The photoelectric effect.

1. Title and aim

Determination of Planck’s constant using the photoelectric effect. Measure stopping potential at several known light frequencies for one emitting material, fit a straight line, and estimate Planck’s constant. Estimate the effective work function and threshold frequency from the intercept.

2. Abstract

Five simulated stopping potentials were obtained for wavelengths from 500 to 300 nm at fixed target material and 50% intensity. Linear regression gave a gradient of 4.15224 × 10⁻¹⁵ V s and intercept −2.28878 V, yielding h = 6.65263 × 10⁻³⁴ J s. The estimate differs by 0.401% from the reference value. These values demonstrate the analysis procedure; the simulator itself uses the reference h and is not an independent measurement.

3. Apparatus and settings

• OpenLab ideal photoelectric simulator; sodium target with representative work function φ = 2.28 eV.
• Monochromatic source: selected wavelengths 500, 450, 400, 350 and 300 nm; intensity fixed at 50%.
• Evacuated photocell, collector-bias control and sensitive current meter represented schematically.
• Collector voltage range: −5 to +2 V; control spacing: 0.01 V. There is no simulated instrument noise.
• A real laboratory additionally needs the actual photocell, specified light source / filters, variable supply, voltmeter, current meter, connecting leads and the instrument manual.

4. Theory and working equations

A photon carries energy hf. Emission occurs when this energy can overcome the surface work function. The most energetic emitted electrons satisfy Einstein’s energy balance. A retarding voltage stops collection when its electric potential-energy barrier equals that maximum kinetic energy.

V_s is a positive magnitude. The collector control is negative at the stopping point. For a fit V_s = mf + b with f in Hz, h = em and φ = −eb in joules. Numerically, φ in eV equals −b in volts. Threshold frequency f₀ = −b/m.

Constants: c = 299792458 m s⁻¹; e = 1.602176634 × 10⁻¹⁹ C; reference h = 6.62607015 × 10⁻³⁴ J s. The SI values of these constants are exact; the fitted estimate is limited by the observations.

5. Procedure

1. Select sodium, set intensity to 50%, and set collector voltage to 0 V.
2. Select 500 nm and check that emission produces a nonzero current.
3. Make the collector voltage increasingly negative until the collected current first becomes zero.
4. Increase voltage one 0.01 V step toward zero to verify that current returns. Return to the first zero-current step.
5. Record the wavelength, frequency, last nonzero voltage and current, first zero voltage, and V_s.
6. Repeat at 450, 400, 350 and 300 nm while keeping material and intensity fixed.
7. Plot V_s against f, perform a least-squares fit, calculate h, and inspect residuals and measurement limitations.

6. Observations: raw and derived readings

The current values below are calculated from the simulator’s illustrative current curve. Tiny nonzero currents appear as “<0.01 nA” on the workbench; extra digits here explain the bracket and do not represent real meter precision. One bracket per wavelength is used; no repeated physical trials are claimed.

Download these sample observations as CSV ↓

7. Worked calculation for one reading

For λ = 400 nm = 400 × 10⁻⁹ m:

The boundary is bracketed by −0.81 V with a small nonzero current and −0.82 V with zero current. Thus 0.81 V < V_s,true ≤ 0.82 V for this ideal monotonic model. The notebook records the upper endpoint, 0.82 V.

8. Graph and least-squares calculation

Let x = f / (10¹⁴ Hz) and y = V_s. Fit y = ax + b using all five readings, rather than only two selected points.

Orange points: simulated observations. Green line: equal-weight least-squares fit.

The displayed frequency axis is scaled. Therefore m = a / 10¹⁴ = 4.152244415 × 10⁻¹⁵ V s.

9. Uncertainty and percentage difference

Percentage difference = |h_fit − h_reference| / h_reference × 100 = 0.4008%. This is a comparison with a reference, not an uncertainty estimate.

Fit scatter only: for n = 5, Σr² = 2.27609 × 10⁻⁵ V². The residual standard deviation is s = √[Σr²/(n − 2)] = 0.00275444 V. The slope standard error on the scaled axis is s_a = s/√Σ(x − x̄)² = 0.00087008, yielding s_h = es_a/10¹⁴ = 1.39402 × 10⁻³⁶ J s.

The regression-only result can be written h = (6.653 ± 0.014) × 10⁻³⁴ J s, where ± denotes one slope standard error. It assumes independent, equal-variance voltage errors and exact frequencies. These assumptions do not describe a complete uncertainty budget for these deterministic rounded data.

Resolution and endpoint bias: the true boundary lies in a 0.01 V interval. Recording its first zero-current endpoint biases each observation upward by between 0 and 0.01 V. Under a uniform-within-bracket approximation, a midpoint estimate would subtract 0.005 V and have standard uncertainty 0.01/√12 ≈ 0.00289 V. Subtracting the same 0.005 V from every reading changes the intercept, but not the fitted slope. This report retains the notebook’s endpoint convention.

In a physical lab, include wavelength uncertainty and bandwidth, voltage calibration, current-meter zero, dark current, contact potentials, surface condition and repeatability. They are omitted from this demonstration; the quoted fit error must not be reported as the full physical uncertainty.

10. Results and discussion

• Estimated Planck’s constant: 6.653 × 10⁻³⁴ J s; regression-only standard error: 0.014 × 10⁻³⁴ J s.
• Fitted effective work function: 2.289 eV; threshold frequency: 5.512 × 10¹⁴ Hz.
• Stopping potential increases approximately linearly with frequency, consistent with the model’s energy balance.
• The small residuals arise from the voltage step and first-zero convention. They do not demonstrate realistic instrument performance.
• The fitted work function differs from the model input 2.28 eV because of rounding and the endpoint convention.

11. Precautions and sources of error

Keep the same emitting surface for all readings. Use one known wavelength at a time, distinguish a tiny current from exactly zero in this simulation, bracket the stopping boundary, and maintain a consistent sign convention. In a real laboratory, zero and calibrate instruments, account for background current, follow the equipment manual, and avoid direct exposure to UV sources or laser beams.

12. Conclusion

This worked example shows how stopping-potential observations lead to a graph-based estimate of Planck’s constant. The five simulated readings yield 6.653 × 10⁻³⁴ J s, differing by 0.401% from the reference. The exercise demonstrates the method and its analysis; because the simulator contains the reference constant, it does not independently verify its value.

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