The double slit, diffraction gratings, the photoelectric effect and time dilation: the four ideas of modern physics

The double slit experiment shows that light behaves as a wave: two slits make a pattern of bright and dark bands that only adding waves can explain, with the bright bands spaced Δy = λL/d. A diffraction grating is the same idea with thousands of slits and sharper lines. The photoelectric effect shows the opposite face: light arrives in packets, photons, each with energy hf, and an electron is freed only if one packet carries more than the metal's work function. Time dilation says a moving clock runs slow by the factor γ. Below, one calculation for each, the double slit worked line by line, and the four errors that lose the marks.

What the double slit experiment shows, and the one formula

Shine one colour of light through two narrow slits very close together, and the screen beyond does not show two bright stripes. It shows a row of many bright bands with dark bands between them. Particles fired through two openings could never make that: they would pile up in two heaps. Waves can, because a wave from one slit and a wave from the other arrive at each point on the screen having travelled different distances, and they add.

Where the path difference is a whole number of wavelengths, mλ, the two waves arrive crest on crest and add to a bright band: constructive interference. Where it is a half-odd number, (m + ½)λ, crest meets trough and they cancel to a dark band: destructive interference. Young's double slit, first done in 1801, is the experiment that settled light's wave nature for a century.

The geometry gives the bright bands at angles θ with d sin θ = mλ, where d is the slit separation and m = 0, 1, 2, … counts the bands out from the centre. On a screen a distance L away, when the angles are small, the bands are evenly spaced with Δy = λL/d between neighbours. "Small" means sin θ and tan θ are the same to the figures you keep, which is true when d is thousands of wavelengths; the small-angle approximation is the trig fact behind it.

Three lengths go into the formula and all three must be in metres before the first line: λ arrives in nanometres (10⁻⁹ m), d in millimetres (10⁻³ m), L in metres.

Worked example: 630 nm light, slits 0.250 mm apart, a screen 2.00 m away

Red light of wavelength 630 nm passes through two slits 0.250 mm apart and falls on a screen 2.00 m away. Find the spacing between neighbouring bright fringes.

Givens, converted: λ = 630 × 10⁻⁹ m, d = 0.250 × 10⁻³ m = 2.50 × 10⁻⁴ m, L = 2.00 m. Three significant figures.

Line 1, the numerator.

• λL = 630 × 10⁻⁹ m × 2.00 m = 1.26 × 10⁻⁶ m² (multiplying; the unit is m²)

Line 2, the fringe spacing.

• Δy = λL/d = 1.26 × 10⁻⁶ m² ÷ 2.50 × 10⁻⁴ m = 5.04 × 10⁻³ m (dividing; m² ÷ m = m)
• Δy = 5.04 mm between neighbouring bright fringes

Line 3, the check with the exact formula. Was the small-angle form allowed?

• sin θ₁ = λ/d = 630 × 10⁻⁹ m ÷ 2.50 × 10⁻⁴ m = 2.52 × 10⁻³ (the first-order angle, from d sin θ = mλ with m = 1)
• θ₁ = sin⁻¹(2.52 × 10⁻³) = 0.144° (a very small angle)
• y₁ = L tan θ₁ = 2.00 m × 2.52 × 10⁻³ = 5.04 × 10⁻³ m (the first bright band's distance from the centre; tan of 0.144° equals its sine to three figures)

The same 5.04 mm, which is why the small-angle form was allowed here and why it will not be allowed for the grating below.

Line 4, what moves what. Read the formula as three dials. A longer wavelength (red rather than blue) spreads the fringes. Slits farther apart (bigger d) close them up. A farther screen spreads them. A question that changes one of the three and asks what happens is answered by this line without a number.

Diffraction gratings: the same formula, sharper, and the angle is no longer small

A diffraction grating is a plate ruled with hundreds or thousands of slits per millimetre. The condition for a bright line is the same, d sin θ = mλ, but with thousands of slits the lines are far sharper and far brighter, which is what makes gratings the tool for splitting light into its wavelengths. The slit spacing is the reciprocal of the line density: d = 1/(lines per metre).

Because d is now only a few wavelengths, sin θ is not small and the full sine is kept. A grating with 600 lines/mm, with the same 630 nm light:

• d = 1 ÷ (600 lines/mm) = 1.667 × 10⁻³ mm = 1.667 × 10⁻⁶ m (the spacing between neighbouring slits; the fourth figure is kept because 1/600 is exact)
• First order: sin θ₁ = λ/d = 630 × 10⁻⁹ ÷ 1.667 × 10⁻⁶ = 0.378, so θ₁ = 22.2° (nowhere near small)
• Second order: sin θ₂ = 2λ/d = 0.756, so θ₂ = 49.1°
• Third order: sin θ₃ = 3λ/d = 1.13, which is greater than 1, so there is no third-order line for this light on this grating

That last line is a question setters like: the highest order you can see is the largest m for which mλ/d is still at most 1.

The photoelectric effect: an energy ledger in electron volts

Shine light on a clean metal surface and, under the right conditions, electrons come off it. The conditions are the surprise. Below a certain frequency no electrons come off at all, however bright the light is; above it they come off at once, and making the light brighter produces more electrons but not faster ones. A wave model cannot explain that, because a wave's energy depends on its brightness and arrives continuously. Einstein's 1905 explanation does: light arrives in packets, photons, each carrying energy E = hf = hc/λ, and one photon frees one electron, or none.

The ledger is three lines in electron volts. The photon's energy comes in. The work function W, the energy it costs to free an electron from that metal, goes out. What is left is the electron's maximum kinetic energy, KE_max = hf − W. The stopping potential, the voltage that just stops the fastest electrons, is V₀ = KE_max/e, which in electron volts is the same number read as volts.

The constant that makes the ledger quick is hc = 1240 eV·nm. It follows from the SI values: hc = 6.63 × 10⁻³⁴ J·s × 3.00 × 10⁸ m/s = 1.99 × 10⁻²⁵ J·m, and dividing by 1.60 × 10⁻¹⁹ J/eV gives 1.24 × 10⁻⁶ eV·m, which is 1240 eV·nm. With it, E in eV is simply 1240 divided by λ in nm.

Light of wavelength 250 nm falls on a metal whose work function is 2.30 eV (a number chosen for this page, not a named metal's):

• E_photon = 1240 eV·nm ÷ 250 nm = 4.96 eV (the energy in)
• KE_max = 4.96 eV − 2.30 eV = 2.66 eV (the work function paid, the rest kept by the electron)
• In joules: 2.66 eV × 1.60 × 10⁻¹⁹ J/eV = 4.26 × 10⁻¹⁹ J (the conversion, if the question wants SI)
• V₀ = 2.66 V (the stopping potential, the same number)
• Threshold wavelength: λ₀ = 1240 ÷ 2.30 = 539 nm (the longest wavelength that frees any electron: E = W exactly)

So 600 nm light, with E = 1240/600 = 2.07 eV, frees nothing from this metal at any brightness, while 250 nm light frees electrons at 2.66 eV however dim it is. That is the whole case for the photon.

Time dilation: γ, and which clock is the slow one

Special relativity says that two observers moving relative to each other at speed v disagree about how much time passes between two events, by the factor γ = 1/√(1 − v²/c²). γ is 1 at rest and grows without limit as v approaches c.

The proper time Δt₀ is the time measured by the clock that is present at both events, the clock carried with the thing whose "before" and "after" we are timing. Every other observer, for whom that clock is moving, measures a longer time: Δt = γΔt₀. In words, "the moving clock runs slow", where "moving" means moving relative to the observer doing the measuring. There is no absolutely moving clock; each observer sees the other's clock as the slow one, and the proper time is the one on the clock that was at both events.

At v = 0.80c:

• v²/c² = 0.80² = 0.64 (squaring the fraction of c)
• 1 − 0.64 = 0.36, and √0.36 = 0.60 (the root)
• γ = 1 ÷ 0.60 = 1.67 (the factor)

A clock on a ship passing at 0.80c ticks off 10.0 s between two events on the ship, so Δt₀ = 10.0 s. An observer the ship passes measures Δt = 1.67 × 10.0 s = 16.7 s between the same two events. Length contraction is the companion result, one line: a 100 m ship, measured by the observer it passes at 0.80c, reads L = L₀/γ = 100 m ÷ 1.67 = 60.0 m, shorter along the direction of motion only.

The four wrong lines a marker sees

1. Nanometres or millimetres left unconverted in Δy = λL/d. The symptom is a fringe spacing in the wrong power of ten, sometimes in kilometres. Write the three lengths in metres on their own line first.
2. The photon energy subtracted from the work function, or the two added. KE_max = hf − W, the energy in minus the energy owed. The symptom is a negative kinetic energy for light above threshold, or a kinetic energy larger than the photon brought.
3. The small-angle form used for a grating. When sin θ is 0.4 or more, tan θ and sin θ differ badly, and Δy = λL/d is wrong by tens of percent. For a grating use d sin θ = mλ with the full sine.
4. Dividing by γ for the dilated time, or calling the observer's time the proper time. The proper time is on the clock present at both events, and it is the shortest; every other observer multiplies it by γ. The symptom is a dilated time shorter than the proper time.

A fifth, conceptual: thinking brighter light gives faster electrons. It gives more electrons at the same maximum speed. Only a higher frequency gives faster ones.

A sibling to try cold

Cover the example. Light of wavelength 500 nm passes through slits 0.200 mm apart onto a screen 1.50 m away; find the fringe spacing. Then one photon line: 400 nm light falls on the same 2.30 eV metal; find the photon energy and KE_max. Write the three lengths in metres before any line, and write the ledger as "in, out, left" before any number. The example above was the demonstration and this is the attempt; Sweller and Cooper (1985) found that pairing of a worked example with a matching problem teaches a procedure faster than problems alone.

Check

• Δy = λL/d = 500 × 10⁻⁹ m × 1.50 m ÷ 2.00 × 10⁻⁴ m = 7.50 × 10⁻⁷ ÷ 2.00 × 10⁻⁴ = 3.75 × 10⁻³ m = 3.75 mm
• E_photon = 1240 ÷ 400 = 3.10 eV
• KE_max = 3.10 − 2.30 = 0.80 eV, so V₀ = 0.80 V

If your fringe spacing was 3.75 m, a millimetre stayed a millimetre. If your KE_max was 5.40 eV, the ledger added instead of subtracting.

Where this sits in the course

This page covers four of the last seven ideas of SPH4U: idea 19, Interference of light; Young's double slit, and idea 20, Diffraction and diffraction gratings, in strand E; idea 22, Special relativity: time dilation and length contraction, and idea 24, The photoelectric effect and photon energy, in strand F. Between them in the course sit idea 21, Polarization of light, and idea 23, Mass–energy equivalence, and the course closes with idea 25, Energy levels and matter waves, where the particle-and-wave story is turned round and applied to electrons.

Before idea 19 is idea 18, Magnetic fields and the force on moving charges and currents, the last of the fields strand. The magnetic force, the idea before the light strand is written the same way as this page. The SPH4U course map shows all 25 ideas in teaching order, and every idea of the three science courses has a page like this one.

The four ideas on one screen

• Four formulas. Double slit: d sin θ = mλ, and Δy = λL/d for small angles. Grating: the same with the full sine and d = 1/(lines per metre). Photoelectric: KE_max = hf − W, with hc = 1240 eV·nm. Time dilation: Δt = γΔt₀, γ = 1/√(1 − v²/c²).
• Three conversions. nm → m (× 10⁻⁹), mm → m (× 10⁻³), eV → J (× 1.60 × 10⁻¹⁹).
• Wave here, particle there, and both are light. The double slit needs waves; the photoelectric effect needs photons; neither picture alone is the whole of it.
• The four wrong lines. Lengths unconverted; the ledger added or reversed; the small-angle form on a grating; γ divided or the proper time misassigned.
• The habit. This strand carries its marks in vocabulary as much as in numbers: proper time, work function, threshold frequency, order, constructive. How to study physics for this strand is a recall card for each word and one calculation of each kind, and flashcards for the vocabulary keeps the definition off the front of the card.