) points into the page. Its \(0.20\,\mathrm m\) leading side crosses the boundary at \(3.0\,\mathrm{m/s}\), and the loop resistance is \(2.0\,\Omega\).

\(\mathcal E=B\ell v=(0.80)(0.20)(3.0)=0.48\,\mathrm V\)
\(I=\frac{0.48}{2.0}=0.24\,\mathrm A\)

Inward flux is increasing, so the induced field points outward and the current is counterclockwise. When the loop is fully inside the field, the current becomes zero in the ideal model.

10. Rotating Coil and Generator Emf

Let a coil rotate at constant angular speed \(\omega\) in a uniform field. If its normal is initially aligned with the field, the flux per turn and induced emf are

\(\Phi_B(t)=BA\cos(\omega t)\)
\(\mathcal E(t)=-N\frac{d\Phi_B}{dt}=NBA\omega\sin(\omega t)\)

The peak emf is \(\mathcal E_0=NBA\omega\). Notice the phase relationship: emf is zero when flux is at a maximum or minimum, and emf magnitude is greatest when flux passes through zero most rapidly.

Emf is the negative time derivative of flux linkageA graph of flux gives emf through its slope: horizontal tangents mean zero emf, while the steepest flux changes produce the largest emf magnitude.
Worked Example 8 · Rotating Generator

Build the Emf Function

A \(100\)-turn coil of area \(0.020\,\mathrm{m^2}\) rotates at \(60\,\mathrm{rad/s}\) in a \(0.30\,\mathrm T\) field. Its normal is aligned with the field at \(t=0\).

\(\mathcal E_0=NBA\omega=(100)(0.30)(0.020)(60)=36\,\mathrm V\)
\(\mathcal E(t)=(36\,\mathrm V)\sin(60t)\)

The emf starts at zero because the flux initially has zero slope, even though the flux itself is maximum.

11. Induced Electric Fields

A changing magnetic field can create a circulating electric field even when no wire is present. A conductor merely gives charges a path on which that field can act. The field form of Faraday's law is

\(\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}\)

This induced electric field is nonconservative: its closed-loop integral can be nonzero. Therefore induced emf should not always be treated as an ordinary electrostatic potential difference between two uniquely defined endpoints.

12. Graph and Data Reasoning

  • The signed slope of an \(N\Phi_B\)-versus-\(t\) graph is \(-\mathcal E\).
  • A flat flux graph means zero induced emf, even if the flux value is large.
  • A steeper flux graph means a larger emf magnitude.
  • A corner in an idealized flux graph represents an abrupt emf change; real systems usually smooth it.
  • The area under an emf-versus-time graph equals \(-\Delta(N\Phi_B)\).

13. Common Traps

  • Measuring \(\theta\) from the plane instead of the area normal.
  • Claiming any magnetic field creates an induced current.
  • Opposing the external field rather than its change.
  • Multiplying by \(N\) twice after already using flux linkage.
  • Using total area when only part of a loop lies in the field.
  • Assuming emf guarantees current in an open circuit.
  • Ignoring the sign convention or reporting a negative magnitude.
  • Using \(B\ell v\) when velocity, rod, and field are not mutually perpendicular.

14. Reliable Problem-Solving Workflow

  1. Choose a surface and positive normal; pair it with a positive loop direction by the right-hand rule.
  2. Write the signed flux through one turn, using an integral if the field is nonuniform.
  3. Identify which of \(B\), \(A\), or \(\theta\) changes and over what interval.
  4. Apply \(\mathcal E=-d(N\Phi_B)/dt\) or its average form.
  5. Use Lenz's law to interpret direction and check energy consistency.
  6. If the circuit is closed, use its resistance and network rules to find current.
  7. Check units, limiting cases, and whether the result changes correctly if the rate doubles.
Mastery Check

1. A uniform magnetic field lies in the plane of a flat loop. What is the magnetic flux?

Show reasoning and answer

The field is \(90^\circ\) from the area normal, so \(\Phi_B=BA\cos90^\circ=0\).

2. A rigid loop moves at constant velocity entirely inside a uniform, steady magnetic field without rotating. Is there a net induced emf?

Show reasoning and answer

No. \(B\), \(A\), and \(\theta\) remain constant, so \(d\Phi_B/dt=0\) and the net loop emf is zero.

3. Magnetic flux into the page is decreasing. What direction is the induced current?

Show reasoning and answer

The induced field must point into the page to oppose the decrease. The right-hand rule gives a clockwise current.

4. The same flux change occurs in half the original time. What happens to average emf magnitude?

Show reasoning and answer

\(\lvert\mathcal E_{\mathrm{avg}}\rvert=N\lvert\Delta\Phi_B\rvert/\Delta t\), so halving the time doubles the magnitude.

5. Can a changing flux produce emf in an open loop? Can it sustain current?

Show reasoning and answer

It can produce an emf and charge separation, but the open path cannot sustain a circulating current.

6. A rotating coil has maximum positive flux. What is its instantaneous emf at that moment?

Show reasoning and answer

Zero. At a flux maximum, \(d\Phi_B/dt=0\), so \(\mathcal E=-N\,d\Phi_B/dt=0\).

Investigation: What Controls Induced Emf?

Use the open educational PhET Faraday's Law simulation. Before each trial, predict the meter direction and relative bulb brightness.

  1. Move the same magnet through the coil slowly, then quickly. Compare peak response.
  2. Stop the magnet at the center. Compare magnetic flux with induced emf.
  3. Reverse the pole entering the coil and predict the sign reversal.
  4. Move the magnet into, hold it, and pull it out. Sketch qualitative flux-versus-time and emf-versus-time graphs.
  5. Explain every observation using rate of flux change rather than magnet position alone.

For a fuller virtual laboratory with coil turns and generator models, explore PhET Faraday's Electromagnetic Lab.

Official curriculum reference: College Board AP Physics 2 course page. The explanation and worked example are independently written for this study site.