Photoelectric Effect & the Threshold

Shine light of one colour on a metal and measure what comes off, or run Millikan's experiment with the lines of a mercury lamp. The table under the drawing works out the photon energy, the fastest electron and the current.

Worked example — Photoelectric Effect & the Threshold

Frequency	740.9 THz
Photon energy	3.064 eV
Work function	2.750 eV
Threshold frequency	664.9 THz
Threshold wavelength	450.9 nm
Maximum kinetic energy	0.314 eV
Stopping potential	0.314 V
Fastest electron	332 km/s

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Photoelectric Effect & the Threshold

Light falling on a metal can free electrons from its surface: the photoelectric effect. Light arrives as photons of energy E = hf, and each can give its whole energy to one electron. Freeing the electron costs the metal's work function φ, so the fastest electron leaves with K = hf − φ. Below the threshold frequency f₀ = φ ÷ h almost nothing comes off: the threshold is sharp only at absolute zero, and warmth leaves a faint tail below it. The stopping potential V_s is the reverse voltage that turns the fastest electron back, eV_s = K, and it is how K is measured.

What each symbol means

f: the light's frequency, c ÷ λ, in THz. It alone sets each photon's energy
φ: the work function, in eV. What the metal charges to release one electron
K: the fastest electron's kinetic energy, hf − φ. Independent of how bright the light is
V_s: the stopping potential, in volts. Numerically equal to K in electronvolts
η: quantum efficiency, the share of photons that free an electron. It scales the current and nothing else

Breaks when: Work functions depend on the surface as well as the element. The 2.75 eV used here for sodium is Michaelson's 1977 figure; many textbooks print 2.28 or 2.36 eV, and at 2.28 eV green light would free electrons from sodium. Measured values scatter, partly with the crystal face and partly because a surface that has touched air is a different surface, which is why Millikan cut fresh metal inside the vacuum.

Brighter light makes more electrons, not faster ones 🖖

Press Violet on sodium and then Twice as bright. The current doubles, from 326.4 to 652.8 nA, and the fastest electron carries 0.314 eV both times, moving at 332 km/s both times. A wave picture of light predicts something else: a stronger wave shakes the electrons harder and throws them out faster. Einstein's explanation is light arriving in packets, each carrying hf. Brighter light is more packets per second. Each packet still frees at most one electron and hands it what is left after paying the work function. That is Einstein's 1905 equation, K = hf − φ, and it is the result his Nobel Prize was awarded for, not relativity.

Below a threshold, no amount of light frees a single electron 🖖

Press Red on sodium. The lamp is ten times as powerful as the violet one and delivers sixteen times as many photons, and the current is 0.0 nA at every anode voltage and every efficiency the controls allow. Each red photon carries 1.907 eV, and sodium will not let an electron go for less than 2.75. At lamp brightness two photons practically never combine their energy on one electron, which takes a focused pulsed laser, so 3.27 × 10¹⁶ weak photons a second free exactly as many electrons as one does: none. The wavelength bar under the slider hatches everything past sodium's threshold, 450.9 nm, which puts the whole visible spectrum except violet out of reach. Press Green on caesium and green light works, because caesium gives up an electron for 2.14 eV. This is why so many photocells used caesium, usually as a compound such as silver–oxygen–caesium or caesium antimonide, whose response reaches further into the red than the bare metal's.

Every metal draws the same slope, and the slope is Planck's constant 🖖

The lower plot draws stopping potential against frequency for all six metals. Each is a straight line, and all six are parallel: the metal only decides where its line crosses zero, and the slope is h ÷ e for every one. Press Millikan's sodium run. Five mercury lines, each stopping potential measured with a little noise, and a least-squares line through them gives h = 6.610 × 10⁻³⁴ J·s, 0.24% below the defined value. Robert Millikan published a measurement of this kind in 1916, expecting, by his own later account, to find Einstein's equation wrong. His slope agreed with Planck's constant from radiation measurements, and in the same paper he called the theory behind the equation untenable.

Problem solved in full

Ultraviolet light on a zinc plate 5 steps
The 253.65 nm line of a mercury lamp falls on a zinc plate, whose work function is 4.33 eV. Find the photon energy, the energy and speed of the fastest electron, the stopping potential, and the longest wavelength that frees any electron from zinc. Press UV on zinc to check each figure.

Open this problem in the tool
1. The photon energy in electronvolts, using hc = 1239.84 eV·nm:
  
  $E = \frac{h c}{λ} = \frac{1239 . 84 eV nm}{253 . 65 nm} = 4.888 eV$
2. Take away the work function to get the fastest electron's energy:
  
  $K_{m a x} = E - φ = 4.888 - 4.33 = 0.558 eV$
3. The stopping potential is that energy in volts:
  
  $V_{s} = \frac{K _{m a x}}{e} = 0.558 V$
4. The speed follows from K = ½mv², with K in joules:
  
  $v = \frac{2 K _{m a x}}{m _{e}} = \frac{2 \cdot 0 . 558 \cdot 1 . 602 \times 1 0 ^{- 19}}{9 . 109 \times 1 0 ^{- 31}} = 4.43 \times 1 0^{5} m/s$
5. The threshold wavelength is where the photon energy equals the work function:
  
  $λ_{0} = \frac{h c}{φ} = \frac{1239 . 84}{4 . 33} = 286.3 nm$
Answer

The photon carries 4.888 eV, the fastest electron leaves with 0.558 eV at 443 km/s, and a reverse voltage of 0.558 V stops it. Zinc responds to nothing longer than 286.3 nm, which is ultraviolet. Sunlight frees almost nothing from zinc even outdoors: ozone removes nearly everything shorter than about 295 nm, and zinc needs 286.3 nm or shorter.

$K_{m a x} = 0.558 eV, λ_{0} = 286.3 nm$

Learning path

Quantum mechanics: matter as waves

References (3)

Einstein's paper, which proposes that light is absorbed in quanta of energy hf and derives the equation this page plots: A. Einstein, "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt." Annalen der Physik 322(6), 1905, pp. 132-148. DOI 10.1002/andp.19053220607
Millikan's measurement of h from the slope of the stopping potential against frequency, the experiment the Millikan mode reproduces: R. A. Millikan, "A Direct Photoelectric Determination of Planck's h." Physical Review 7(3), 1916, pp. 355-388. DOI 10.1103/PhysRev.7.355
Where the six work functions come from, and why sodium's differs from many textbooks: H. B. Michaelson, "The work function of the elements and its periodicity." Journal of Applied Physics 48(11), 1977, pp. 4729-4733. DOI 10.1063/1.323539

By the 42Math team · Reviewed against published sources. Corrections welcome.