Threshold Wavelength And Work Function

Photoelectric emission occurs only when each photon carries at least the work function ${\varphi }_0$, the minimum energy needed to liberate an electron from the metal surface. The threshold (longest) wavelength ${\lambda }_0$ corresponds to a photon energy exactly equal to ${\varphi }_0$, so ${\varphi }_0=\frac{hc}{{\lambda }_0}$. Using the convenient product $hc=1240\text{eV nm}$, we get ${\lambda }_0=\frac{1240}{2.0}=620\text{nm}$. Light of longer wavelength has lower photon energy and cannot eject any electron, no matter how intense it is. The option $310\text{nm}$ wrongly doubles the work function. The option $1240\text{nm}$ takes the work function as $1\text{eV}$. The option $413\text{nm}$ corresponds to $3\text{eV}$. This threshold concept is central to the NCERT treatment of Hertz's and Lenard's observations and to Einstein's photoelectric equation. A sanity check: $620\text{nm}$ lies in the orange-red region, so only photons bluer than this can release electrons, consistent with a modest $2.0\text{eV}$ barrier.