Photoelectric effect: trusting the gradient but not the intercept
A student uses a vacuum photocell with a sodium cathode and a set of interference filters, measuring the stopping potential by increasing the reverse voltage until the photocurrent falls to zero: 365 nm gives 1.22 V, 405 nm gives 0.89 V, 436 nm gives 0.67 V, 480 nm gives 0.41 V, and at 546 nm no current is detected at any lamp intensity. A least-squares fit of stopping potential against frequency gives a gradient of 4.10 × 10⁻¹⁵ V s and an intercept of −2.15 V. The accepted work function of sodium is 2.28 eV and the accepted value of h/e is 4.136 × 10⁻¹⁵ V s. The anode is made of a different metal, and the photocurrent never quite reaches zero but flattens at about 2 % of its saturation value. The student claims to have measured Planck's constant and to have confirmed the work function to within 6 %.
1. Obtain h from the gradient and assess its agreement with the accepted value. Explain why the gradient is the reliable quantity here while the intercept is not.
Read the worked answer
Gradient = h/e = 4.10 × 10⁻¹⁵ V s, so h = 4.10 × 10⁻¹⁵ × 1.602 × 10⁻¹⁹ = 6.57 × 10⁻³⁴ J s, against 6.626 × 10⁻³⁴ J s - 0.9 % low.
Is that within uncertainty? The stopping potentials are read to about ±0.02 V and the frequency span is (8.21 − 6.25) × 10¹⁴ = 1.97 × 10¹⁴ Hz, so δ(h/e) ≈ 0.02/1.97 × 10¹⁴ ≈ 1.0 × 10⁻¹⁶ V s, about 2.5 % of the gradient. The 0.9 % deviation is therefore not significant; with five points the fit uncertainty is smaller, but the point stands that the data support the accepted h.
Why the gradient and not the intercept: any constant voltage offset - the contact potential between the sodium cathode and the dissimilar anode, amplifier zero error, the finite current threshold used to define 'zero' - adds the same amount to every stopping potential. A constant added to all V_s is absorbed by the intercept and cancels exactly in the gradient. The same offset shifts the deduced work function, which is why 2.15 eV must not be read as sodium's work function.
2. Explain why no current flows at 546 nm even at maximum intensity, and show how this observation diagnoses the intercept.
Read the worked answer
A single photon is absorbed by a single electron, so the electron can escape only if hf ≥ φ. At 546 nm, hc/λ = 1240/546 = 2.27 eV, which is just below sodium's 2.28 eV. No individual photon carries enough energy, and turning up the lamp only delivers more of the same photons, so brighter light gives still zero current. The 480 nm point (2.58 eV) is above the threshold and does give current, so the cutoff wavelength for sodium sits between 480 and 546 nm, i.e. φ is between 2.27 and 2.58 eV - consistent with 2.28 eV and inconsistent with the fitted 2.15 eV.
The fitted intercept, taken at face value, would put the threshold at 1240/2.15 = 577 nm and would predict a small current at 546 nm; none is seen. The observed cutoff is a direct, offset-free measurement of the cathode work function, while the intercept carries the contact-potential offset of about 0.13 eV. The student has measured h/e well and mis-assigned the intercept.
3. The current flattens at about 2 % of saturation rather than reaching zero. Explain its effect on the measured stopping potentials and on the value of h obtained, and suggest a better procedure.
Read the worked answer
Defining the stopping potential as the voltage where the current reaches 2 % of saturation means the true zero is at a slightly more negative voltage, so every V_s is biased in the same direction by roughly the same amount. The bias may also come from photoelectrons released from the anode by stray light: those give a reverse current that never vanishes, so the measured null is where the net current is zero, again a constant offset.
Because the bias is constant across wavelengths, it cancels in the gradient, so h/e is essentially unaffected - the 0.9 % agreement confirms this. It does not cancel at a single wavelength, so any work function taken from the intercept or from one filter is unreliable.
Better procedure: use the same current criterion at every wavelength (already done, which is why the gradient is good), extrapolate the linear falling part of each I–V curve to zero current rather than eyeballing a threshold, shield the anode from stray light, and check that the fitted intercept changes systematically when a small fixed offset is added - a healthy data set shows the gradient unchanged while the intercept moves.