Work Function And Threshold

The threshold wavelength marks the longest wavelength that can still just eject an electron, and NCERT defines it through $\varphi =\frac{hc}{{\lambda }_0}$, where $\varphi$ is the work function. Rearranging gives ${\lambda }_0=\frac{hc}{\varphi }$. Using the convenient constant $hc=1242$ eV·nm and $\varphi =2.0$ eV, we obtain ${\lambda }_0=\frac{1242}{2.0}=621$ nm. Any wavelength longer than this carries too little energy per photon to overcome the work function, so emission ceases. The value 310 nm is wrong because it doubles the work function or halves the wavelength incorrectly, giving too short a threshold. The value 414 nm is wrong because it uses a work function of 3.0 eV rather than the stated 2.0 eV. The value 1242 nm is wrong because it forgets to divide by the 2.0 eV work function, leaving the bare constant. As stated in NCERT Class 12, Chapter 11, a smaller work function corresponds to a longer threshold wavelength, since less energy is needed to free an electron. Physically, the threshold wavelength is simply the wavelength at which each photon carries exactly the escape energy, leaving the electron with zero kinetic energy. Alkali metals such as caesium and sodium have low work functions and therefore long threshold wavelengths reaching into the visible, whereas metals like platinum need ultraviolet light to emit at all. A magnitude check confirms that a 2 eV barrier corresponds to red-orange light near 621 nm, a physically reasonable threshold for many metals.