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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 14 · Topic 14.3

Boundary Changes and Polarization

Superposition, boundary conditions, diffraction, and phase explain sound and physical-optics patterns.

1. Topic Lens

Boundary Changes and Polarization is studied through waves, sound, and physical optics. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[v=f\lambda\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

  1. A repeating wave advances one wavelength during one period T.
  2. Speed is distance over time, so v=λ/T.
  3. Frequency is 1/T, giving v=fλ.
\[\lambda=\frac{v}{f}\]

3. Detailed Visual Model

Pixel diagram for Boundary Changes and PolarizationOriginal schematic connecting Boundary Changes and Polarization to Waves, Sound, and Physical Optics.
Boundary Changes and Polarization: an original pixel-style model. Use it as a schematic, not a literal scale drawing.

4. Worked Example and Lab Link

A 340 Hz tone travels at 340 m/s. Find its wavelength.

Answer: λ=v/f=1.00 m.

Investigation idea: Measure standing-wave nodes in an air column or string and compare allowed wavelengths with boundary conditions.

Common trap: Wave speed is determined by the medium; changing frequency usually changes wavelength, not the medium's speed.

Expanded Topic 14.3 Lesson

5. Learning Targets

  • Predict whether a reflected pulse is inverted at a fixed end, free end, or string junction.
  • Track frequency, speed, wavelength, phase, and energy when a wave reaches a stationary boundary.
  • Use string impedance to quantify reflected and transmitted displacement and energy fractions.
  • Explain why transverse waves can be polarized and identify light's polarization direction.
  • Apply component reasoning and Malus's law to one or more ideal polarizers.

6. Boundary Accounting

When an incident wave reaches a change in its medium, the boundary conditions must be satisfied at every instant. In general, part of the disturbance returns and part continues:

\[y_{\mathrm{boundary}}=y_i+y_r=y_t\]
  • Incident: approaches the boundary in medium 1.
  • Reflected: carries energy back through medium 1.
  • Transmitted: carries energy forward through medium 2.

The reflected and transmitted amplitudes are not normally equal to the incident amplitude. Amplitude signs encode orientation or phase; energy fractions are nonnegative.

7. Fixed and Free Ends

Fixed endThe endpoint must remain at zero displacement. A crest reflects as a trough: \(r=-1\), a phase shift of \(\pi\).
Free endThe endpoint can move and has no transverse force from beyond the end. A crest reflects as a crest: \(r=+1\), no inversion.

For ideal complete reflection, reflected amplitude magnitude equals incident amplitude. “Inverted” describes displacement, not a reversal of energy sign.

8. Original Boundary-Reflection Model

Pulse reflection at fixed, free, and joined-string boundaries A positive incident pulse reflects inverted at a fixed end, upright at a free end, and splits into reflected and transmitted parts at a string junction. light → heavy string fixed end: reflected pulse is invertedfree end: reflected pulse remains uprightjunction: reflection and transmission share the energyfixedfree
The reflected orientation follows the boundary condition. At a partial boundary, use both phase and energy accounting; the smaller drawing amplitude is not by itself a complete energy calculation.

9. Crossing into a New Medium

A stationary boundary does not create a new oscillation rate, so incident, reflected, and transmitted waves share one frequency:

\[f_i=f_r=f_t\]

The speed is set by each medium. Consequently, wavelength adjusts:

\[\lambda_1=\frac{v_1}{f},\qquad \lambda_2=\frac{v_2}{f}\]
  • Low-density string \(\rightarrow\) high-density string: transmitted speed and wavelength decrease; the reflected displacement is inverted.
  • High-density string \(\rightarrow\) low-density string: transmitted speed and wavelength increase; the reflected displacement remains upright.

10. Quantitative String-Junction Model

For strings under the same steady tension \(F_T\), define transverse wave impedance

\[Z=\frac{F_T}{v}=\sqrt{F_T\mu}\]

For displacement amplitude, the reflection and transmission coefficients at a lossless junction are

\[r=\frac{A_r}{A_i}=\frac{Z_1-Z_2}{Z_1+Z_2},\qquad t=\frac{A_t}{A_i}=1+r=\frac{2Z_1}{Z_1+Z_2}\]

The corresponding energy fractions are

\[\mathcal R=r^2,\qquad \mathcal T=\frac{4Z_1Z_2}{(Z_1+Z_2)^2},\qquad \mathcal R+\mathcal T=1\]

The factor \(Z_2/Z_1\) in transmitted power is why energy cannot be compared by squaring transmitted displacement alone.

11. Worked Boundary Example

Two strings share tension \(64\,\mathrm N\). A wave of frequency \(8.0\,\mathrm{Hz}\) travels from \(\mu_1=0.0040\,\mathrm{kg/m}\) into \(\mu_2=0.016\,\mathrm{kg/m}\).

\[v_1=\sqrt{\frac{64}{0.0040}}=126.5\,\mathrm{m/s},\qquad v_2=\sqrt{\frac{64}{0.016}}=63.2\,\mathrm{m/s}\]
\[\lambda_1=\frac{126.5}{8.0}=15.8\,\mathrm m,\qquad \lambda_2=\frac{63.2}{8.0}=7.91\,\mathrm m\]

Because \(Z\propto\sqrt{\mu}\), \(Z_2=2Z_1\). Therefore

\[r=\frac{Z_1-2Z_1}{Z_1+2Z_1}=-\frac13,\qquad \mathcal R=\frac19,\qquad \mathcal T=\frac89\]

The negative \(r\) means inversion. One ninth of incident energy reflects and eight ninths transmits in the ideal lossless model.

12. What Polarization Means

Polarization describes the orientation of a transverse disturbance. For an electromagnetic wave, the polarization direction is defined by the electric-field direction.

  • A transverse wave has more than one possible displacement direction perpendicular to propagation, so an orientation can be selected.
  • An ideal longitudinal wave oscillates along its propagation direction and cannot be linearly polarized in the same way.
  • Unpolarized light contains rapidly changing or randomly distributed electric-field orientations.
  • Linearly polarized light has a definite electric-field axis.

This optical polarization is different from charge polarization inside a dielectric.

13. A Polarizer Projects the Field

An ideal polarizer transmits the electric-field component parallel to its transmission axis. If \(\theta\) is the angle between the incident polarization and that axis,

\[E_{\mathrm{out}}=E_{\mathrm{in}}\cos\theta\]

Intensity is proportional to field amplitude squared, producing Malus's law for already polarized incident light:

\[I_{\mathrm{out}}=I_{\mathrm{in}}\cos^2\theta\]

For ideal unpolarized light entering its first polarizer, averaging all input orientations gives \(I_1=I_{\mathrm{unpol}}/2\). Apply Malus's law only after that first step.

14. Original Polarizer Model

Electric-field projection through two polarizers A diagonally polarized electric field reaches a vertical polarizer, leaving only its vertical component, then reaches a second polarizer at an angle. incident Eaxis 1projected Eaxis 2θE cos θpropagation
A polarizer changes the transmitted component; it does not rotate every incident field vector intact. The second field amplitude is a projection, so intensity follows a cosine squared.

15. Worked Polarization Examples

Two polarizers: unpolarized light of intensity \(120\,\mathrm{W/m^2}\) enters an ideal vertical polarizer and then a second whose axis is \(30^\circ\) away.

\[I_1=\frac{120}{2}=60\,\mathrm{W/m^2},\qquad I_2=60\cos^2 30^\circ=45\,\mathrm{W/m^2}\]

Three polarizers: place a \(45^\circ\) polarizer between crossed \(0^\circ\) and \(90^\circ\) polarizers.

\[I_{\mathrm{out}}=\frac{I_0}{2}\cos^2 45^\circ\cos^2 45^\circ=\frac{I_0}{8}\]

The middle sheet creates a new \(45^\circ\) polarization state, allowing a component through the final sheet. This does not violate energy conservation; each polarizer absorbs part of the incident energy.

16. Reflection Can Select Polarization

At many nonmetallic surfaces, reflected light is partially polarized. At Brewster's angle, the ideal reflected and refracted rays are perpendicular and the reflected component is maximally linearly polarized:

\[\tan\theta_B=\frac{n_2}{n_1}\]

This helps explain why appropriately oriented polarizing sunglasses reduce glare from horizontal water or road surfaces. The result concerns reflected light at a dielectric boundary, not every reflection from every material.

17. Evidence-Building Investigations

A. Mechanical boundary
  1. Open PhET Wave on a String and select Pulse with low damping.
  2. Compare Fixed End and Loose End using identical pulse settings.
  3. Pause after reflection and record orientation, amplitude, direction, and return time.
  4. Explain each result with the endpoint boundary condition.
B. Optical polarization
  1. Place two polarizing sheets in front of a steady lamp or bright screen; never view the Sun.
  2. Rotate the analyzer from \(0^\circ\) to \(90^\circ\) in \(10^\circ\) steps and record a light-sensor value.
  3. Plot normalized intensity against \(\cos^2\theta\).
  4. Use the slope, intercept, and residuals to evaluate the ideal Malus-law model.

18. AP-Style Reasoning Checks

  • A crest reaches a fixed end. Describe the reflected displacement and phase change.
  • A wave enters a slower medium at a stationary boundary. Which of \(f,v,\lambda\) change?
  • Why can light be polarized but an ideal longitudinal sound wave in air cannot?
  • Polarized light meets an analyzer at \(60^\circ\). What intensity fraction passes?
  • Why is transmitted energy not generally equal to \(t^2\) at a string junction?

Answers: it returns inverted with phase shift \(\pi\); \(f\) stays fixed while \(v\) and \(\lambda\) decrease; polarization requires transverse orientation freedom; \(1/4\); the two media have different impedances, so power also depends on \(Z_2/Z_1\).

Checkpoint · Topic 14.3

Explain how boundary changes and polarization supports or limits this conclusion: λ=v/f=1.00 m.

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