Photoelectric effect: zero error and a false threshold
Students illuminate a sodium photocathode with work function about 2.3 eV in a vacuum tube and measure the stopping potential with a high-impedance electrometer. At λ=400 nm they record a stopping potential of 0.85 V; at λ=500 nm they record 0.25 V. At λ=600 nm they see a very small current even with the stopping voltage set to 1.0 V, and one student claims this proves that 600 nm photons can eject electrons. The electrometer has a zero error: with the light blocked it reads 0.05 V. The wavelength calibration is ±2 nm. The accepted values are h=6.63×10^-34 J s, c=3.00×10^8 m/s, and e=1.60×10^-19 C. The students want to determine h from their data and to test the intensity dependence predicted by the photon model.
1. Use the two stopping potentials to calculate h and the work function, correcting for the zero error. Compare with the accepted values and comment on the wavelength uncertainty.
Read the worked answer
Correct the electrometer reading by subtracting the 0.05 V zero error: Vs1 = 0.85 − 0.05 = 0.80 V and Vs2 = 0.25 − 0.05 = 0.20 V.
Photon energies: E1 = hc/λ1 = 1240 eV nm/400 nm = 3.10 eV; E2 = 1240/500 = 2.48 eV. Since Kmax = eVs, the work function is φ = E1 − eVs1 = 3.10 − 0.80 = 2.30 eV, and from the second point φ = 2.48 − 0.20 = 2.28 eV. The average is about 2.29 eV, close to the sodium value.
For h, use e(Vs1 − Vs2) = hc(1/λ1 − 1/λ2). The stopping-potential difference is 0.60 V, and 1/λ1 − 1/λ2 = (2.50 − 2.00)×10^6 = 5.00×10^5 m^-1. Thus h = (1.60×10^-19×0.60)/(3.00×10^8×5.00×10^5) = 6.40×10^-34 J s. This is about 3.5% below the accepted 6.63×10^-34 J s. The zero error cancels in the difference, so it does not affect h, but it does affect φ. A ±2 nm wavelength uncertainty is about 0.5% in each photon energy; it is smaller than the 3.5% discrepancy, so the discrepancy likely comes from calibration, the cathode surface, or the assumed work function, not from the wavelength reading alone.
2. Evaluate the claim that the small current at 600 nm proves photoemission by 600 nm photons. Calculate the threshold wavelength and identify likely causes of the small current.
Read the worked answer
At 600 nm the photon energy is 1240/600 ≈ 2.07 eV, which is less than the work function of about 2.3 eV. The threshold wavelength is λth = hc/φ = 1240/2.3 ≈ 539 nm. Therefore 600 nm photons do not have enough energy to eject electrons from sodium in a one-photon process, regardless of intensity.
The small current persisting at 1.0 V stopping potential is likely not from 600 nm photoemission. Possible causes are leakage current in the electrometer or tube, stray ultraviolet or violet light reaching the cathode, photoemission from the anode or from impurity patches with a lower work function, or a small two-photon process. Two-photon emission would be extremely weak at ordinary intensities and would scale nonlinearly with intensity, so it can be tested by varying the light intensity and checking whether the current disappears when the tube is carefully shielded. The claim is not supported without those controls.
3. The students double the intensity of the 400 nm light. Predict what happens to the stopping potential and to the photocurrent, and explain how this tests the photon model against a classical wave model.
Read the worked answer
The stopping potential should remain at about 0.80 V. The stopping potential measures the maximum kinetic energy of the emitted electrons, Kmax = hc/λ − φ, which depends on the photon energy and the work function, not on the number of photons per second. Doubling the intensity doubles the number of photons per second, so the saturation photocurrent should approximately double, provided the setup is not limited by space charge or collection efficiency.
A classical wave model would predict that a more intense wave carries more energy and should give electrons more energy, raising the stopping potential, and it would not naturally predict a threshold frequency. The observed constancy of the stopping potential with intensity, together with the existence of a threshold wavelength, supports the photon model. The students should plot photocurrent versus intensity at fixed wavelength and stopping potential versus frequency at fixed intensity to display both tests.