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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 14.8

Two-Slit and Grating Pattern Geometry

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

1. Topic Lens

Two-Slit and Grating Pattern Geometry 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 Two-Slit and Grating Pattern GeometryOriginal schematic connecting Two-Slit and Grating Pattern Geometry to Waves, Sound, and Physical Optics.
Two-Slit and Grating Pattern Geometry: 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 14.8 Lesson

5. Learning Targets

  • Relate two-slit path difference to bright and dark interference directions.
  • Convert angular fringe conditions into screen positions and fringe spacing.
  • Separate the roles of slit spacing \(d\) and individual slit width \(a\).
  • Explain why many equally spaced slits produce narrow principal maxima.
  • Use grating line density, order limits, and angular dispersion to analyze spectra.

6. One Wavefront Creates Two Coherent Sources

In Young's arrangement, one monochromatic wave illuminates two narrow slits. Because the two emerging waves come from the same incident wavefront, they have a stable phase relation and can form a persistent interference pattern.

  • The center is bright when both slits are illuminated in phase.
  • A change in path length produces a phase difference at the screen.
  • Alternating constructive and destructive directions form bright and dark fringes.
  • Independent ordinary light sources generally do not maintain the phase stability needed for stationary optical fringes.

7. Coherence Conditions

A clear pattern requires sufficiently coherent illumination: nearly one frequency for temporal coherence and a predictable phase relation across the two slits for spatial coherence.

Unequal slit illumination reduces fringe visibility but does not move the ideal fringe positions. If the intensities from the individual slits are \(I_1\) and \(I_2\),

\[I_{\max}=(\sqrt{I_1}+\sqrt{I_2})^2,\qquad I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2\]

8. Original Two-Slit Geometry

Path geometry from two slits to a screen point Two slits separated by distance d send waves to a point P on a distant screen. Nearly parallel rays make angle theta with the central axis, and the path difference is d sine theta. S₁S₂dPθr₁r₂central axisscreen
When screen distance \(L\gg d\), the two rays are nearly parallel. Projecting the slit separation onto their direction gives the path difference \(\Delta r=d\sin\theta\).

9. Bright and Dark Directions

For two slits that begin in phase, the far-field path difference is

\[\Delta r=r_2-r_1\approx d\sin\theta\]

Whole wavelengths preserve phase, while half-integer wavelengths reverse it:

\[\text{bright: }d\sin\theta_m=m\lambda,\qquad m=0,\pm1,\pm2,\ldots\]
\[\text{dark: }d\sin\theta_m=\left(m+\frac12\right)\lambda,\qquad m=0,\pm1,\pm2,\ldots\]

If the slits begin with phase difference \(\phi_0\), add it to \(2\pi\Delta r/\lambda\); the central point is not necessarily bright.

10. From Angle to Screen Position

Exact screen geometry gives \(y=L\tan\theta\). When \(|y|\ll L\),

\[\sin\theta\approx\tan\theta\approx\frac{y}{L}\]

Thus the bright-fringe position and adjacent bright-fringe spacing are approximately

\[y_m\approx\frac{m\lambda L}{d},\qquad \Delta y\approx\frac{\lambda L}{d}\]

The angular condition is fundamental. The screen-spacing equation is a small-angle approximation and should not be used automatically for high orders or large angles.

11. Pattern-Trend Predictions

Increase \(\lambda\)Fringes move farther apart.
Increase \(L\)Linear spacing grows; angular spacing is unchanged.
Increase \(d\)Fringes move closer together.
Dim one slitVisibility decreases; ideal positions stay fixed.
Block one slitTwo-source fringes disappear; one-slit diffraction remains.
Enter a mediumFrequency stays fixed and \(\lambda=\lambda_0/n\), so angular spacing decreases.

12. Worked Two-Slit Example

Light of wavelength \(520\,\mathrm{nm}\) illuminates slits separated by \(d=0.250\,\mathrm{mm}\). A screen is \(L=2.00\,\mathrm m\) away.

\[\Delta y\approx\frac{(520\times10^{-9})(2.00)}{2.50\times10^{-4}}=4.16\times10^{-3}\,\mathrm m\]

The third bright fringe lies approximately

\[y_3=3\Delta y=12.5\,\mathrm{mm}\]

Check the approximation: \(y_3/L=0.00624\ll1\), so the small-angle model is well justified.

13. Finding an Unknown Wavelength

In an experiment, six bright-to-bright intervals span \(18.0\,\mathrm{mm}\), so \(\Delta y=3.00\,\mathrm{mm}\). With \(L=1.50\,\mathrm m\) and \(d=0.300\,\mathrm{mm}\),

\[\lambda=\frac{d\Delta y}{L}=\frac{(3.00\times10^{-4})(3.00\times10^{-3})}{1.50}=6.00\times10^{-7}\,\mathrm m\]

Measuring across many intervals reduces the fractional uncertainty compared with measuring a single narrow gap.

14. Real Slits: Interference Inside a Diffraction Envelope

Each slit has finite width \(a\), so it diffracts. The two-slit interference fringes are multiplied by the single-slit envelope:

\[I(\theta)=I_{\mathrm{peak}}\cos^2\left(\frac{\pi d\sin\theta}{\lambda}\right)\left[\frac{\sin\left(\pi a\sin\theta/\lambda\right)}{\pi a\sin\theta/\lambda}\right]^2\]
Separation \(d\)Controls the spacing of the narrow interference fringes.
Width \(a\)Controls the broad diffraction envelope that modulates their brightness.

A two-slit bright order disappears if it coincides with a single-slit minimum. This is a missing order, not a failure of the interference equation.

15. Original Envelope-and-Fringe Graph

Two-slit fringes contained by a single-slit diffraction envelope Many narrow interference peaks lie beneath a broad diffraction envelope. Peaks become lower away from the central direction and disappear at envelope minima. single-slit envelopetwo-slit maximaθ
The individual peak heights are schematic. The key geometry is a dense fringe scale set by \(d\) inside a broader envelope scale set by \(a\).

16. Missing-Order Example

Interference maxima occur at \(d\sin\theta=m\lambda\), while diffraction minima occur at \(a\sin\theta=p\lambda\). If \(d/a=4\), then

\[m\frac{\lambda}{d}=p\frac{\lambda}{a}\quad\Rightarrow\quad m=p\frac{d}{a}=4p\]

Every fourth interference order \(m=\pm4,\pm8,\ldots\) lies at an envelope minimum and is absent. The central order remains bright because the minimum index \(p=0\) is not allowed.

17. From Two Slits to a Grating

A diffraction grating has many equally spaced slits. Adjacent slits still differ in path by \(d\sin\theta\), so all \(N\) contributions align at

\[d\sin\theta_m=m\lambda\]

The principal maxima occur at the same angles as for two slits with the same \(d\), but increasing \(N\) makes them narrower and the regions between them darker. At exact alignment, field amplitudes add roughly as \(N\), so ideal peak intensity scales as \(N^2\) relative to one slit.

18. Why Many Slits Sharpen the Peaks

At a principal maximumEach adjacent phase step is \(2\pi m\); all slit phasors align.
Slightly off the maximumSmall phase steps accumulate across many slits, spreading the phasors around an arc.
ResultCancellation becomes strong after a smaller angular change, producing a narrower peak.

An ideal array factor can be written

\[I_N\propto\left(\frac{\sin N\alpha}{\sin\alpha}\right)^2,\qquad \alpha=\frac{\pi d\sin\theta}{\lambda}\]

Real finite-width slits still multiply this narrow multi-slit structure by a broader single-slit envelope.

19. Converting Grating Line Density

If a grating has \(g\) lines per unit length, its center-to-center spacing is

\[d=\frac{1}{g}\]

For \(600\) lines/mm,

\[g=6.00\times10^5\,\mathrm{m^{-1}},\qquad d=1.67\times10^{-6}\,\mathrm m\]

Do not insert “600” directly into a meter-based equation. Convert the line density before taking its reciprocal.

20. Worked Grating Example

Light of wavelength \(500\,\mathrm{nm}\) reaches a \(600\)-lines/mm grating. With \(d=1.67\,\mu\mathrm m\), first order occurs at

\[\sin\theta_1=\frac{500\times10^{-9}}{1.67\times10^{-6}}=0.300,\qquad \theta_1=17.5^\circ\]

The maximum possible order satisfies \(|m|\lambda\le d\):

\[|m|\le\frac{d}{\lambda}=3.34\quad\Rightarrow\quad m_{\max}=3\]

Orders \(m=0,\pm1,\pm2,\pm3\) are geometrically possible. Envelope strength and detector range may still limit which are observed.

21. White Light and Angular Dispersion

At \(m=0\), all wavelengths overlap. For \(m\ne0\), longer wavelengths satisfy the grating equation at larger angles for a fixed order. Differentiating gives

\[\frac{d\theta}{d\lambda}=\frac{m}{d\cos\theta}\]

Higher order, smaller spacing, and larger angle increase angular dispersion. Different orders can overlap—for example, second-order \(\lambda\) can share an angle with first-order \(2\lambda\).

22. Resolving Nearby Wavelengths

For an ideal grating with \(N\) illuminated slits, a common resolving-power result is

\[R=\frac{\lambda}{\Delta\lambda}=mN\]

A larger illuminated slit count narrows the principal maxima; a higher order also improves ideal spectral resolution. Dispersion and resolution are related but not identical: separation angle alone does not specify whether finite-width peaks can be distinguished.

23. Evidence-Building Investigation

A. Two-slit geometry
  1. Open PhET Wave Interference and select two openings.
  2. Hold \(\lambda\) and screen distance fixed while changing slit separation.
  3. Measure several fringe intervals and test \(\Delta y\propto1/d\).
  4. Then change wavelength and test \(\Delta y\propto\lambda\).
B. Grating measurement
  1. Use a classroom-approved low-power laser and a grating with known line density under instructor supervision.
  2. Measure symmetric \(+m\) and \(-m\) positions to locate the central axis and reduce alignment bias.
  3. Use \(\theta=\tan^{-1}(y/L)\), then calculate \(\lambda=d\sin\theta/m\).
  4. Report uncertainty from screen distance, peak width, ruler resolution, and grating specification.

Safety: never look into a laser beam or aim it toward people, reflective surfaces, vehicles, or aircraft.

24. Common Reasoning Traps

  • Swapping \(a\) and \(d\): slit width sets the envelope; slit separation sets interference spacing.
  • Using the single-slit minimum equation for bright double-slit fringes: the symbols may look similar, but the conditions differ.
  • Assuming all orders exist: require \(|m|\lambda\le d\).
  • Using \(y=m\lambda L/d\) at large angles: return to \(d\sin\theta=m\lambda\) and \(y=L\tan\theta\).
  • Claiming a grating moves the principal maxima solely because it has more slits: if \(d\) is unchanged, their ideal angles stay the same while peaks sharpen.
  • Confusing dispersion and resolution: angular separation and peak width are different performance measures.

25. AP-Style Reasoning Checks

  1. What path difference produces the second-order bright fringe for in-phase slits?
  2. If \(d\) doubles, what happens to small-angle fringe spacing?
  3. Why can a predicted double-slit maximum be absent?
  4. A grating changes from \(N=20\) to \(N=100\) illuminated slits without changing \(d\). What changes about the principal maxima?
  5. Which color appears at a larger first-order angle, red or violet?
  6. A grating has \(500\) lines/mm. Find \(d\) in meters.

Answers: \(2\lambda\); it halves; the order can coincide with a finite-slit diffraction minimum; the angles remain fixed while peaks become narrower and ideally stronger; red; \(d=1/(5.00\times10^5)=2.00\times10^{-6}\,\mathrm m\).

Checkpoint · Topic 14.8

Explain how two-slit and grating pattern geometry 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.