Photoelectric Effect with Changing Intensity and Wavelength
Light from a variable-wavelength source falls on a clean metal surface with work function 2.20 eV. At wavelength 400 nm, a stopping potential of 0.90 V is measured. The intensity is then doubled at the same wavelength, and the stopping potential is again measured. Next the wavelength is changed to 300 nm while the intensity is kept at the higher value. A student predicts that doubling the intensity will double the stopping potential because more photons arrive per second, and that the shorter wavelength will increase the photocurrent but not the stopping potential. You are asked to test these predictions with the photon model. Use hc = 1240 eV·nm.
1. Calculate the maximum kinetic energy and stopping potential at 400 nm, and confirm consistency with the work function.
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Photon energy is E = hc/λ = 1240/400 = 3.10 eV. The photoelectric equation gives K_max = E − φ = 3.10 − 2.20 = 0.90 eV. The stopping potential is the voltage that just stops the most energetic electrons, so eV_s = K_max; numerically V_s = 0.90 V. This is consistent with the quoted measurement. The current at this wavelength is set by the number of photons per second and the quantum efficiency, not by the stopping potential. A smaller work function would increase K_max and V_s for the same wavelength; a larger work function would reduce them.
2. Predict the stopping potential after the intensity is doubled and after the wavelength is changed to 300 nm.
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Doubling the intensity at 400 nm doubles the number of photons arriving per second, so the saturation photocurrent doubles, but each photon still has 3.10 eV. Therefore K_max remains 0.90 eV and the stopping potential remains 0.90 V. At 300 nm, E = 1240/300 = 4.13 eV, so K_max = 4.13 − 2.20 = 1.93 eV and V_s = 1.93 V. The intensity is unchanged in this second change, so the photon flux is lower than at 400 nm for the same power, but the stopping potential is higher because each photon carries more energy.
3. Evaluate the student's two predictions using the photon model.
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Both predictions are wrong. Intensity is photon flux at fixed wavelength; it changes the number of photoelectrons per second, not the energy of each photon. The stopping potential measures the maximum kinetic energy, so it is independent of intensity once the frequency is above threshold. The shorter wavelength increases the photon energy, so it increases K_max and the stopping potential. The photocurrent after the wavelength change is not guaranteed to increase; it depends on the photon flux and on how the metal absorbs at 300 nm, and it could fall if the source power is fixed. The photon model therefore separates intensity (current) from wavelength (energy) in the way the student did not.