Observations of the Photoelectric Effect

Spec mapping: OCR H556 Module 4.5 — Quantum physics (the photoelectric effect; existence of threshold frequency; instantaneous emission; dependence of photocurrent on intensity; independence of $KE_{\text{max}}$KEmax​ from intensity; classical wave-theory failure). Refer to the official OCR H556 specification document for exact wording.

The photoelectric effect is the emission of electrons from a metal surface when light shines on it. First observed by Heinrich Hertz in 1887 (ironically, while he was working on experiments that confirmed Maxwell's wave theory of electromagnetism), it was studied carefully by Philipp Lenard in the years that followed. By around 1902, Lenard had established a set of striking experimental facts about the effect — facts which, taken together, were completely incompatible with the classical wave theory of light.

This lesson presents those observations qualitatively, explains why they were so shocking in 1902, and prepares the way for Einstein's 1905 explanation in terms of photons (lesson 4). It also sets up the standard OCR photocell apparatus: cathode, anode, variable retarding p.d., ammeter, and stopping-potential measurement. The story is not just about a particular experiment — it is about why the wave theory of light, however successful for interference and diffraction, has to be supplemented by a particle description before atomic-scale photon–electron interactions can be understood at all.

Understanding the four observations qualitatively — what was observed, and why classical wave theory cannot explain it — is just as important as the numerical work that follows in lessons 4 and 5. OCR examiners routinely test the discriminator: a candidate who can rearrange $hf = \phi + KE_{\text{max}}$hf=ϕ+KEmax​ but cannot articulate why classical theory fails is at most a mid-band answer.

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The Experimental Set-up

The classic photoelectric experiment uses a vacuum photocell: a sealed glass envelope from which the air has been pumped out, containing two metal electrodes connected to an external circuit.

• The cathode (or photocathode) is a clean metal plate (typically caesium, potassium or zinc, depending on the wavelength range under study) on which light is shone.
• The anode is a collecting electrode held at a positive (accelerating) or negative (retarding) variable potential relative to the cathode.

When light of sufficiently high frequency strikes the cathode, electrons are ejected from the metal surface. These electrons are called photoelectrons — they are ordinary electrons, the prefix merely recording how they were liberated. If the anode is held at a positive potential relative to the cathode, the photoelectrons are attracted to it and a current flows in the external circuit. This photocurrent is measured by an ammeter.

flowchart LR
    L[Light source<br/>variable f, I] --> C[Photocathode<br/>clean metal]
    C -- photoelectrons --> A[Anode<br/>collector]
    A --> I[Ammeter<br/>reads photocurrent]
    I --> V[Variable p.d.<br/>±V_s region]
    V --> C

By varying (a) the frequency $f$f of the incident light, (b) the intensity $I$I of the incident light, and (c) the potential difference $V$V between the electrodes, Lenard discovered four distinct experimental facts that together constitute the observations of the photoelectric effect. Each one is, in turn, in flat contradiction with the classical wave theory of light.

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Observation 1: Threshold Frequency

No photoelectrons are emitted unless the frequency of the incident light is above a certain minimum — the threshold frequency $f_0$f0​ — regardless of how intense the light is.

Shine red light ($\lambda = 650$λ=650 nm) on a clean zinc plate for an hour: not a single electron is liberated. Switch on a UV lamp, even a very dim one: photoelectrons are emitted immediately. The threshold is sharp: a tiny change in wavelength around the threshold is the difference between no current at all and a measurable current.

The threshold frequency depends on the metal:

Metal	$f_0$f0​ (Hz)	$\lambda_0 = c/f_0$λ0​=c/f0​ (nm)	Region
Caesium	$\approx 5.1 \times 10^{14}$≈5.1×1014	$\approx 590$≈590	visible (yellow)
Potassium	$\approx 5.6 \times 10^{14}$≈5.6×1014	$\approx 540$≈540	visible (green)
Sodium	$\approx 5.8 \times 10^{14}$≈5.8×1014	$\approx 520$≈520	visible (green)
Zinc	$\approx 1.0 \times 10^{15}$≈1.0×1015	$\approx 290$≈290	UV
Copper	$\approx 1.1 \times 10^{15}$≈1.1×1015	$\approx 270$≈270	UV
Platinum	$\approx 1.5 \times 10^{15}$≈1.5×1015	$\approx 200$≈200	UV

These values correspond, via $f_0 = \phi/h$f0​=ϕ/h, to the work function values you will meet in lesson 4. Caesium has $\phi \approx 2.1$ϕ≈2.1 eV; platinum has $\phi \approx 6.4$ϕ≈6.4 eV.

Why is this strange classically?

On the classical wave picture, light of any frequency delivers energy continuously to the metal. A sufficiently bright red lamp ought, eventually, to pump enough energy into a surface electron for it to escape — the process might take a while, but it should happen. The experimental absence of emission at low frequencies, no matter how bright the light or how long you wait, is completely at odds with this expectation. Classical theory predicts a slow drip; experiment shows a hard threshold.

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Observation 2: Instantaneous Emission

When the frequency is above threshold, photoelectrons are emitted immediately — within less than $10^{-9}$10−9 s of the light being switched on — no matter how dim the light.

Even at the most extreme low intensities (single-photon experiments performed in the 1920s and 30s), there is no detectable delay between illumination and photoemission. The first electron flies off as soon as the first photon of sufficient energy arrives. There is no "energy accumulation" phase.

Why is this strange classically?

On the classical wave picture, a very dim light would deposit energy very slowly into the electrons of the metal. For a surface electron to absorb enough energy to escape might take minutes or even hours, because the energy of a wave at a given point is distributed across all the electrons in the illuminated region.

A rough estimate of the classically predicted delay:

$t_{\text{delay}} \sim \frac{\phi}{P_{\text{absorbed per electron}}}$tdelay​∼Pabsorbed per electron​ϕ​

For a typical work function $\phi = 4$ϕ=4 eV $= 6.4 \times 10^{-19}$=6.4×10−19 J, illuminated at $1$1 μW/m$^2$2 over an electron's effective cross-section of $\sim 10^{-19}$∼10−19 m$^2$2, the delay should be of order seconds to minutes. The observed delay is less than a nanosecond — at least nine orders of magnitude faster than the wave theory allows. Classical theory predicts a measurable delay; experiment shows no delay at all.

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Observation 3: Intensity Controls Photocurrent, Not Energy

Above the threshold, the number of photoelectrons emitted per second is proportional to the intensity of the light. But the maximum kinetic energy of each photoelectron is independent of intensity.

This was perhaps the most astonishing of Lenard's findings. Doubling the brightness, tripling it, increasing it by a factor of a hundred — the photocurrent rises correspondingly, but the fastest photoelectrons still come out at exactly the same speed.

To measure the maximum kinetic energy, Lenard used the stopping-potential method: reverse the polarity of the external p.d. and increase its magnitude until the photocurrent just falls to zero. At this point, no photoelectron has enough kinetic energy to climb the potential hill between cathode and anode, and the work done against the retarding field equals the maximum initial kinetic energy:

$eV_s = KE_{\text{max}}$eVs​=KEmax​

The stopping potential $V_s$Vs​ is therefore a direct measure of $KE_{\text{max}}$KEmax​. And — most strikingly — this stopping potential is independent of intensity. Doubling the brightness doubles the photocurrent (more photons in, more electrons out per second) but leaves $V_s$Vs​ unchanged.

Why is this strange classically?

On the classical wave picture, a more intense light has a larger amplitude of the electric field. A larger electric field exerts a larger force on each electron, so each electron should oscillate more vigorously and carry away more kinetic energy when it escapes. The observation that $KE_{\text{max}}$KEmax​ is fixed regardless of intensity is simply inexplicable. Classical theory predicts $KE_{\text{max}}$KEmax​ rises with intensity; experiment shows it does not.

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Observation 4: Frequency Controls Maximum Kinetic Energy

The maximum kinetic energy of photoelectrons increases linearly with the frequency of the incident light, and is zero at the threshold frequency $f_0$f0​.

If you plot $KE_{\text{max}}$KEmax​ (or equivalently $eV_s$eVs​) against frequency $f$f, you get a straight line:

• Slope = $h$h (Planck's constant — the same $h$h that appears in $E = hf$E=hf)
• x-intercept = $f_0$f0​ (the threshold frequency)
• y-intercept = $-\phi$−ϕ (the negative of the work function)

This linear relationship, which holds with astonishing precision across many metals, is one of the most striking results in physics. It is the experimental smoking gun for Einstein's photon theory.

flowchart LR
    F["Increase f above f₀"] --> E["KE_max rises linearly"]
    I["Increase intensity (fixed f)"] --> N["Photocurrent rises<br/>(more e⁻ per s)"]
    I --> NE["KE_max unchanged"]
    F0["Below f₀"] --> Z["No emission<br/>regardless of intensity"]

Why is this strange classically?

On the classical wave picture, the kinetic energy of the emitted electrons should depend on the amplitude (intensity) of the wave, not on its frequency. Frequency, in the wave view, affects how often the field oscillates — it has no obvious connection to the energy delivered to any single electron. The linear dependence of $KE_{\text{max}}$KEmax​ on $f$f, and the absence of intensity dependence, is the most decisive single piece of evidence for the quantisation of light. Classical theory predicts $KE_{\text{max}}$KEmax​ rises with intensity and is unrelated to frequency; experiment shows exactly the opposite.

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Summary Table: Classical Predictions versus Observations

Feature	Classical wave prediction	Observation
Threshold frequency	No threshold; any frequency should work given enough time	Sharp threshold $f_0$f0​ below which nothing happens
Time delay	Should be detectable at low intensity	None — emission instantaneous ($< 10^{-9}$<10−9 s)
Dependence of $KE_{\text{max}}$KEmax​ on intensity	$KE_{\text{max}}$KEmax​ should rise with intensity	$KE_{\text{max}}$KEmax​ is independent of intensity
Dependence of photocurrent on intensity	Should rise with intensity	Rises linearly with intensity (the only correct prediction)
Dependence of $KE_{\text{max}}$KEmax​ on frequency	No clear prediction	Rises linearly with frequency, slope $h$h

Four predictions wrong, one right (partially, for the wrong reason). Classical wave theory is comprehensively a poor fit to the data — and that is the conclusion that forced Einstein to the photon hypothesis.

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The Photocurrent–Voltage Characteristic

A particularly informative measurement is the $I$I–$V$V characteristic of a photocell: the photocurrent as a function of the p.d. between anode and cathode, for fixed light frequency and intensity.

Key features (moving from positive to negative anode p.d.):

1. At large positive anode p.d., the photocurrent saturates at a constant value $I_{\text{sat}}$Isat​. Every photoelectron emitted is collected — the rate of emission (photons per second × quantum efficiency) is the bottleneck.
2. Reducing the anode p.d. towards zero, the current stays roughly constant until small p.d. (most photoelectrons are emitted with enough KE to reach the anode unaided).
3. At zero anode p.d., a small current still flows because photoelectrons have enough KE to reach the anode on their own.
4. As the anode p.d. becomes negative (retarding), fewer photoelectrons make it across the gap; the current drops steeply.
5. At a particular negative p.d. — the stopping potential $V_s$Vs​ — the current falls to exactly zero. No photoelectron has enough KE to climb the potential hill.

flowchart LR
    V1[+V anode potential] --> S[I sat: every e⁻ collected]
    S --> Z[V = 0: current still flows]
    Z --> R[−V retarding]
    R --> D[Current decreases]
    D --> Vs[V = −V_s: I = 0<br/>eV_s = KE_max]

The position of $V_s$Vs​ (horizontally on the $I$I–$V$V graph) tells you $KE_{\text{max}}$KEmax​; the height of $I_{\text{sat}}$Isat​ (vertically) tells you the photon-arrival rate.

By varying intensity and frequency independently, you verify all four observations:

• Increase intensity at fixed frequency $\Rightarrow$⇒ $I_{\text{sat}}$Isat​ rises; $V_s$Vs​ does not change. (More photons in, more electrons out per second, but each photoelectron carries the same maximum KE.)
• Increase frequency at fixed intensity $\Rightarrow$⇒ $V_s$Vs​ rises (more negative — each electron has more KE to shed); $I_{\text{sat}}$Isat​ may fall or rise depending on how the total power is held constant.

This dual independence — frequency controls $V_s$Vs​, intensity controls $I_{\text{sat}}$Isat​ — is exactly what one would expect if light arrived as discrete photons each carrying $hf$hf, and exactly what one would not expect on a wave picture.

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Worked Example 1: Reading a Photocell Graph

A student illuminates a caesium cathode with monochromatic light and measures the photocurrent as a function of anode p.d. The current falls to zero at $V = -0.80$V=−0.80 V. Calculate the maximum kinetic energy of the photoelectrons in (a) joules and (b) eV.

Solution.

The defining relation is $eV_s = KE_{\text{max}}$eVs​=KEmax​, with $V_s = 0.80$Vs​=0.80 V (taking the magnitude of the retarding p.d.).

(a) $KE_{\text{max}} = eV_s = (1.60 \times 10^{-19} \text{ C})(0.80 \text{ V}) = 1.28 \times 10^{-19}$KEmax​=eVs​=(1.60×10−19 C)(0.80 V)=1.28×10−19 J.

(b) $KE_{\text{max}} = 0.80$KEmax​=0.80 eV — by definition of the electronvolt, no conversion needed.