Stopping Potentials That Give the Wrong Work Function
In a photoelectric-effect lab, students illuminate a cesium cathode (work function 2.10 eV per the tube datasheet) using the 365 nm and 405 nm lines of a mercury lamp, and find the stopping potential by adjusting a retarding voltage until the photocurrent, read on a 1 nA scale, falls to zero. Dark current of about 0.3 nA persists even at large retarding voltages, so 'zero' is judged by eye. The students record stopping potentials of 1.06 V at 365 nm and 0.72 V at 405 nm. They compute a slope and an intercept, find that h/e is close to the accepted value, but obtain a work function near 2.35 eV. They conclude that the datasheet is wrong and that cesium's work function has been miscalculated for decades.
1. Compute h/e and the apparent work function from the two points. Then explain what kind of error can shift the work function while leaving h/e almost correct, and identify its physical source in this apparatus.
Read the worked answer
Frequencies: f1 = c/lambda = 3.00e8/365e-9 = 8.22e14 Hz; f2 = 3.00e8/405e-9 = 7.41e14 Hz.
Slope: (1.06 - 0.72)/(8.22e14 - 7.41e14) = 0.34/8.1e13 = 4.2e-15 V s, against the accepted h/e = 4.136e-15 V s, an agreement of about 1%. That is within the reading precision of a hand-set voltage.
Intercept: using V = (h/e)f - (phi/e), phi = (4.136e-15)(8.22e14) - 1.06 = 3.40 - 1.06 = 2.34 eV. The datasheet says 2.10 eV, so the apparent work function is about 0.24 eV too high.
Now notice the shape of the discrepancy. The measured potentials are both low by the same amount: the true stopping potentials from hc/lambda - phi are 1.30 V and 0.96 V, and each measurement is 0.24 V lower. A constant additive offset shifts every point equally, so it changes the intercept (hence phi) while leaving the slope (hence h/e) completely untouched. That is exactly what is seen.
The physical source is the contact potential difference. When no external voltage is applied, the cathode and the collector are already at different electrostatic potentials because they are made of different metals, and the voltmeter reads the applied voltage, not the internal potential difference. The student's zero on the meter is not the zero of the electron's energy. An offset of a few tenths of an electron volt is entirely normal for dissimilar metals, and it is why careful experiments determine phi only from a set of lines and a calibration against a known cathode, or use the same metal for both electrodes.
2. Using the datasheet value of 2.10 eV rather than the student's intercept, compute the maximum kinetic energy and the de Broglie wavelength of the emitted electrons at 365 nm. Predict the stopping potential the students would measure for the 313 nm mercury line.
Read the worked answer
Photon energy at 365 nm: E = hc/lambda = 1240/365 = 3.40 eV. Then K_max = 3.40 - 2.10 = 1.30 eV = 2.08e-19 J.
De Broglie wavelength: p = sqrt(2 m K) = sqrt(2 x 9.11e-31 x 2.08e-19) = sqrt(3.79e-49) = 6.16e-25 kg m/s. Then lambda = h/p = 6.63e-34/6.16e-25 = 1.08e-9 m, about 1.1 nm. This is comparable to atomic spacings, which is why photoelectrons at these energies can diffract from crystal surfaces.
At 313 nm: E = 1240/313 = 3.96 eV, so the true stopping potential is 3.96 - 2.10 = 1.86 V. With the same 0.24 V offset the apparatus displayed at the other two lines, the students would read about 1.62 V.
The key discipline is to compute K_max from the accepted work function, not from the student's fitted intercept. Using 2.35 eV would give 1.05 eV and a 1.2 nm wavelength, which is wrong by roughly the contact potential. The experiment measures differences in photon energy with high accuracy; it measures absolute energies only after the offset is established independently.
3. The students propose to test their conclusion by repeating the experiment with a lamp of twice the intensity. Predict the result and explain why it does or does not resolve the discrepancy.
Read the worked answer
Doubling the intensity doubles the number of photons per second and therefore doubles the photocurrent at any retarding voltage below the stopping point. It does not change the photon energy, so the maximum kinetic energy of each emitted electron is unchanged and the stopping potential stays put. One photon is absorbed by one electron; making more photons per second makes more electrons, not faster ones.
So the intensity test is a good check of the model but does not resolve the disagreement. If the students find that the stopping potential does shift with intensity, that is diagnostic of an artefact rather than a real effect: with a photocurrent of a few nanoamperes and a dark current of 0.3 nA, changing the light level changes the current the student is trying to null, and judging 'zero' by eye on a 1 nA scale will bias the voltage at which they declare the current gone. Space charge and leakage paths can do the same.
A test that would actually resolve the discrepancy is to measure a third and fourth spectral line and check whether the offset is constant across all of them, then to swap in a tube with a known photocathode and see whether the offset is the same. If it is a property of the electrode pair, the offset follows the tube; if it is a property of the electrometer or the nulling method, it moves with the electronics.