science-2018:P.W.2
science-2018:P.W.2 Light phenomena
Learning goal: Predict and calculate diffraction patterns from a double slit and a diffraction grating using d sin θ = mλ, describe how single-slit patterns change with slit width and wavelength, determine the wavelength of a laser from a measured pattern, compare the electromagnetic-wave model and the photon model of light, and explain the visible spectrum and the color of objects.
Diffraction patterns are interference made visible. When light passes through an opening comparable in size to its wavelength it spreads out (diffracts), and when it passes through two or more openings the spreading waves overlap and interfere. On a distant screen the result is a pattern of alternating bright lines (constructive interference) and dark lines (destructive interference). For two narrow slits a distance d apart, light leaving the slits toward an angle θ has a path difference of d sin θ; the bright fringes occur where that path difference is a whole number of wavelengths: d sin θ = mλ, with m = 0, 1, 2, ... the order of the fringe. The central bright fringe (m = 0) is straight ahead; the first-order fringes sit on each side at sin θ = λ/d, and so on. A diffraction grating is thousands of slits per centimeter; the same condition applies with d the spacing between adjacent lines, and the many slits make the bright lines much sharper and brighter, which is what makes gratings useful for measuring wavelengths. For small angles the fringe spacing on a screen at distance L is approximately Δy = λL/d. A single slit of width a also produces a pattern: a wide central bright band with weaker bands beside it, with dark minima where a sin θ = mλ (m = 1, 2, ...).
Predictable changes. The patterns change in ways you can predict from the equations. Longer wavelength (red rather than blue) means larger angles and wider spacing; this is why a grating spreads white light into a spectrum with red bent most. Closer slits (smaller d) push the fringes farther apart; a narrower single slit (smaller a) widens the central band. A grating with more lines per millimeter has a smaller d and therefore larger angles. There is also a limit: because sin θ cannot exceed 1, the highest order visible is the largest m with mλ/d ≤ 1.
Worked model: a laser and a grating. A red laser (λ = 650 nm = 6.50 × 10⁻⁷ m) shines through a grating with 500 lines per millimeter, so d = 1.00 × 10⁻³ m/500 = 2.00 × 10⁻⁶ m. First order: sin θ₁ = 6.50 × 10⁻⁷/2.00 × 10⁻⁶ = 0.325, θ₁ = 19.0°. Second order: sin θ₂ = 0.650, θ₂ = 40.5°. Third order: sin θ₃ = 0.975, θ₃ = 77.2°, just possible; a fourth order would need sin θ = 1.30, impossible, so only three orders appear on each side. Now reverse the logic to measure a wavelength. A different laser through the same grating produces first-order spots 0.52 m either side of the center on a screen 1.50 m away: tan θ = 0.52/1.50 = 0.347, θ = 19.1°, sin θ = 0.328, so λ = d sin θ = (2.00 × 10⁻⁶)(0.328) = 6.55 × 10⁻⁷ m = 655 nm, a red laser. Note that the angle is large enough that the small-angle formula would be off by a few percent; use tan θ for the geometry and sin θ in the grating equation.
Two models of light. Physicists describe light, which crosses empty space at about 3.00 × 10⁸ m/s, with two models, and each is backed by strong experimental evidence. In the electromagnetic-wave model, radiant energy travels as a wave of changing electric and magnetic fields spreading outward from its source; it explains interference, diffraction, refraction and polarization. In the photon model, light travels as discrete packets called photons, each one a single quantum of energy (how many are emitted, when and in which directions depends on the source); each photon's energy is directly proportional to the frequency of the radiation, E = hf, where h = 6.63 × 10⁻³⁴ J·s is Planck's constant. A red photon (λ = 650 nm, f = c/λ = 4.6 × 10¹⁴ Hz) carries 3.1 × 10⁻¹⁹ J; a violet photon (400 nm, 7.5 × 10¹⁴ Hz) carries 5.0 × 10⁻¹⁹ J, more energy per photon because its frequency is higher. Scientists use whichever model suits the problem: the wave model to design a grating, the photon model to explain why ultraviolet light, but not red light of any brightness, ejects electrons from a clean calcium surface (calcium needs light shorter than about 459 nm), and why ultraviolet can break molecules in skin one photon at a time (very intense visible light can still harm tissue, by heating it). You are not expected to know the detailed evidence for either model at this level, only that both are used and what each says.
Misconception: the photon model means light is a stream of tiny balls. A photon is a packet of energy, not a solid particle with a position you can track, and the same light that arrives in photons also produces interference fringes. The two models are descriptions of different aspects of one phenomenon; neither is "the truth" with the other wrong.
The visible spectrum and color. Human eyes respond to just a thin slice of the electromagnetic spectrum, wavelengths of roughly 400 nm (violet) to 700 nm (red), with blue, green, yellow and orange between. Approximate bands, whose edges are not sharp and are drawn a little differently in different sources (this lesson uses one common convention): violet 400 to 450 nm, blue 450 to 495 nm, green 495 to 570 nm, yellow 570 to 590 nm, orange 590 to 620 nm and red 620 to 700 nm. Different colors correspond to different frequencies and therefore different radiant energies; the radiation from the Sun or a hot lamp filament is a mixture of all the visible colors, which together appear white. An object lit by white light has pigments that absorb some colors and scatter the remaining ones back in every direction; a leaf looks green because its pigments reflect green wavelengths and absorb most of the red and blue. A red filter does not add red; it absorbs the other colors and transmits red. Beyond the visible band, infrared, microwaves and radio waves have longer wavelengths and lower photon energies; ultraviolet, X-rays and gamma rays have shorter wavelengths and higher photon energies, and they are all the same kind of wave traveling at c.
Investigation: measure a laser's wavelength. This is a teacher-supervised activity with a low-power (class 2) laser pointer. Materials: the laser in a clamp, a diffraction grating slide of known lines per millimeter, a screen, a meter stick. Steps: fix the laser so it passes through the grating at right angles and strikes the screen at least 1.0 m away; measure L from grating to screen and the distance from the central spot to each first-order spot on both sides, averaging the two; calculate θ from tan θ, then λ = d sin θ; repeat with the second order if it is on the screen. A result that supports the model: both orders give the same wavelength to within your measurement uncertainty (a few percent is a reasonable classroom target), and it lies in the range printed on the laser. Safety: never look into the beam or its reflections, never point it at anyone, keep it below eye level and switch it off when not measuring; remove reflective jewelry. What the model cannot show: the photons; the pattern on the screen looks continuous even though the energy arrives in packets.
Worked example 1 — orders from a grating. Grating 500 lines/mm (d = 2.00 × 10⁻⁶ m), λ = 650 nm. m = 1: sin θ = 0.325, θ = 19.0°. m = 2: sin θ = 0.650, θ = 40.5°. m = 3: sin θ = 0.975, θ = 77.2°. m = 4 would need sin θ = 1.30, so it does not exist; three orders on each side of the center.
Worked example 2 — wavelength from a screen measurement. Same grating, first-order spots 0.52 m from the center on a screen 1.50 m away. tan θ = 0.347, θ = 19.1°, sin θ = 0.328, λ = (2.00 × 10⁻⁶)(0.328) = 6.55 × 10⁻⁷ m = 655 nm. A 3 mm error in the 0.52 m measurement changes λ by about 0.5 %, so the measurement is precise to a few nanometers.
Worked example 3 — double-slit fringe spacing. Two slits 0.25 mm apart, green laser λ = 532 nm, screen 2.0 m away. Small-angle spacing Δy = λL/d = (5.32 × 10⁻⁷)(2.0)/(2.5 × 10⁻⁴) = 4.3 × 10⁻³ m = 4.3 mm between adjacent bright fringes. Halving the slit separation to 0.125 mm doubles the spacing to 8.5 mm; switching to a red 650 nm laser at the original separation widens it to 5.2 mm.
Worked example 4 — single-slit width. A slit 0.10 mm wide with 650 nm light: first minimum at sin θ = λ/a = 6.5 × 10⁻⁷/1.0 × 10⁻⁴ = 0.0065, θ = 0.37°. On a screen 2.0 m away the central bright band extends 2.0 tan θ = 13 mm each side, 26 mm wide in total. Narrowing the slit to 0.05 mm doubles the width of the central band; a wider slit narrows it toward a plain shadow of the slit.
Worked example 5 — photon energies. Green light, λ = 500 nm: f = c/λ = 3.00 × 10⁸/5.00 × 10⁻⁷ = 6.0 × 10¹⁴ Hz, E = hf = (6.63 × 10⁻³⁴)(6.0 × 10¹⁴) = 4.0 × 10⁻¹⁹ J per photon. A 1 W green source emits about 1/(4.0 × 10⁻¹⁹) = 2.5 × 10¹⁸ photons per second, which is why the beam looks continuous.
Representation task: predict the pattern. For a grating with 300 lines/mm and a 532 nm laser, calculate d, the first- and second-order angles (9.2° and 18.6°) and the highest order that can appear, and draw the pattern on a screen 1.0 m away to scale with the spot positions labeled in centimeters. Then draw, without numbers, how the pattern would change for (a) a red laser, (b) a grating with 600 lines/mm, and (c) a single slit instead of a grating, with one sentence of reasoning for each.
Independent performance: wavelength measurement and model comparison. Carry out the supervised grating investigation and write a report. Success criteria: a labeled diagram of the set-up with L, the spot separations and θ marked; measurements for at least two orders with λ calculated from each and the percent difference between them; a comparison with the wavelength stated on the laser; an uncertainty estimate based on the smallest length you could measure; a paragraph comparing the wave model and the photon model of light, stating what each explains and giving the photon energy for your laser's wavelength; and one paragraph explaining, in terms of reflected and absorbed wavelengths, why the screen looked white under room light but showed a colored spot from the laser.
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
A grating has 400 lines per millimeter. Find the angle of the first-order bright line for light of wavelength 600 nm and explain what happens to that angle for a grating with 800 lines per millimeter.
d = 1.00 × 10⁻³ m/400 = 2.50 × 10⁻⁶ m. sin θ = mλ/d = (1)(6.00 × 10⁻⁷)/(2.50 × 10⁻⁶) = 0.240, so θ = 13.9°. With 800 lines/mm, d halves to 1.25 × 10⁻⁶ m, sin θ doubles to 0.480 and θ = 28.7°: closer lines spread the pattern to larger angles.
Light of unknown wavelength through a grating with d = 3.00 × 10⁻⁶ m gives a first-order line at 12.0°. Calculate the wavelength and name its color.
λ = d sin θ/m = (3.00 × 10⁻⁶)(sin 12.0°)/1 = (3.00 × 10⁻⁶)(0.208) = 6.24 × 10⁻⁷ m = 624 nm, at the orange end of the red band in the lesson's list of approximate bands (orange 590 to 620 nm, red 620 to 700 nm). Because the band edges are not sharp, red or orange-red are both acceptable names.
In a double-slit experiment the bright fringes are 3.0 mm apart. Predict the new spacing if (a) the slit separation is doubled, (b) the wavelength is changed from 450 nm to 675 nm, and explain each prediction.
Fringe spacing Δy = λL/d for small angles. (a) Doubling d halves the spacing to 1.5 mm, because the path difference of one wavelength is reached at a smaller angle when the slits are farther apart. (b) Changing λ from 450 nm to 675 nm multiplies the spacing by 675/450 = 1.5, giving 4.5 mm, because a longer wavelength needs a larger angle to produce a path difference of one wavelength.
Explain why the highest order visible from a grating is limited, and find the highest order for 650 nm light with d = 2.00 × 10⁻⁶ m.
The grating condition is sin θ = mλ/d, and sin θ cannot exceed 1, so m ≤ d/λ. With d/λ = 2.00 × 10⁻⁶/6.50 × 10⁻⁷ = 3.08, the largest whole number is m = 3 (at sin θ = 0.975, θ = 77°). A fourth order would require sin θ = 1.30, which is impossible, so only three orders appear on each side.
Compare the electromagnetic-wave model and the photon model of light: what does each say about how radiant energy travels, and name one phenomenon each is used for.
The wave model says radiant energy travels as a spreading wave of changing electric and magnetic fields; it is used to explain and calculate interference, diffraction and refraction, such as the fringe pattern from a grating. The photon model says radiant energy travels in discrete packets, photons, each with energy proportional to frequency (E = hf); when and in which directions they are emitted depends on the source. It is used where light exchanges energy in lumps, such as why higher-frequency ultraviolet light can eject electrons from a metal such as calcium or damage skin when dimmer red light cannot. Strong evidence supports both, and scientists use whichever fits the problem.
Calculate the energy of a photon of blue light with wavelength 450 nm and compare it with a photon of red light at 700 nm.
f = c/λ = 3.00 × 10⁸/4.50 × 10⁻⁷ = 6.67 × 10¹⁴ Hz, so E = hf = (6.63 × 10⁻³⁴)(6.67 × 10¹⁴) = 4.42 × 10⁻¹⁹ J. For red light, f = 4.29 × 10¹⁴ Hz and E = 2.84 × 10⁻¹⁹ J. The blue photon carries about 1.56 times the energy of the red photon because photon energy is directly proportional to frequency, and blue light has the higher frequency (shorter wavelength).
A shirt looks yellow in white light. Explain in terms of the visible spectrum what its pigments are doing, and predict how it would look under pure blue light.
White light from the Sun or a lamp is a mixture of all visible colors. The shirt's pigments reflect the yellow region (and typically the neighboring red and green wavelengths that together appear yellow) in all directions and absorb the other colors, mainly blue and violet. Under pure blue light there is no yellow to reflect and the blue is absorbed, so very little light returns to the eye and the shirt looks dark or black; the color of an object depends on both the light falling on it and the wavelengths its pigments reflect.