Threshold Frequency From Work Function

The threshold frequency ${\nu }_0$ is the smallest frequency whose photon energy just matches the work function, so that $h{\nu }_0={\varphi }_0$ and any lower frequency cannot eject electrons. Rearranging, ${\nu }_0=\frac{{\varphi }_0}{h}$. First convert the work function to joules: ${\varphi }_0=3.3\times 1.6\times 10^{-19}=5.28\times 10^{-19}\text{J}$. Then ${\nu }_0=\frac{5.28\times 10^{-19}}{6.6\times 10^{-34}}=8.0\times 10^{14}\text{Hz}$. The option $4.0\times 10^{14}\text{Hz}$ corresponds to half the work function. The option $1.6\times 10^{15}\text{Hz}$ doubles it. The option $2.0\times 10^{14}\text{Hz}$ underestimates the work function fourfold. Radiation of frequency below ${\nu }_0$, however intense, produces no photoelectrons, a frequency threshold that classical physics could not justify and which Einstein's photon model resolved, as covered in the NCERT dual-nature chapter. A check shows $8.0\times 10^{14}\text{Hz}$ lies just beyond the visible into the near-ultraviolet, consistent with a moderately high work function of $3.3\text{eV}$. A key conceptual point is that the existence of a sharp frequency threshold, completely independent of how bright the source is, was utterly inexplicable on the classical wave theory, which predicted that any frequency should eventually eject electrons if the light were intense enough. The experimental fact that dim blue light works while brilliant red light fails was decisive evidence for the photon picture, and the threshold frequency is simply the work function rendered in frequency units.