Photoelectric Effect: Slope, Threshold, and a Stubborn Student
Students shine monochromatic light of several frequencies on a metal cathode and measure the stopping voltage needed to zero the photocurrent. Their data give stopping voltages of 0.50, 1.05, 1.60, and 2.15 V for frequencies of 6.0, 7.0, 8.0, and 9.0 x 10^14 Hz. One student fits a line and reports a slope of 6.5 x 10^-34 J s and a threshold frequency of 5.5 x 10^14 Hz. The accepted Planck constant is 6.63 x 10^-34 J s. A classmate argues that brighter light should increase the stopping voltage.
1. Use the data to extract the slope and threshold frequency, compare with Planck's constant, and explain what the slope and intercept physically represent.
Read the worked answer
Einstein's equation: eV_stop = hf - phi, so V_stop = (h/e) f - phi/e. A plot of V_stop vs. f is linear with slope h/e and x-intercept f0 = phi/h.
From the data: between 6.0 and 9.0 x 10^14 Hz, dV = 2.15 - 0.50 = 1.65 V over df = 3.0 x 10^14 Hz. Slope = 1.65/3.0e14 = 5.5 x 10^-15 V s. Multiply by e = 1.60e-19 C: h = 5.5e-15 x 1.60e-19 = 8.8 x 10^-34 J s - noticeably above 6.63e-34. (The student's 6.5e-34 comes from a different fit; checking: 6.5e-34/1.6e-19 = 4.06e-15 V s, which does not match the data spacing of 5.5e-15 V s, so the student's slope is inconsistent with the tabulated points.)
Intercept: extrapolating, V = 0 at f0. Using slope 5.5e-15: f0 = 6.0e14 - 0.50/5.5e-15 = 6.0e14 - 0.91e14 = 5.1 x 10^14 Hz. The slope gives h/e; the intercept gives the work function phi = h f0.
2. Evaluate the classmate's claim that brighter light raises the stopping voltage, and explain how you would use this apparatus to distinguish intensity effects from frequency effects.
Read the worked answer
The claim is incorrect for the photoelectric effect. Stopping voltage depends on the maximum kinetic energy of emitted electrons, which is set by photon energy hf minus the work function - not by intensity. Increasing intensity increases the number of photons (and thus the saturation current) but not the energy per photon, so V_stop is unchanged. This is a core piece of evidence for quantization.
Experimental test: hold frequency fixed and vary intensity (neutral-density filters). The stopping voltage should stay constant while the saturation photocurrent scales with intensity. Then vary frequency at fixed intensity and confirm V_stop rises linearly. If V_stop changed with intensity, that would indicate a problem (e.g., space charge, heating, or a non-ideal detector) rather than a real violation. Also note the student's slope (6.5e-34) is closer to accepted than the direct calculation (8.8e-34); the discrepancy suggests the fit or the voltage calibration needs checking before claiming agreement with Planck's constant.