Zinc Plate and the Photoelectric Threshold
A student mounts a cleaned zinc plate on a gold-leaf electroscope and charges it negatively. She shines light from a mercury lamp through filters onto the plate. Zinc has a work function of 4.3 eV. With a 250 nm filter the leaf collapses in a few seconds; with a 400 nm filter the leaf stays down for several minutes. She then repeats the 250 nm test with the lamp moved to twice its original distance and finds the leaf collapses more slowly. Take hc = 1.24 × 10⁻⁶ eV m, h = 6.63 × 10⁻³⁴ J s, c = 3.0 × 10⁸ m/s and e = 1.6 × 10⁻¹⁹ C.
A classmate says: 'Brighten the 400 nm lamp enough and the electrons will come out, because more photons means more energy arriving.' A second student suggests that the rate at which the leaf collapses measures the kinetic energy with which the electrons leave.
1. Calculate the photon energy at each wavelength, the maximum kinetic energy of the photoelectrons and the stopping potential for 250 nm, and explain the 400 nm result.
Read the worked answer
Use E = hc/λ = (1.24 × 10⁻⁶ eV m)/λ.
250 nm: E = 1 240/250 = 4.96 eV. 400 nm: E = 1 240/400 = 3.10 eV.
Einstein's equation: KE_max = hf − W. 250 nm: KE_max = 4.96 − 4.30 = 0.66 eV = 0.66 × 1.6 × 10⁻¹⁹ = 1.06 × 10⁻¹⁹ J. Stopping potential: eV_s = KE_max, so V_s = 0.66 V.
The 400 nm result. One photon carries 3.10 eV, less than the 4.30 eV needed to escape. The photon is absorbed or not; it cannot be topped up by a second photon, because emission must occur within about 10⁻¹⁵ s of absorption. Hence no electrons, no discharge, however long the plate is illuminated.
The threshold wavelength is λ₀ = 1 240/4.30 = 288 nm: only ultraviolet shorter than about 288 nm works with zinc.
2. Answer the classmate, and then say what changes, and what does not change, when the 250 nm lamp is moved to twice the distance.
Read the worked answer
To the classmate. Brightness means photons per second, not energy per photon. Every photon in the 400 nm beam still carries only 3.10 eV, below the 4.30 eV work function, so making the beam a hundred times brighter just supplies a hundred times more useless photons. The emission remains exactly zero. Light of any intensity at 400 nm will never eject electrons from zinc; only shortening the wavelength below 288 nm will. This is the observation that forced the photon model: in the wave model, a bright enough beam of any frequency should eventually supply the energy.
Moving the lamp to 2r.
- Intensity falls as 1/r², i.e. to one quarter: the number of photons per second per unit area on the plate falls fourfold.
- The photocurrent falls to about one quarter, so the leaf collapses about four times more slowly - exactly what she sees.
- KE_max stays at 0.66 eV and V_s stays at 0.66 V, because these depend only on hc/λ − W, not on intensity. That contrast - current set by intensity, electron energy set by frequency - is the clean experimental signature of the photon model.
3. Correct the second student: what does the collapse rate really measure, and what apparatus change is needed to measure KE_max?
Read the worked answer
What the leaf measures. The electroscope is only a charge detector. Electrons leaving the zinc leave it positively charged, and the leaf falls once enough charge has gone. The rate of collapse therefore measures the number of electrons per second (the photocurrent), not their speeds. A rapid collapse with 250 nm light means a large photocurrent; it says nothing about KE_max.
What is needed instead. A photocell in an evacuated envelope, so the electrons can cross to a collector without being stopped by air molecules (in air they are absorbed within a fraction of a millimetre). A variable reverse p.d. is applied with the collector negative; as the voltage is increased the photocurrent falls, and the voltage at which it just reaches zero is the stopping potential V_s, with KE_max = eV_s.
Repeating for several wavelengths and plotting V_s against 1/λ gives a straight line of gradient hc/e and intercept −W/e, from which h and the work function are obtained. The electroscope cannot do this: there is no known retarding field, the oxide layer on the zinc alters the effective work function, and the leaf gives only a yes/no answer about emission.