IInsomniaStudyroom
Focus music
Pomodoro
25:00

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 14.7

Diffraction from Openings and Edges

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

1. Topic Lens

Diffraction from Openings and Edges 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 Diffraction from Openings and EdgesOriginal schematic connecting Diffraction from Openings and Edges to Waves, Sound, and Physical Optics.
Diffraction from Openings and Edges: 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 14.7 Lesson

5. Learning Targets

  • Explain diffraction at openings and edges using wavefronts and secondary wavelets.
  • Predict how the ratio of opening width to wavelength controls angular spreading.
  • Derive and apply the single-slit minimum condition.
  • Interpret single-slit intensity graphs and calculate the central-maximum width.
  • Distinguish near-field, far-field, single-edge, slit, and circular-aperture diffraction models.

6. Diffraction Is Wave Spreading

Diffraction is the redistribution of a wave after it passes through an opening or encounters an obstacle edge. It occurs for mechanical and electromagnetic waves.

  • The wave continues into regions that a straight-ray model would label as geometric shadow.
  • Different parts of the same wavefront can interfere after the boundary.
  • The medium and source frequency need not change.
  • Diffraction is strongest and easiest to observe when the relevant opening or obstacle dimension is comparable to the wavelength.

A finite opening always produces some diffraction. When the opening is enormous compared with \(\lambda\), the angular spreading is simply too small to notice easily.

7. Huygens–Fresnel Picture

Treat every unobstructed point on a wavefront as a source of a forward-moving secondary wavelet. The later field is the superposition of those wavelets.

Across a wide opening, most wavelets reinforce straight ahead and only the edges noticeably curve. Across a narrow opening, fewer wavelets span the aperture and the emerging wave spreads over a larger range of directions. Interference between these contributions produces the observed bright and dark structure.

8. Original Opening-Size Model

Diffraction for wide, comparable, and narrow openings Plane wavefronts approach three barriers. A wide opening produces a mostly straight transmitted wave, an opening comparable to the wavelength produces strong curved spreading, and a narrow opening produces broad spreading with reduced transmitted energy. a ≫ λ · weak angular spreadinga ≈ λ · strong diffractiona < λ · broad spreading, less transmitted poweraaa
The three rows use the same incident wavelength. They compare angular shape, not absolute brightness; shrinking an opening generally also reduces the transmitted power.

9. The Dimensionless Control Ratio

\(a/\lambda\gg1\)Ray behavior is a useful approximation; spreading is confined to small angles.
\(a/\lambda\sim1\)Strong curvature and broad angular redistribution are visible.
\(a/\lambda<1\)The opening acts roughly like a small source, but transmitted amplitude may be small.

This explains why audible sound bends around a doorway more noticeably than visible light: their wavelengths differ enormously while the doorway size is unchanged.

10. Slit Orientation Predicts Spread Direction

Diffraction is controlled by the narrow dimension. A tall vertical slit that is narrow horizontally produces wide horizontal spreading and little vertical spreading. Rotating the slit rotates the diffraction pattern.

This inverse relationship is important: making an aperture narrower in one direction makes its far-field pattern wider in that direction.

11. Deriving Single-Slit Minima

Consider a slit of width \(a\). At observation angle \(\theta\), the path difference between wavelets from the two slit edges is

\[\Delta r=a\sin\theta\]

If \(a\sin\theta=\lambda\), divide the slit into two equal halves. Every point in the upper half can be paired with a point \(a/2\) away in the lower half. Each pair has path difference \(\lambda/2\) and cancels. Extending this pairing gives

\[a\sin\theta_m=m\lambda,\qquad m=\pm1,\pm2,\pm3,\ldots\]

These are minimum conditions. \(m=0\) is excluded because \(\theta=0\) is the central maximum.

12. Which Orders Can Exist?

Because \(|\sin\theta|\le1\), a minimum of order \(m\) exists only if

\[|m|\lambda\le a\]

For \(a=2.5\lambda\), minima \(m=\pm1\) and \(m=\pm2\) exist, but \(m=\pm3\) do not. A missing mathematical order is not the same as a detector failing to see a very dim fringe.

13. Original Single-Slit Intensity Graph

Single-slit far-field intensity pattern A broad bright central maximum lies between the first minima. Narrower, weaker side maxima occur symmetrically outside it. central maximumwidth between first minima−2−1+1+2θI₀
The central maximum is twice the angular width between adjacent side minima. Side maxima are weaker and are not located exactly halfway between minima.

14. Intensity Envelope

In the far-field model for a uniformly illuminated slit, define

\[\beta=\frac{\pi a\sin\theta}{\lambda}\]

The angular intensity is

\[I(\theta)=I_0\left(\frac{\sin\beta}{\beta}\right)^2\]

At \(\theta=0\), use the limit \(\sin\beta/\beta\to1\), so \(I=I_0\). The formula assumes coherent monochromatic illumination across the slit and far-field observation.

15. Central-Maximum Width

The first minima occur at \(\sin\theta_1=\lambda/a\). For small angles and a screen distance \(L\), \(y_1\approx L\theta_1\), giving

\[W_{\mathrm{central}}=2y_1\approx\frac{2L\lambda}{a}\]
Increase \(\lambda\)The central maximum becomes wider.
Decrease \(a\)The central maximum becomes wider.
Increase \(L\)The linear pattern expands, but its angular width stays the same.

16. Worked Single-Slit Example

Light of wavelength \(600\,\mathrm{nm}\) passes through a slit of width \(0.120\,\mathrm{mm}\) and reaches a screen \(2.00\,\mathrm m\) away.

\[\sin\theta_1=\frac{600\times10^{-9}}{1.20\times10^{-4}}=5.00\times10^{-3}\]

The angle is small, so \(\theta_1\approx5.00\times10^{-3}\,\mathrm{rad}\). Therefore

\[y_1\approx L\theta_1=(2.00)(5.00\times10^{-3})=1.00\times10^{-2}\,\mathrm m\]
\[W_{\mathrm{central}}=2y_1=2.00\,\mathrm{cm}\]

Halving the slit width would double the width of the pattern in this approximation, even though less total light would pass through.

17. Sound Through a Doorway

A \(500\,\mathrm{Hz}\) sound in air at \(343\,\mathrm{m/s}\) has

\[\lambda=\frac{v}{f}=\frac{343}{500}=0.686\,\mathrm m\]

For a \(0.90\,\mathrm m\)-wide doorway, \(a/\lambda\approx1.31\), so broad spreading is expected. Visible light has wavelength near \(10^{-7}\,\mathrm m\), making the same doorway millions of wavelengths wide and its diffraction angle extremely small.

18. Diffraction at a Single Edge

An opaque straight edge blocks part of a wavefront. Wavelets from the unblocked region extend into the geometric shadow and interfere, producing a softened boundary and, under suitable coherent conditions, alternating bright and dark structure near the edge.

The single-slit condition \(a\sin\theta=m\lambda\) does not apply directly because there is no finite slit width \(a\). Edge diffraction is usually treated with near-field wavefront geometry or Fresnel-zone methods.

19. Near Field and Far Field

Fresnel diffractionSource or screen is close enough that wavefront curvature matters. The pattern changes noticeably with distance.
Fraunhofer diffractionIncident and observed waves are approximately planar, achieved far away or with lenses. The angular pattern has a stable form.

The single-slit angular formulas above describe the Fraunhofer regime. A classroom screen at finite distance can approximate it when the geometry and slit size make the far-field condition adequate.

20. Circular Openings and Resolution

A circular aperture produces a central Airy disk surrounded by weaker rings. Its first minimum is approximately

\[\theta_{\min}\approx1.22\frac{\lambda}{D}\]

where \(D\) is aperture diameter and \(\theta\) is in radians. The Rayleigh criterion uses this angle as a practical estimate of when two point images are just resolvable.

\[\text{larger }D\Rightarrow\text{smaller diffraction spot and better angular resolution}\]

This is a circular-aperture result; do not replace the slit width \(a\) with \(D\) and keep the single-slit coefficient.

21. Evidence-Building Investigation

A. Virtual aperture study
  1. Open PhET Wave Interference and select one opening.
  2. Hold wavelength fixed while comparing at least three opening widths.
  3. Measure an angular or screen-width indicator of spreading and graph it against \(1/a\).
  4. Repeat after changing wavelength and state which dimensionless ratio organizes all trials.
B. Pattern-model audit
  1. Sketch the predicted central maximum and first two minima before collecting data.
  2. Identify the symmetry axis and test whether minima appear in \(\pm m\) pairs.
  3. Compare the measured central width with \(2L\lambda/a\).
  4. Discuss slit-width tolerance, screen distance, alignment, finite detector size, and ambient light.

Safety: use only a classroom-approved low-power laser under instructor supervision. Never look into a beam or aim it toward people, reflective surfaces, vehicles, or aircraft.

22. Common Reasoning Traps

  • “Diffraction begins only when \(a=\lambda\).” It always occurs; the ratio controls how noticeable the angular spreading is.
  • “The wave speed decreases because the wave bends.” Directional redistribution in the same medium does not by itself change speed or frequency.
  • Using \(m=0\) as a minimum: the center of a single-slit pattern is bright.
  • Confusing slit and double-slit equations: \(a\sin\theta=m\lambda\) identifies single-slit minima.
  • Assuming a narrower slit makes a narrower beam: it produces broader angular spreading.
  • Applying small-angle formulas at large angles: use \(y=L\tan\theta\) and the exact sine condition when needed.

23. AP-Style Reasoning Checks

  1. An opening changes from \(10\lambda\) wide to \(2\lambda\) wide. Predict the angular spreading.
  2. A vertical slit is made narrower horizontally. In which direction does the far-field pattern expand?
  3. Why is \(m=0\) absent from the single-slit minimum equation?
  4. If wavelength doubles while \(a\) and \(L\) remain fixed, what happens to the central width?
  5. Adjacent first minima are \(3.0\,\mathrm{cm}\) apart on a screen \(1.5\,\mathrm m\) away. Estimate \(\lambda/a\).
  6. Why does increasing telescope diameter improve diffraction-limited resolution?

Answers: it becomes broader; horizontally, perpendicular to the long slit direction; \(\theta=0\) is constructive and forms the central maximum; it doubles in the small-angle model; \(W=2L\lambda/a\) gives \(\lambda/a=0.030/(3.0)=0.010\); a larger aperture decreases the minimum resolvable angle.

Checkpoint · Topic 14.7

Explain how diffraction from openings and edges 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.