Einstein's Photoelectric Equation

Spec mapping: OCR H556 Module 4.5 — Quantum physics (Einstein's photoelectric equation $hf = \phi + KE_{\text{max}}$hf=ϕ+KEmax​; work function $\phi$ϕ; threshold frequency $f_0 = \phi/h$f0​=ϕ/h; PAG 5 anchor — Planck constant determination). Refer to the official OCR H556 specification document for exact wording.

In 1905 — his annus mirabilis, the year in which he also published the special theory of relativity and his paper on Brownian motion — Albert Einstein wrote a short paper entitled On a Heuristic Point of View Concerning the Production and Transformation of Light. In it he proposed a simple but revolutionary idea: light consists of discrete quanta of energy $hf$hf, and these quanta interact with matter one at a time.

Applied to the photoelectric effect, this hypothesis gave what is now called the Einstein photoelectric equation. In one line it resolves every puzzle Lenard had raised. It was for this result, not for relativity, that Einstein was awarded the Nobel Prize in Physics in 1921.

This lesson presents Einstein's equation, explains the physical meaning of each term, and shows how it accounts for all four observations of the previous lesson. We then work three numerical examples, lay out the $KE_{\text{max}}$KEmax​-vs-$f$f Millikan graph (the same plot that anchors PAG 5 in lesson 5), and finish with a specimen H556-format question that combines Einstein's equation with the stopping-potential relation. The equation is central to OCR H556 Module 4.5 and reappears in lessons 5, 6, 8 and 9; understanding the derivation, dimensional structure and limits of the equation here pays dividends for the rest of the topic.

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The Einstein Hypothesis

Einstein's starting assumption is simple. When light of frequency $f$f is absorbed at a metal surface, the absorption happens one photon at a time. Each photon delivers its entire energy $hf$hf to a single electron, and the photon ceases to exist. There is no gradual "soaking up" of wave energy; there is no sharing of the photon's energy among many electrons. It is a single, discrete, all-or-nothing event.

This is the crucial conceptual leap. Classical wave theory pictures energy being absorbed smoothly and shared among many electrons. Einstein's picture is utterly different: one photon in, one excited electron out, total transfer $hf$hf. Once this principle is accepted, the rest of the photoelectric phenomenology follows from a one-line application of energy conservation.

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The Work Function

Some of the energy delivered by the photon is needed just to liberate the electron from the metal. Conduction electrons in a metal are not free — they are held inside the metal by the combined electrostatic attraction of the positive ions and the potential well formed by the crystal lattice. To escape the metal surface, an electron must do work against this binding.

Definition. The work function $\phi$ϕ of a metal is the minimum energy required to release an electron from the metal surface.

It is an intrinsic surface property of the metal — and is highly sensitive to surface cleanliness. Typical values:

Metal	$\phi$ϕ (eV)	$\phi$ϕ (J)	Corresponding $\lambda_0$λ0​ (nm)
Caesium	2.1	$3.4 \times 10^{-19}$3.4×10−19	$\approx 590$≈590 (visible yellow)
Potassium	2.3	$3.7 \times 10^{-19}$3.7×10−19	$\approx 540$≈540 (visible green)
Sodium	2.4	$3.8 \times 10^{-19}$3.8×10−19	$\approx 520$≈520 (visible green)
Zinc	4.3	$6.9 \times 10^{-19}$6.9×10−19	$\approx 290$≈290 (UV)
Copper	4.7	$7.5 \times 10^{-19}$7.5×10−19	$\approx 265$≈265 (UV)
Platinum	6.4	$1.0 \times 10^{-18}$1.0×10−18	$\approx 195$≈195 (UV)

The variation is large: caesium has the smallest work function of any common metal and is therefore the standard photocathode material for visible-light photocells. Platinum's work function corresponds to a deep-UV wavelength and is essentially insensitive to visible light. Copper sits in between but its $\phi$ϕ still exceeds the energy of any visible photon, which is why ordinary visible light cannot liberate electrons from a clean copper surface.

The work function $\phi$ϕ is a minimum — electrons deep inside the metal are bound more strongly. Only electrons right at the surface escape with $\phi$ϕ; those liberated from below the surface lose some energy to collisions on the way out, so they emerge with less than $hf - \phi$hf−ϕ. This is why $KE_{\text{max}}$KEmax​ (not $KE$KE) appears in the equation.

It is also important to note that $\phi$ϕ is per electron, not per atom. The photoelectric effect liberates one electron from the metallic conduction band, leaving the lattice essentially unchanged. (Ionising a whole atom would require considerably more energy — typically tens of eV.)

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Deriving the Einstein Equation

Consider a photon of frequency $f$f striking a clean metal surface. It carries energy $hf$hf. When the photon is absorbed by a single electron, energy conservation demands:

$\text{(energy in)} = \text{(work to liberate electron)} + \text{(kinetic energy after escape)}$(energy in)=(work to liberate electron)+(kinetic energy after escape)

An electron right at the surface needs only the minimum energy $\phi$ϕ to escape, leaving the maximum possible kinetic energy:

$hf = \phi + KE_{\text{max}}$hf=ϕ+KEmax​

This is the Einstein photoelectric equation, the form printed on the OCR H556 data sheet. Writing $KE_{\text{max}} = (1/2) m v_{\text{max}}^2$KEmax​=(1/2)mvmax2​ explicitly:

$hf = \phi + \frac{1}{2} m v_{\text{max}}^2$hf=ϕ+21​mvmax2​

where $m = m_e = 9.11 \times 10^{-31}$m=me​=9.11×10−31 kg is the electron mass and $v_{\text{max}}$vmax​ is the maximum speed of the emitted photoelectrons.

Rearranging:

$KE_{\text{max}} = hf - \phi$KEmax​=hf−ϕ

This single line accounts for all four photoelectric observations from the previous lesson — and more, including the linear $V_s$Vs​-vs-$f$f Millikan plot that anchors PAG 5.

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The Threshold Frequency

Setting $KE_{\text{max}} = 0$KEmax​=0 in the Einstein equation gives the minimum frequency at which emission is just possible:

$h f_0 = \phi \quad \Rightarrow \quad f_0 = \frac{\phi}{h}$hf0​=ϕ⇒f0​=hϕ​

Below this frequency, the photon does not carry enough energy even to overcome the work function — no emission occurs, and the photon's energy is simply absorbed as heat in the metal (lattice vibration). At $f = f_0$f=f0​, electrons are just barely liberated with vanishing KE. Above $f_0$f0​, the excess energy $hf - \phi$hf−ϕ appears as kinetic energy.

For caesium ($\phi = 2.1$ϕ=2.1 eV $= 3.4 \times 10^{-19}$=3.4×10−19 J):

$f_0 = \frac{\phi}{h} = \frac{3.4 \times 10^{-19}}{6.63 \times 10^{-34}} = 5.1 \times 10^{14} \text{ Hz}$f0​=hϕ​=6.63×10−343.4×10−19​=5.1×1014 Hz

The corresponding threshold wavelength:

$\lambda_0 = \frac{c}{f_0} = \frac{3.00 \times 10^{8}}{5.1 \times 10^{14}} \approx 590 \text{ nm}$λ0​=f0​c​=5.1×10143.00×108​≈590 nm

Yellow light. So yellow or any shorter-wavelength (higher-frequency) light can liberate electrons from caesium, but red light (e.g. $650$650 nm) cannot. Note the easy memory aid: the threshold wavelength is the longest wavelength that causes emission (shortest wavelengths above are well above threshold; the photon energy is more than enough).

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Explaining the Four Observations

Einstein's equation explains every feature of the photoelectric effect with one-line ease:

1. Threshold frequency. If $f < f_0 = \phi/h$f<f0​=ϕ/h, then $hf < \phi$hf<ϕ and a single photon cannot supply enough energy to free an electron. No matter how many photons arrive (intensity), none has the energy needed. No emission. Above $f_0$f0​, a single photon is sufficient; emission begins immediately.

2. Instantaneous emission. A single photon liberates a single electron. There is no need to accumulate energy slowly from a wave. As soon as a photon arrives and is absorbed, the electron is freed — the delay is bounded by the photon-arrival rate, not by any classical energy-accumulation time.

3. Independence of $KE_{\text{max}}$KEmax​ from intensity. Each photon individually carries $hf$hf, regardless of how many photons are present. More intensity = more photons per second = more photoelectrons per second, but the energy delivered to each individual electron is unchanged. $KE_{\text{max}} = hf - \phi$KEmax​=hf−ϕ depends only on $f$f.

4. Linear dependence on frequency. From $KE_{\text{max}} = hf - \phi$KEmax​=hf−ϕ, a plot of $KE_{\text{max}}$KEmax​ against $f$f is a straight line with slope $h$h and y-intercept $-\phi$−ϕ, crossing the x-axis at $f_0 = \phi/h$f0​=ϕ/h. This is exactly Millikan's result.

flowchart TB
    P[Photon arrives, E = hf]
    P --> D{f ≥ f₀ = φ/h?}
    D -- no --> X[No emission<br/>Energy → lattice heat]
    D -- yes --> E[Electron absorbs hf]
    E --> W[Work φ done<br/>against surface binding]
    W --> K[Escape with KE_max = hf − φ<br/>Stopping potential V_s = KE_max/e]

The flowchart captures the entire content of Einstein's 1905 paper in five boxes. The conceptual move that took six lines of arithmetic — one-photon-one-electron, energy conservation, threshold frequency — replaced thirty years of classical-wave-theory frustration.

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Worked Example 1: Sodium Photocathode

The work function of sodium is $\phi = 2.28$ϕ=2.28 eV. Calculate (a) the threshold frequency, (b) the threshold wavelength, and (c) the maximum kinetic energy of photoelectrons when light of wavelength $\lambda = 400$λ=400 nm falls on the cathode.

Solution.

(a) Threshold frequency. Convert $\phi$ϕ to joules first:

$\phi = (2.28)(1.60 \times 10^{-19}) = 3.65 \times 10^{-19} \text{ J}$ϕ=(2.28)(1.60×10−19)=3.65×10−19 J

$f_0 = \frac{\phi}{h} = \frac{3.65 \times 10^{-19}}{6.63 \times 10^{-34}} = 5.50 \times 10^{14} \text{ Hz}$f0​=hϕ​=6.63×10−343.65×10−19​=5.50×1014 Hz

(b) Threshold wavelength.

$\lambda_0 = \frac{c}{f_0} = \frac{3.00 \times 10^{8}}{5.50 \times 10^{14}} = 5.45 \times 10^{-7} \text{ m} \approx 545 \text{ nm}$λ0​=f0​c​=5.50×10143.00×108​=5.45×10−7 m≈545 nm

This is in the green part of the visible spectrum. Light with $\lambda < 545$λ<545 nm (i.e. green or shorter) causes emission from sodium; longer-wavelength light does not.

(c) Photon energy at 400 nm:

$E_{\text{photon}} = \frac{hc}{\lambda} = \frac{1.99 \times 10^{-25}}{400 \times 10^{-9}} = 4.97 \times 10^{-19} \text{ J}$Ephoton​=λhc​=400×10−91.99×10−25​=4.97×10−19 J

Maximum KE:

$KE_{\text{max}} = E_{\text{photon}} - \phi = 4.97 \times 10^{-19} - 3.65 \times 10^{-19} = 1.32 \times 10^{-19} \text{ J}$KEmax​=Ephoton​−ϕ=4.97×10−19−3.65×10−19=1.32×10−19 J

In electronvolts: $1.32 \times 10^{-19} / 1.60 \times 10^{-19} = 0.825$1.32×10−19/1.60×10−19=0.825 eV.

Or directly using the $1240$1240 eV nm shortcut: photon energy $= 1240/400 = 3.10$=1240/400=3.10 eV, so $KE_{\text{max}} = 3.10 - 2.28 = 0.82$KEmax​=3.10−2.28=0.82 eV. Same answer in two lines.

Exam Tip: The shortcut $hc \approx 1240$hc≈1240 eV nm is a fantastic time-saver. Given wavelength in nm, photon energy in eV is just $1240/\lambda$1240/λ. Subtract the work function in eV and read $KE_{\text{max}}$KEmax​ directly in eV — without ever converting to joules.

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Worked Example 2: Finding the Work Function

Light of frequency $f = 1.0 \times 10^{15}$f=1.0×1015 Hz falls on a metal surface. The photoelectrons have maximum kinetic energy $KE_{\text{max}} = 1.6 \times 10^{-19}$KEmax​=1.6×10−19 J. Calculate the work function of the metal in joules and in eV.

Solution. Rearrange Einstein's equation:

$\phi = hf - KE_{\text{max}} = (6.63 \times 10^{-34})(1.0 \times 10^{15}) - (1.6 \times 10^{-19})$ϕ=hf−KEmax​=(6.63×10−34)(1.0×1015)−(1.6×10−19)

$\phi = (6.63 \times 10^{-19}) - (1.60 \times 10^{-19}) = 5.03 \times 10^{-19} \text{ J}$ϕ=(6.63×10−19)−(1.60×10−19)=5.03×10−19 J

In eV: $\phi = (5.03 \times 10^{-19})/(1.60 \times 10^{-19}) \approx 3.1$ϕ=(5.03×10−19)/(1.60×10−19)≈3.1 eV.

This is consistent with magnesium ($\phi \approx 3.7$ϕ≈3.7 eV) or one of the lower-work-function transition metals such as titanium ($\phi \approx 4.3$ϕ≈4.3 eV) — the exact match depends on surface preparation.

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Worked Example 3: Electron Speed

For the photoelectric effect described in Worked Example 1, calculate the maximum speed of the photoelectrons.

Solution. Using $KE_{\text{max}} = (1/2) m_e v_{\text{max}}^2$KEmax​=(1/2)me​vmax2​ with $m_e = 9.11 \times 10^{-31}$me​=9.11×10−31 kg and $KE_{\text{max}} = 1.32 \times 10^{-19}$KEmax​=1.32×10−19 J:

$v_{\text{max}} = \sqrt{\frac{2 KE_{\text{max}}}{m_e}} = \sqrt{\frac{2 \times 1.32 \times 10^{-19}}{9.11 \times 10^{-31}}} = \sqrt{2.90 \times 10^{11}} = 5.38 \times 10^{5} \text{ m s}^{-1}$vmax​=me​2KEmax​​​=9.11×10−312×1.32×10−19​​=2.90×1011​=5.38×105 m s−1

This is about $0.18\%$0.18% of the speed of light — fast by everyday standards, but comfortably non-relativistic (Lorentz factor $\gamma - 1 \sim 10^{-6}$γ−1∼10−6), so the classical formula $(1/2) m v^2$(1/2)mv2 applies without correction. For all practical A-Level photoelectric problems, photoelectron speeds are non-relativistic.

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The $KE_{\text{max}}$KEmax​ vs $f$f Graph (Millikan Plot)

A plot of $KE_{\text{max}}$KEmax​ against $f$f is the signature of the Einstein equation. From $KE_{\text{max}} = hf - \phi$KEmax​=hf−ϕ, the graph is a straight line with:

• Slope = $h$h (Planck's constant). The slope is the same for all metals — a universal constant of Nature. This is how Millikan measured $h$h directly in 1916, confirming the value Planck had inferred from blackbody radiation to better than $1\%$1%.
• x-intercept = $f_0 = \phi/h$f0​=ϕ/h (the threshold frequency of the particular metal).
• y-intercept = $-\phi$−ϕ (the negative of the work function — extrapolating to $f = 0$f=0, which has no physical meaning because emission requires $f \geq f_0$f≥f0​).

Different metals give parallel lines, shifted horizontally by their differing work functions. The slope — Planck's constant — is a genuine property of Nature, not of any particular metal. This universality is the strongest evidence for the photon model: the same $h$h appears in blackbody radiation, the photoelectric effect, atomic spectra and de Broglie's matter waves.

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