Photocell: two stopping potentials and a current that is not photoelectric
In a photoelectric-effect experiment a photocell with a potassium cathode (work function 2.30 eV) is illuminated by the 365 nm line of a mercury lamp. A reverse potential difference is applied and increased until the photocurrent just falls to zero. The student repeats the measurement with 435 nm and 546 nm filters. Use hc = 1240 eV nm. At 546 nm the microammeter still reads 0.4 µA even with the largest reverse p.d. available, and the same 0.4 µA is still there when the lamp is switched off. The student's write-up states: 'the 546 nm light is also ejecting electrons, because current flows'. A second student argues that if light were a classical wave, a brighter beam should give faster electrons.
1. Calculate the maximum kinetic energy and the stopping potential for the 365 nm and 435 nm lines, and say what the two results show.
Read the worked answer
365 nm: photon energy = 1240/365 = 3.40 eV. Maximum kinetic energy = 3.40 - 2.30 = 1.10 eV. The stopping potential V_s is the reverse p.d. that just stops the fastest electrons, so eV_s = KE_max, giving V_s = 1.10 V.
435 nm: photon energy = 1240/435 = 2.85 eV. KE_max = 2.85 - 2.30 = 0.55 eV, so V_s = 0.55 V.
What this shows: the two stopping potentials differ by exactly the difference in photon energies, and the work function is the constant offset. Halving the wavelength slightly more than doubles V_s (1.10 versus 0.55) while the photon energy rises by only a factor 1.19. That is a direct measurement of the work function and it confirms that electron energy is set by photon energy, not by the brightness or the duration of illumination.
2. Explain the 0.4 µA at 546 nm and describe a quick experimental test that settles the question.
Read the worked answer
Photon energy at 546 nm = 1240/546 = 2.27 eV, which is below the 2.30 eV work function. No single photon has enough energy to free an electron, so genuine photoemission is impossible at this wavelength, whatever the intensity.
The 0.4 µA is a dark (leakage) current: stray light leaks around the filter, and there is also thermionic emission from the cathode and finite insulation resistance between the electrodes, all of which drive a small current that does not care about the retarding voltage.
Test: switch off the lamp and read the meter. If it is still 0.4 µA, the current is not photoelectric. A second test is to vary the intensity with neutral filters: the genuine photocurrent is proportional to intensity and vanishes when the filter blocks the beam completely, whereas leakage is independent of the light. The student should subtract this background before quoting any photocurrent.
3. The intensity of the 365 nm line is doubled. Predict what happens to the stopping potential and to the saturation current, and explain how this distinguishes the photon model from a classical wave model.
Read the worked answer
Stopping potential: unchanged at 1.10 V. It depends only on photon energy (hence wavelength) and the work function, and doubling the intensity does not change the energy of any individual photon.
Saturation current: approximately doubles. Twice the intensity means twice the number of photons arriving per second, hence twice as many emitted electrons per second once the anode collects them all.
Why this discriminates: a classical wave model predicts the opposite pattern. In a wave picture the energy absorbed by an electron depends on the amplitude of the field and on how long it has been illuminated, so a brighter beam should eject faster electrons (a larger stopping potential) and should show a measurable delay at low intensity while energy accumulates. In practice the stopping potential is independent of intensity and emission starts within about 10⁻⁹ s, so the energy must arrive in discrete packets whose size is fixed by the frequency.