Photoelectric Effect Stopping Potentials That Are Uniformly Low
A student uses a phototube with a potassium cathode (accepted work function 2.30 eV) to measure stopping potentials for two mercury lines isolated with interference filters: 435.8 nm and 404.7 nm. The retarding potential comes from a calibrated ten-turn potentiometer and is read with a digital voltmeter; the photocurrent is monitored with a picoammeter, and the student records the voltage at which the current first reads zero on the most sensitive range. Results: 0.30 V at 435.8 nm and 0.52 V at 404.7 nm, with repeat runs agreeing to ±0.01 V. Using KEmax = hc/λ − W with W = 2.30 eV, the predictions are 0.545 V and 0.764 V. Noting that both readings are low by roughly the same amount, the student plans to report a measured work function of 2.54 eV and to state that the cathode 'cannot be potassium after all.' The instructor points out that the slope of a stopping-potential-versus-frequency graph, not the intercept, determines h, and asks the student to reconsider before writing the conclusion.
1. From the two data points, decide whether the discrepancy points to an error in h or in the work function. Show the arithmetic that supports your conclusion.
Read the worked answer
The photoelectric equation gives eV_s = hc/λ − W, i.e. V_s = (h/e)f − W/e, so a plot of V_s against f has slope h/e and intercept −W/e.
Photon energies (using hc = 1240 eV·nm):
1240/435.8 = 2.845 eV, so predicted V_s = 2.845 − 2.30 = 0.545 V 1240/404.7 = 3.064 eV, so predicted V_s = 3.064 − 2.30 = 0.764 V
Predicted difference: 0.764 − 0.545 = 0.219 V. Measured difference: 0.52 − 0.30 = 0.220 V. These agree to about 0.5%, so the slope - and therefore the measured h/e - is correct.
What is wrong is the intercept. The measured points sit about 0.24 V below prediction at both wavelengths. Extracting W from each point: 2.845 − 0.30 = 2.545 eV and 3.064 − 0.52 = 2.544 eV. The two agree to 1 meV, which is the signature of a constant additive offset rather than random error or a bad Planck constant. A wrong h would tilt the line; this line has the right slope and a shifted intercept. The student's arithmetic (2.54 eV) is right but the physical inference is premature.
2. Give a physical mechanism inside the tube that would shift every stopping-potential reading by a nearly constant amount, and explain why the shift is nearly the same at both wavelengths.
Read the worked answer
The offset is a contact potential difference, compounded by anode photoemission.
Contact potential. The cathode and anode are different metals. When they are joined, their Fermi levels equalise and a built-in potential of order a volt appears between them. The external retarding voltage must overcome that built-in potential as well as the photoelectrons' kinetic energy, so every reading of V_s is displaced by an amount fixed by the two work functions and the tube geometry - not by the light. That is precisely why the shift is the same at both wavelengths: it is a property of the electrodes, so it translates the whole line rather than rotating it. The measured intercept becomes −(W/e + offset/e), and any single-point work-function extraction is wrong by the offset.
Anode photoemission. Some light reaches the anode, which is a different metal and may have a lower work function. The anode then emits its own electrons, which the circuit counts as a reverse current. The picoammeter reads zero not when the cathode current vanishes but when cathode and anode currents cancel, and that cancellation occurs at a smaller retarding potential than the true stopping potential.
Both effects are stable properties of the tube, which also explains the excellent ±0.01 V reproducibility: the offset is systematic, not noise, so repeating the measurement cannot reveal it.
3. Describe two procedures that would remove or measure the offset without any knowledge of the tube's construction.
Read the worked answer
Procedure 1 - intensity variation. Record the full I–V curve at three or four light levels using neutral-density filters, keeping the wavelength fixed. The true stopping potential is the voltage at which the curves converge. The cathode photocurrent scales with intensity, so the apparent zero-crossing walks as the light level changes, while the contact potential and the anode's geometric contribution do not. Reading the intersection of the family of curves cancels most of the offset without any model of the tube. Repeating at a second wavelength and taking the slope then gives h/e cleanly.
Procedure 2 - many-wavelength slope and intercept, plus a better feature than 'current equals zero' on the I–V curve. Collect data at five or more mercury lines, fit V_s against f, and report h from the slope, which no constant offset can affect, while reporting the intercept explicitly as (W/e + offset/e). For W itself, use the reverse-current plateau: raise the retarding voltage until the photocurrent flattens into its reverse-saturation value, and identify the knee where the fastest photoelectrons are just stopped, rather than the point where the meter happens to read zero. That knee is far less sensitive to ammeter threshold and anode current than the zero crossing is.
Either procedure converts the problem from 'the cathode is the wrong metal' to 'the measurement has a known, removable systematic,' which is the correct diagnosis.