Photoelectric effect and the wave theory
What the wave theory predicts
By the end of the nineteenth century, everyone agreed that light is a wave. Interference, diffraction and polarisation were all explained by waves, and Maxwell showed that light is an electromagnetic wave.
In the wave picture, the energy of light is spread continuously over the whole wavefront. A brighter beam has a larger amplitude, and carries more energy each second through each square metre.
So picture a wave arriving at a metal surface. Every free electron near the surface is shaken by it, and soaks up energy from it, little by little. When an electron has gathered enough to overcome the work function, it escapes.
Prediction one. Brighter light pours in energy faster, so the electrons should gather more of it. The maximum kinetic energy of the photoelectrons should increase with intensity.
Experiment says no. Make the light ten times brighter, and the stopping potential does not change at all. K max is independent of intensity.
Prediction two. Any wave, of any frequency, keeps feeding energy to the electrons. If the light is bright enough, or if we wait long enough, electrons should come out whatever the frequency. There should be no threshold.
Experiment says no again. Red light on zinc frees no electrons at all, however intense it is and however long we wait. Below the threshold frequency, nothing happens.
And the frequency matters in the experiment: higher frequency gives faster electrons. In the wave picture, frequency plays no special role in how much energy an electron soaks up.
How long would an electron wait?
Prediction three is about time. In the wave picture, the energy is spread over the whole surface. Each electron can only take in the energy falling on a tiny patch around it, about the size of one atom.
Let us estimate. Take that patch as 10⁻²⁰ square metre. Shine dim light of intensity 10⁻³ watt per square metre on sodium.
The power reaching one electron is intensity times area: 10⁻³ times 10⁻²⁰, which is 10⁻²³ watt.
Sodium’s work function is 2.75 electron volt, or 2.75 × 1.6 × 10⁻¹⁹, which is 4.4 × 10⁻¹⁹ joule.
Time needed equals energy divided by power: 4.4 × 10⁻¹⁹ divided by 10⁻²³, which is 4.4 × 10⁴ seconds. That is about twelve hours before the first electron could come out.
The experiment: switch on the dim light, and the photocurrent starts within about 10⁻⁹ second. Not hours, not minutes. Instantly.
Let us score the wave theory. Photocurrent proportional to intensity: fine. K max independent of intensity: fails. Threshold frequency: fails. No time lag: fails.
The way out is to stop spreading the energy around. Suppose light energy arrives in small, concentrated packets, and one electron takes in a whole packet at once.
Then the electron does not wait, and what it gets depends on the size of the packet, not on how many packets arrive. That is Einstein’s idea of the photon, coming up next.
Try it: wave theory or photons?
- In “Wave theory”, the energy bars fill up continuously. Lower the intensity and see how long the wait becomes in real time.
- Still in wave theory, change the wavelength to 650 nm (red). The bars fill just as fast: the wave picture says the colour does not matter.
- Switch to “Photon picture” with red light on sodium. However bright you make it, no electron leaves: each photon has too little energy.
- In the photon picture, change only the intensity. More photons arrive, so more electrons leave, but each one leaves with the same K_max.
- Compare the two scorecards with the experimental facts.
Bright red light
Bright red light falls on a metal whose threshold frequency is in the green. What does the wave theory predict, and what is actually observed?
- Wave theory: no electrons. Observed: electrons after a delay
- Wave theory: electrons, after enough energy builds up. Observed: no electrons at all
- Both say electrons come out instantly
- Both say no electrons come out
Answer
B. In the wave picture the energy delivered depends on the intensity, not the frequency, so bright red light should eventually give the electrons enough energy to escape. In the experiment, light below the threshold frequency frees no electrons, however intense it is and however long it shines.
The waiting time
Light of intensity 10⁻² W/m² falls on potassium (φ₀ = 2.30 eV). In the wave picture, one electron collects energy from an area of 10⁻²⁰ m². About how long must it wait before it can escape?
- About 10⁻⁹ s
- 3.7 × 10⁻³ s
- 3.7 × 10³ s, about an hour
- 2.3 × 10²² s
Answer
C. Power on one electron = 10⁻² × 10⁻²⁰ = 10⁻²² W. Energy needed = 2.30 × 1.6 × 10⁻¹⁹ = 3.68 × 10⁻¹⁹ J. Time = 3.68 × 10⁻¹⁹ / 10⁻²² ≈ 3.7 × 10³ s, about an hour. Forgetting to change eV into joules gives 2.3 × 10²² s. 10⁻⁹ s is what the experiment shows, and the wave theory cannot explain it.
Summary
- Wave picture: power on one atom:
P = I × A, A ≈ 10⁻²⁰ m² - Wave picture: waiting time:
t = φ₀ / (I A)
- In the wave picture, energy is spread continuously over the wavefront, and every electron absorbs it gradually.
- It predicts that K_max should grow with intensity; experiment shows it does not.
- It predicts emission at any frequency if the light is bright enough; experiment shows a threshold frequency.
- It predicts a time lag of minutes to hours for dim light; experiment shows emission within about 10⁻⁹ s.
- So the wave picture fails to explain the photoelectric effect. Light energy must arrive in packets, each absorbed by one electron.