Assignment Sample: Photoelectric Effect Data Analysis Exercise

Model Answer

Introduction

The photoelectric effect – the emission of electrons from a metal surface when illuminated by light above a threshold frequency – provided one of the earliest and clearest demonstrations that light carries energy in discrete quanta rather than as a continuous wave, and Einstein’s 1905 explanation of the effect, itself rooted in Planck’s quantum hypothesis, remains a standard vehicle for introducing photon energy in undergraduate physics (Einstein, 1905; Eisberg and Resnick, 1985). This assignment analyses a set of stopping-voltage measurements recorded for a sodium photocathode under five different illuminating wavelengths, with the aim of extracting an experimental value for Planck’s constant, h, and for the work function, φ, and threshold frequency, f₀, of the sodium surface, and of evaluating the precision of the resulting values against the accepted CODATA figures.

The result is significant because a classical, wave-based picture of light makes qualitatively different predictions from Einstein’s photon model. Classically, the energy delivered to the metal surface should depend on the intensity of the light, so increasing the brightness of a sub-threshold light source should eventually supply enough cumulative energy to eject electrons, and the maximum kinetic energy of any emitted electrons should rise smoothly with intensity rather than with frequency. Neither prediction is observed experimentally: below the threshold frequency, no photoelectrons are emitted no matter how intense the illumination, while above the threshold, the maximum electron kinetic energy is found to depend only on frequency and to be completely independent of intensity, which instead controls only the number of electrons emitted per second (Tipler and Llewellyn, 2012). This assignment’s data analysis is therefore not simply a curve-fitting exercise but a direct quantitative test of the quantum prediction embodied in Einstein’s equation.

The measurement also draws together several distinct experimental skills that are examined explicitly below: controlling a single independent variable, wavelength, while other factors are held broadly constant; adopting a consistent operational definition for a stopping voltage that in practice approaches zero asymptotically rather than sharply; and converting five independent readings into two physically meaningful constants by linear regression, together with an explicit comparison against the accepted CODATA values.

Apparatus and Method

The stopping-voltage data supplied were recorded using a vacuum photocell containing a sodium photocathode and a collecting anode, connected in series with a sensitive picoammeter and a variable, reversible bias supply. Monochromatic illumination at each of the five wavelengths in Table 1 was obtained using narrow-band interference filters placed in front of a stabilised light source, with each filter’s centre wavelength independently verified against the manufacturer’s specification. For each wavelength, the bias voltage was first set to zero, and the anode was made increasingly negative relative to the cathode while the photocurrent was recorded; the stopping voltage was taken as the bias at which the photocurrent first fell to a negligible, pre-defined threshold close to zero, rather than to exactly zero, since photocurrent approaches zero asymptotically rather than abruptly. Ambient light was excluded from the photocell housing throughout, and the apparatus was allowed to reach thermal equilibrium before each set of readings to minimise drift in the picoammeter zero. To limit the influence of electrical noise on the current readings close to the estimated stopping point, three consecutive readings were taken and averaged at each bias setting in this region, where the photocurrent is most sensitive to small changes in retarding voltage. Each interference filter was held in a fixed mount at normal incidence throughout, since transmitted centre wavelength shifts measurably at oblique incidence.

Theoretical Background

Einstein’s photoelectric equation states that a single photon of frequency f transfers its entire energy hf to a single bound electron, part of which, the work function φ, is used to overcome the binding energy holding the electron in the metal, with any remainder appearing as the kinetic energy of the emitted photoelectron:

hf = φ + KE_max

In a photocell circuit, the maximum kinetic energy of the emitted electrons can be measured by applying a retarding (stopping) voltage, V_s, between the photocathode and the collecting anode and finding the minimum voltage at which the photocurrent falls to zero, at which point eV_s = KE_max. Substituting into Einstein’s equation and rearranging gives:

V_s = (h/e) f − φ/e

This is the equation of a straight line when V_s is plotted against f: the gradient of the line is h/e, from which Planck’s constant can be found once the electronic charge e is known, and the y-intercept is −φ/e, from which the work function follows directly. The line also crosses the frequency axis (V_s = 0) at the threshold frequency f₀ = φ/h, below which no photoelectrons are emitted regardless of light intensity, a result that cannot be explained by a classical, wave-based model of light and that is one of the central pieces of evidence for photon quantisation (Krane, 1996).

Experimental Data

Table 1 gives the recorded stopping voltage for five wavelengths of light, each converted to a corresponding frequency using f = c/λ with c = 2.998 × 10⁸ m/s.

Table 1. Measured stopping voltage against frequency for the sodium photocathode.

Data Analysis and Worked Calculations

The five (f, V_s) pairs were fitted with a least-squares straight line. Working in units of 10¹⁴ Hz for x = f to keep the arithmetic manageable, the mean values are:

x̄ = (8.214+7.402+6.876+6.094+5.743)/5 = 6.8658   |   ȳ = (1.12+0.78+0.56+0.24+0.10)/5 = 0.560

Tabulating the deviations from the mean for each point and forming the two sums required for the least-squares gradient gives Σ(x_i−x̄)(y_i−ȳ) = 1.6365 (in units of 10¹⁴ Hz·V) and Σ(x_i−x̄)² = 3.9617 (in units of 10²⁸ Hz²), so the gradient of the best-fit line is:

gradient = 1.6365 × 10¹⁴ ÷ 3.9617 × 10²⁸ = 4.131 × 10⁻¹⁵ V·s

Since the gradient of the V_s–f line equals h/e, the experimental value of Planck’s constant is obtained by multiplying by the electronic charge, e = 1.602 × 10⁻¹⁹ C:

h_exp = gradient × e = 4.131 × 10⁻¹⁵ × 1.602 × 10⁻¹⁹ = 6.618 × 10⁻³⁴ J·s

Comparing this with the accepted CODATA value of h = 6.626 × 10⁻³⁴ J·s gives a percentage difference of:

% difference = |6.626−6.618| ÷ 6.626 × 100 = 0.12%

The y-intercept of the best-fit line follows from c = ȳ − gradient × x̄, giving c = 0.560 − (4.131×10⁻¹⁵×6.8658×10¹⁴) = 0.560 − 2.836 = −2.276 V. Since the intercept equals −φ/e, the work function of the sodium sample is:

φ = 2.276 × e = 2.276 eV

which is very close to the widely quoted textbook value of 2.28 eV for sodium (Kittel, 2005). The threshold frequency follows from f₀ = φ/h_exp = 2.276/(4.131×10⁻¹⁵) = 5.510 × 10¹⁴ Hz, corresponding to a threshold wavelength of λ₀ = c/f₀ = 544 nm, in the yellow-green part of the visible spectrum, consistent with sodium’s well-known sensitivity to blue and violet light but not to red light. Figure 1 plots the five data points together with the best-fit line and marks the threshold frequency at which the line crosses V_s = 0.

Figure 1. Stopping voltage against frequency, with best-fit line and threshold frequency f₀ marked.

As an independent check on the least-squares result, the gradient can also be estimated from the two most widely separated readings alone, using the 365 nm and 522 nm data points: gradient_2pt = (1.12−0.10)/(8.214−5.743)×10⁻¹⁴ = 4.128×10⁻¹⁵ V·s, giving h_2pt = 4.128×10⁻¹⁵×1.602×10⁻¹⁹ = 6.613×10⁻³⁴ J·s, within 0.1% of the five-point least-squares value. This agreement gives some confidence that the fitted line is not being distorted by any single anomalous reading, although a two-point calculation of this kind discards the information carried by the three intermediate measurements and would in general carry a larger formal uncertainty than the full least-squares fit.

Evaluation of Results

The agreement between the experimental and accepted values is excellent for both quantities: the 0.12% discrepancy in h and the near-exact match in φ are well within what would normally be expected from a five-point stopping-voltage experiment using a simple photocell and a manually adjusted retarding voltage supply, and are consistent with results reported by Millikan (1916) in the original precision measurement of this kind. Nonetheless, several sources of uncertainty should be considered when interpreting the fit. First, the stopping voltage is conventionally defined as the point at which the photocurrent falls to zero, but in practice the current approaches zero asymptotically because of the spread of initial electron kinetic energies caused by electrons being emitted from below the metal surface, so the operational definition used to read V_s from a current–voltage curve (for example, the voltage at which the current falls to some small fraction of its zero-voltage value) can introduce a small, systematic bias into every measurement.

Second, contact potential differences between the cathode and anode materials add a small, wavelength-independent offset to the measured stopping voltage; because this offset is constant across all five measurements, it shifts the intercept of the fitted line and therefore the derived work function, but it does not affect the gradient and so leaves the derived value of h essentially unaffected, which is one reason the gradient method is preferred over reading φ from a single measurement. Third, the interference filters used to select each wavelength are not perfectly monochromatic, and a finite spectral bandwidth smears the true threshold slightly and broadens the current–voltage knee used to identify V_s, an effect that would be expected to introduce more scatter about the fitted line than is seen here, again suggesting a well-controlled apparatus and a consistent operational definition of the stopping point across all five readings. Fourth, the photocathode surface itself is a further source of systematic error: surface contamination, oxidation or non-uniform coating thickness on a sodium photocathode can lower the effective work function in patches, which would tend to broaden the current–voltage knee still further and could shift the apparent stopping voltage downward at every wavelength, again biasing the intercept rather than the gradient.

Finally, with only five data points and no repeat measurements at each wavelength, the uncertainty on the fitted gradient and intercept cannot be estimated rigorously from the data alone; a stronger version of this exercise would repeat each stopping-voltage measurement several times at each wavelength, report the standard error of the least-squares gradient explicitly, and extend the wavelength range further into the ultraviolet to widen the spread of frequencies used in the fit, which would reduce the sensitivity of the fitted gradient and intercept to random error in any single measurement (Taylor, 1997). Extending the dataset to include a wavelength close to but above the threshold value of 544 nm derived here would also provide a useful independent check on the extrapolated threshold frequency, since the current experiment infers f₀ entirely by extrapolation rather than by direct observation of the current disappearing at that wavelength.

A further, subtler source of uncertainty concerns the assumption that every photoelectron considered originates from a single, uniform work function; in a real sodium sample the work function is better described as a narrow distribution, reflecting minor variations in local surface condition, and the measured stopping voltage corresponds to the highest-energy electrons within that distribution rather than to one sharply defined threshold. Such averaging effects are typically small for a well-prepared alkali-metal surface but would be expected to grow with surface roughness or oxidation.

Conclusion

Analysis of the sodium photocathode stopping-voltage data by linear regression gives an experimental Planck’s constant of h = 6.618 × 10⁻³⁴ J·s, within 0.12% of the accepted value, and a work function of φ = 2.276 eV with a corresponding threshold wavelength of 544 nm, both closely matching published values for sodium. The strong linearity of the V_s–f relationship confirms the central prediction of Einstein’s photon model, that photoelectron energy depends on light frequency rather than intensity, and the gradient-based method used here is shown to be robust to the constant systematic offsets, such as contact potential, that would otherwise bias a single-point estimate of the work function. More broadly, the close agreement obtained here between a simple undergraduate apparatus and the CODATA reference value illustrates why the stopping-voltage method remains a standard teaching experiment for recovering a fundamental constant from simple voltage and wavelength readings.

References

• Einstein, A. (1905) ‘Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt’, Annalen der Physik, 17(6), pp. 132–148.
• Millikan, R.A. (1916) ‘A Direct Photoelectric Determination of Planck’s h’, Physical Review, 7(3), pp. 355–388.
• Eisberg, R. and Resnick, R. (1985) Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd edn. New York: Wiley.
• Krane, K.S. (1996) Modern Physics. 2nd edn. New York: Wiley.
• Kittel, C. (2005) Introduction to Solid State Physics. 8th edn. Hoboken: Wiley.
• Tipler, P.A. and Llewellyn, R.A. (2012) Modern Physics. 6th edn. New York: W.H. Freeman.
• Serway, R.A., Moses, C.J. and Moyer, C.A. (2005) Modern Physics. 3rd edn. Belmont: Thomson Brooks/Cole.
• Taylor, J.R. (1997) An Introduction to Error Analysis. 2nd edn. Sausalito: University Science Books.
• National Institute of Standards and Technology (2019) CODATA Recommended Values of the Fundamental Physical Constants. Gaithersburg: NIST.