Determining Planck's Constant From Photoelectric Data

Since $V_0=\frac{h}{e}\nu -\frac{{\varphi }_0}{e}$ is linear in frequency, the slope of $V_0$ versus $\nu$ equals $\frac{h}{e}$, and the unknown work function cancels when we take differences between two data points. The change in stopping potential is $\Delta V_0=2.15-0.50=1.65\text{V}$, while the change in frequency is $\Delta \nu =(1.0\times 10^{15})-(6.0\times 10^{14})=4.0\times 10^{14}\text{Hz}$. The slope is therefore $\frac{\Delta V_0}{\Delta \nu }=\frac{1.65}{4.0\times 10^{14}}=4.125\times 10^{-15}\text{V s}$. Multiplying by the electronic charge gives $h=\frac{\Delta V_0}{\Delta \nu }\times e=4.125\times 10^{-15}\times 1.6\times 10^{-19}=6.6\times 10^{-34}\text{J s}$. The value $3.3\times 10^{-34}$ would result from halving the slope. The value $1.6\times 10^{-19}$ is the electronic charge, not Planck's constant. The value $9.1\times 10^{-31}$ is the electron mass, an unrelated constant. This difference method, which eliminates the work function entirely, is exactly how Millikan extracted $h$ from photoelectric measurements, as recounted in NCERT. The recovered value matches the accepted Planck constant, confirming the data and the linear photoelectric model are mutually consistent. A valuable feature of this difference method is that it is completely insensitive to the particular metal used, because the work function affects only the intercept and is differenced away. This robustness is why the photoelectric determination of Planck's constant is considered one of the cleanest in physics, requiring only accurate stopping-potential and frequency measurements rather than any knowledge of the cathode material.