14.8 Double-Slit Interference and Diffraction Gratings

The double-slit experiment (Young’s experiment) shows light behaves like a wave. Monochromatic light hits two narrow slits separated by distance d; each slit acts as a coherent wave source. Where the waves meet on a screen they interfere: constructive interference (bright fringes) when the path difference ΔD = d sinθ = mλ (m = 0, ±1, ±2…), and destructive when the path difference equals half-odd multiples of λ. For small angles (θ < ~10°) you can use d(y/L) ≈ mλ to get fringe positions y on a screen distance L from the slits. The regular interference peaks sit inside a broader single-slit diffraction envelope (so some expected maxima are dimmed). Young’s experiment was key evidence that light has wave properties. For AP exam prep, practice using ΔD = d sinθ and the small-angle approximation; see the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and extra practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Because light is a wave, the two slits act like two coherent sources that send out wavefronts. Where the waves arrive in phase (their path difference ΔD = d sinθ equals an integer multiple of the wavelength, mλ) they add constructively and make a bright fringe. Where the waves arrive out of phase by half a wavelength (ΔD = (m + ½)λ) they cancel and make a dark fringe. For small angles (θ < 10°) you can use d(y/L) ≈ mλ to find fringe positions on the screen. Also remember the slits themselves diffract each beam, so the uniform interference maxima sit inside a broader single-slit diffraction envelope (some interference peaks can be suppressed). This is exactly what Topic 14.8 in the CED explains—Young’s double-slit shows light’s wave nature and lets you measure λ from the fringe spacing. For a quick review, see the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and try practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Think of ΔD = d sin θ as the extra distance light from one slit travels compared to light from the other when they meet at angle θ on the screen. d is the slit separation; draw the two rays to a distant point at angle θ—the one from the farther slit has a little extra slanted segment. That extra segment’s length = d sin θ (basic right-triangle geometry). Why it matters: interference depends on that path difference. If ΔD = mλ (an integer m) the waves arrive in phase → bright fringe; if ΔD = (m + 1/2)λ they arrive out of phase → dark fringe. For small angles (θ < ~10°) sin θ ≈ tan θ ≈ y/L, so the common classroom formula d(y/L) ≈ mλ pops out, which is what you use on the AP to connect λ, d, L, and fringe spacing. For a quick review, see the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs). For practice, check Fiveable’s AP Physics 2 problem set (https://library.fiveable.me/practice/ap-physics-2-revised).

Interference is the pattern from two coherent wavefronts combining; diffraction is the spread of each wave after passing an opening. In the double-slit setup the bright fringes you count come from interference—constructive interference when the path difference ΔD = d sinθ equals mλ (so for small θ, d(y/L) ≈ mλ) and destructive when it’s (m+½)λ. If you only considered interference, the maxima are uniformly spaced (CED 14.8.A.1.i–iv). Diffraction is the single-slit effect from each slit’s finite width: it produces a broad envelope (an overall intensity modulation) that suppresses some interference maxima. The real double-slit pattern is interference fringes sitting inside that single-slit diffraction envelope (CED 14.8.A.1.vi). For more review and practice tied to the AP expectations, see the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and unit resources (https://library.fiveable.me/ap-physics-2-revised/unit-14). For lots of practice problems, check (https://library.fiveable.me/practice/ap-physics-2-revised).

Use d(y_max/L) = mλ when the screen is far from the slits (small-angle approx: θ ≈ y/L, valid for θ ≲ 10°). Quick recipe: - Know what’s given and what you need: d (slit separation), L (slit-to-screen distance), λ (wavelength), m (order, integer 0, ±1, ±2…), y_max (distance from central bright to mth bright). - Start from path-difference: ΔD = d sinθ = mλ. For small θ, sinθ ≈ tanθ ≈ y/L, so d(y/L) ≈ mλ. - Rearranged forms you’ll use: - y = (mλL)/d (find fringe position) - λ = d(y)/(mL) (find wavelength) - d = mλL/y (find slit separation) - m = d y/(λ L) (determine order—must be integer) - Remember m = 0 is central bright; ±m give symmetric fringes. Check small-angle validity (y ≪ L). If result for m isn’t an integer, that fringe doesn’t exist. Example: d = 0.50 mm, L = 2.0 m, λ = 500 nm. y1 = (1·5e-7·2)/(5e-4) = 2.0×10^-3 m = 2.0 mm. For more practice and AP-aligned review, see the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and the AP Physics 2 practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Young’s double-slit experiment shows light behaves like a wave because it produces interference fringes—a regular pattern of bright and dark bands—that only make sense if light waves add (constructive interference) or cancel (destructive interference). Light from the two slits travels different distances to a point on the screen; the path difference ΔD = d sinθ determines whether the waves arrive in phase (mλ → bright) or out of phase ((m+½)λ → dark). The uniformly spaced maxima predicted by simple interference and the superimposed single-slit diffraction envelope match observations, which a particle model can’t explain. That direct link between path difference, wavelength λ, and fringe spacing (d·y/L ≈ mλ for small θ) is why Young’s result is textbook evidence that light has wave properties (CED Topic 14.8, EK 14.8.A.1–3). For a quick AP-focused review and worked examples, see the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs)—and practice problems for this unit are at (https://library.fiveable.me/practice/ap-physics-2-revised).

If you shine white light through a diffraction grating (many evenly spaced slits) you won’t get a single red fringe like with monochromatic light—you get a white central maximum and separated colored spectra for each higher order. Because the interference condition d sinθ = mλ depends on λ, different wavelengths emerge at different angles: longer wavelengths (red) diffract more and appear farther from center than shorter wavelengths (blue/violet). A grating of many slits makes these orders sharp (bright spectral lines) rather than broad fringes from a double slit. Orders can overlap if the grating or wavelengths allow it, and a grating’s resolving power determines how well nearby wavelengths separate. This is exactly what the CED notes (central max white; higher orders disperse into rainbows; red farthest). For practice and review, check the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and Unit 14 resources (https://library.fiveable.me/ap-physics-2-revised/unit-14).

Constructive interference happens when waves from the two slits arrive in phase—their crests line up—so they add and make a bright fringe. Destructive interference happens when they arrive out of phase (crest meets trough) and cancel, making a dark fringe. The key is the path difference ΔD between the two routes: ΔD = d sinθ (d = slit separation, θ = angle to a fringe). If ΔD = mλ (m = 0, ±1, ±2…), you get constructive (bright) fringes. If ΔD = (m + 1/2)λ you get destructive (dark) fringes. For small angles (θ < 10°) you can use the small-angle approx to get fringe position on a screen: d (ymax / L) ≈ mλ, where y is distance from center and L is screen distance. Young’s double-slit shows light’s wave nature and is tested on the AP; practice applying these equations and sketching fringe patterns. For a focused review and examples, check the Topic 14.8 study guide on Fiveable (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and the Unit 14 overview (https://library.fiveable.me/ap-physics-2-revised/unit-14).

The small-angle approximation (sin θ ≈ tan θ ≈ θ, with θ in radians) is used so you can replace d sin θ with d(y/L) (since tan θ ≈ y/L ≈ θ). The reason AP texts call out θ < 10° is practical: at 10° the approximation errors are tiny—on the order of 0.5–1%—so using θ instead of sin θ or tan θ won’t meaningfully change fringe-spacing answers on the exam. For example, at 10°: sin 10° = 0.17365, θ (radians) = 0.17453 (≈0.5% difference); tan 10° = 0.17633 (≈1.1% difference). Those small errors keep d(y/L) ≈ mλ accurate for typical AP problems. If θ gets much larger, the percent error grows and you must use the exact relation d sin θ = mλ. For extra practice on when to use the approximation and related problems, see the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and the AP unit review (https://library.fiveable.me/ap-physics-2-revised/unit-14).

Short answer: both use interference of waves from multiple openings, but they behave differently because of how many slits and the resulting diffraction. Details you should know for the AP exam (CED language): - Double slit: two narrow, coherent slits separated by d produce evenly spaced interference maxima given by d sin θ = mλ (or d(y/L) ≈ mλ for small θ). The pattern is interference fringes (bright/dark bands) *inside* an overall single-slit diffraction envelope if each slit has finite width (see EK 14.8.A.1.i and 14.8.A.1.vi). - Diffraction grating: many (often hundreds or thousands) evenly spaced slits. The interference peaks are much sharper and brighter because many wavefronts reinforce only at discrete angles satisfying d sin θ = mλ. With white light this also disperses colors (higher orders separate wavelengths), so red is farthest from center (EK 14.8.A.4–5). Why it matters on the AP exam: double-slit questions test path-difference, fringe spacing, and single-slit envelope; gratings test resolving power and wavelength dispersion. For extra review, check the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Read the diagram like a map: identify the central bright (m = 0), count the order m of the bright you want, measure the distance y from center to that m-th bright on the screen, and note the screen distance L from the slits. Core formulas (from the CED): - Path difference: ΔD = d sin θ. - Constructive interference: d sin θ = m λ. - Small-angle approx (θ < 10°): sin θ ≈ tan θ ≈ y / L so d (y / L) ≈ m λ → λ ≈ d y / (m L). Practical uses: - If you measure adjacent bright spots, fringe spacing Δy ≈ λ L / d (use m = 1 difference). - For a diffraction grating, d is the slit spacing (often given as 1/N where N is lines per meter); use the same d sin θ = m λ relation but watch that multiple orders separate colors for white light. - If the pattern shows an envelope (single-slit diffraction), make sure you use maxima inside that envelope and not suppressed peaks. Want worked examples and practice problems? Check the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and more practice at (https://library.fiveable.me/practice/ap-physics-2-revised). These types of questions appear on the AP exam as translation-between-representations and mathematical-routines problems, so practice measuring y, L, and identifying m.

Because a grating separates light by path difference, the interference condition is d sin θ = mλ. For a given order m and fixed slit spacing d, sin θ is proportional to the wavelength λ. So longer wavelengths (red ≈ 700 nm) satisfy the equation at larger angles than shorter wavelengths (blue ≈ 450 nm). That’s why, when white light hits a grating, the central (m = 0) spot stays white but higher-order maxima spread into a rainbow with red appearing farthest from the center (CED 14.8.A.5, 14.8.A.1.iv–v). This is exactly the idea you need for AP questions: identify the diffraction/interference equation and explain how θ depends on λ. For a quick review, check the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

If you increase the slit separation d, the bright fringes move closer together; if you decrease d, the fringes spread farther apart. Reason: constructive interference occurs when d sin θ = mλ, and for small angles y ≈ L sin θ so the small-angle formula gives d (y/L) ≈ mλ → y_m ≈ mλL/d. So y (fringe spacing) is inversely proportional to d. Physically, larger d produces a bigger path-difference for a given angle so adjacent maxima occur at smaller angles (closer on the screen). Keep in mind: the pattern you actually see is interference fringes inside the single-slit diffraction envelope (14.8.A.1.vi). Use the small-angle approximation only for θ < ~10°. This relationship and how to manipulate it is exactly the kind of quantitative reasoning asked on AP Physics 2 free-response and multiple-choice items. For a quick refresher, check the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs) and try practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Scientists measure tiny distances or wavelengths by turning interference into a ruler. In a double-slit or grating experiment, bright fringes occur where path difference ΔD = d sinθ = mλ. For small θ you can use d(ym/L) ≈ mλ, so by measuring fringe spacing y (and knowing slit separation d and screen distance L) you solve for λ or d. Diffraction gratings (many evenly spaced slits) sharpen peaks and let you measure λ more precisely because peaks are narrower and higher-order m are easier to resolve—that increases resolving power. Practically: use higher m, longer L, or larger d (for slit arrays) and average many fringes to reduce error. This directly connects to CED essentials (path difference, fringe spacing, grating dispersion) and shows up in AP problems asking you to derive or use d(y/L)=mλ and explain single-slit envelopes. For extra practice, check the Topic 14.8 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs), the unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-14), and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Think of two effects happening at once. The double slit produces evenly spaced bright and dark interference fringes (constructive when d sinθ = mλ), but each slit has a finite width so each slit also diffracts light, producing a broader pattern whose intensity falls to zero at certain angles (single-slit minima at a sinθ = mλ). “Superimposed within the envelope” means the bright interference lines sit inside that slowly varying single-slit intensity curve—some interference maxima are strong (inside the envelope peaks) and some are suppressed or disappear where the single-slit intensity goes to zero. On the AP exam you should be able to: write d sinθ = mλ for interference, recognize the uniform fringe spacing, and explain that the overall envelope comes from single-slit diffraction (Topic 14.8; see the study guide for examples) (https://library.fiveable.me/ap-physics-2-revised/unit-6/8-double-slit-interference-and-diffraction-gratings/study-guide/K1umdgpBDu7M2fEs). For extra practice try problems in the Unit 14 practice set (https://library.fiveable.me/practice/ap-physics-2-revised).