Photoelectric Effect: One Unsettled Point
A student studies the photoelectric effect using a cesium photocathode with a work function of 2.10 eV. Light from a mercury lamp passes through filters to give five frequencies. The stopping potentials are: 5.5×10¹⁴ Hz, 0.20 V; 6.5×10¹⁴ Hz, 0.60 V; 7.5×10¹⁴ Hz, 1.02 V; 8.5×10¹⁴ Hz, 1.44 V; 9.5×10¹⁴ Hz, 1.55 V. The last reading was taken immediately after changing the filter, before the electrometer had settled. A linear fit to all five points gives a slope of 3.9×10⁻¹⁵ V·s and an intercept of −2.4 V. The accepted value of h/e is 4.14×10⁻¹⁵ V·s. The student concludes that the experiment disproves the accepted value and that the work function is 2.4 eV.
1. Identify the suspect data point and explain how it affects the fitted slope and intercept. Recalculate the slope and intercept using the remaining points, and compare with the accepted h/e and the cesium work function.
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The 9.5×10¹⁴ Hz point is suspect: its 1.55 V stopping potential is much lower than the roughly 1.83 V expected from the first four points and the accepted h/e. A low point at the high-frequency end rotates the fitted line clockwise, reducing the slope and making the intercept more negative. Using the first four points, slope = (1.44 − 0.20)/(8.5 − 5.5)×10¹⁴ = 4.13×10⁻¹⁵ V·s, close to the accepted 4.14×10⁻¹⁵ V·s. The intercept is b = 0.20 − (4.13×10⁻¹⁵)(5.5×10¹⁴) ≈ −2.07 V. Since the stopping-potential equation is eV_s = hf − φ, the work function is φ = −b ≈ 2.07 eV, consistent with the cesium value 2.10 eV within reading uncertainty. The all-point fit's 3.9×10⁻¹⁵ and 2.4 eV are artifacts of the unsettled reading, not evidence against the accepted value.
2. The student wants to improve the experiment. Describe two changes that would reduce this kind of error and explain how each affects the graph. Would changing to a photocathode with a larger work function change the slope?
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First, allow the electrometer to settle at each frequency and take repeated readings until they agree; this removes the time-dependent systematic error that produced the low 9.5×10¹⁴ Hz point. Second, measure the light frequency independently with a spectrometer or use a calibrated monochromator, so frequency uncertainty does not masquerade as a slope error. Repeated readings reduce random scatter and make outliers obvious; settling removes the systematic bias. Changing to a larger-work-function photocathode shifts the entire line downward-the threshold frequency increases and the intercept becomes more negative-but it does not change the slope, because the slope is h/e and is independent of the cathode material.
3. The student claims that because the first four points fit h/e, the fifth point must be wrong. Is that reasoning valid? What independent check would confirm it?
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The reasoning is not valid by itself. Agreement with the accepted value can bias the analysis, and a real deviation would be important. The fifth point should be judged by independent evidence, not by whether it improves the fit. A proper check is to repeat the 9.5×10¹⁴ Hz measurement after the electrometer has settled, take several readings, and compare their scatter with the residual of the original point. If the repeated values converge near 1.83 V and the original 1.55 V lies far outside the repeat scatter, the original point is confirmed as an outlier. Checking the electrometer time constant, reversing the retarding polarity, and testing another filter at the same frequency would further confirm it.