Quantum Mechanics 101 · Unit 1: The Classical World Cracks

Quantum Academy · Chapter 4: The Photoelectric Effect: Light Comes in Packets

Einstein's 1905 photon, and why brightness can't buy you an electron

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Instructions. Work through these on paper before checking the answers. Use E = hf = hc/λ, the shortcut E ≈ 1240/λ (electron-volts and nanometres), KE_max = hf − φ, and e·V₀ = KE_max. Take h = 6.626×10⁻³⁴ J·s, c = 2.998×10⁸ m/s, e = 1.602×10⁻¹⁹ coulombs, 1 eV = 1.602×10⁻¹⁹ joules, and the electron mass m = 9.109×10⁻³¹ kg. Show units at every step. One point for each vocabulary match and each question; the challenge is a bonus.

Part A Vocabulary matching

Write the letter of the matching definition in the blank beside each term.

1.Photoelectric effect
2.Photon
3.Photoelectron
4.Work function
5.Threshold frequency
6.Stopping potential
7.Photocurrent
8.Electron-volt
9.Intensity
10.Quantum efficiency
11.Band gap
12.Photocathode

A.The light-sensitive electrode in a photomultiplier or night-vision tube that emits electrons when struck by photons.
B.The retarding voltage that just stops the fastest photoelectrons, so that e·V₀ = KE_max.
C.The minimum energy needed to free an electron from a material's surface, written φ and usually quoted in electron-volts.
D.The fraction of incoming photons that produce a counted electron in a detector or photocathode.
E.The emission of electrons from a material, usually a metal, when light of sufficiently high frequency falls on it.
F.The energy an electron gains when accelerated through one volt: exactly 1.602176634×10⁻¹⁹ joules.
G.A quantum, or indivisible packet, of light carrying energy E = hf; the name was popularized by Gilbert Lewis in 1926.
H.In a semiconductor, the forbidden range of energies an electron must jump to become free to carry current; 1.12 eV for silicon.
I.The electric current carried by photoelectrons; at fixed frequency it is proportional to the light's intensity.
J.The power a beam of light delivers per unit area, in watts per square metre; for photons, the number arriving per second per square metre times hf.
K.The lowest frequency of light that can eject electrons from a given material, f₀ = φ/h.
L.An electron released from a material by absorbing a photon.

Part B Questions

1.
Which single change will raise the maximum kinetic energy of photoelectrons ejected from a given metal?Circle one
1. AMaking the light brighter
2. BShining the light for longer
3. CUsing light of shorter wavelength
4. DUsing a larger metal plate
2.
The work function of a metal is best described as:Circle one
1. AThe energy needed to heat the metal until it glows
2. BThe minimum energy needed to free an electron from the metal's surface
3. CThe kinetic energy of the fastest photoelectron
4. DThe voltage needed to start a photocurrent
3.
Einstein's Nobel Prize in Physics for 1921 was awarded:Circle one
1. AFor his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect
2. BFor the special theory of relativity
3. CFor the general theory of relativity, confirmed by the 1919 eclipse expeditions
4. DFor the equation E = mc²
4.
Below the threshold frequency, light made a million times brighter will eventually eject electrons through ordinary single-photon photoemission.Circle one
TrueFalse
5.
On a graph of stopping potential against frequency, every metal's straight line has the same slope.Circle one
TrueFalse
6.
The electron-volt is a unit of voltage.Circle one
TrueFalse
7.
The energy of a single photon depends only on the light's .Fill in the blank
8.
The retarding voltage that just prevents the fastest photoelectrons from reaching the collector is called the potential.Fill in the blank
9.
Using the photon picture, explain why doubling the intensity of light above the threshold doubles the photocurrent but leaves the maximum kinetic energy of the photoelectrons unchanged.Short answer
10.
Explain the difference between the external photoelectric effect in a photomultiplier tube and the internal photoelectric effect in a solar cell.Short answer
11.
A green laser pointer emits light of wavelength 532 nm. Find the frequency, and the energy of one photon in joules and in electron-volts.Show your working
Working
Answer:
12.
Zinc has a work function of 4.3 eV. Find its threshold frequency and threshold wavelength. Can any visible light, 380 to 750 nm, eject electrons from zinc?Show your working
Working
Answer:
13.
Ultraviolet light of wavelength 200 nm falls on clean copper, whose work function is 4.7 eV. Find the maximum kinetic energy of the photoelectrons, the stopping potential, and the maximum speed.Show your working
Working
Answer:
14.
Light of wavelength 250 nm shines on an unknown metal, and the photocurrent stops when the retarding voltage reaches 0.88 V. Find the work function and use the values in this chapter to identify the metal: cesium 2.1 eV, sodium 2.28 eV, aluminium 4.08 eV, zinc 4.3 eV, copper 4.7 eV.Show your working
Working
Answer:

Part C Mad scientist challenge bonus

The impatient electron. A clean cesium surface (φ = 2.1 eV) is lit by blue light of wavelength 450 nm at a feeble intensity of 1.0×10⁻⁶ W/m². (a) Classical wave picture: suppose each atom absorbs all the light falling on a disc of radius 0.10 nm, a typical atomic size, and reflects none. How long must an atom wait before one of its electrons has gathered the work function? (b) Photon picture: how many photons per second strike a 1.0 square-centimetre patch of the surface, and what is the average time between their arrivals? (c) What is KE_max for the electrons that are emitted? (d) Lawrence and Beams found that emission begins within 3×10⁻⁹ seconds. Explain why the photon picture predicts prompt emission even though the energy per atom per second is minuscule.

Working

Memory aid

Mnemonic Colour's the coin, brightness the crowd; one coin per electron, no pooling allowed. Pay the toll, pocket the change. Picture a toll booth at the surface of the metal. Every photon is a coin whose value is set only by its colour; the toll is the work function φ; each electron may pay with exactly one coin; and the change it pockets is its kinetic energy. Run the lines in order on any photoelectric problem: find the coin, E ≈ 1240/λ in electron-volts; check that it beats the toll, or nothing happens however large the crowd; the change is KE_max = hf − φ, which equals e·V₀; and the size of the crowd, R = P/(hf), sets only the current.

In one line Light comes in packets of energy hf; each can free one electron, which pays the work function and keeps the change. Colour decides; brightness counts.