Q.When a light of frequency $9 \times 10^{14}$ Hz is incident on a metal surface, photoelectrons are emitted with a maximum speed of $8.14 \times 10^{5}\ \text{ms}^{-1}$. Determine the threshold frequency of the surface.

Building directly on Planck's quantization idea, Einstein in 1905 proposed that light of frequency $\nu$ should itself be thought of as a stream of discrete energy quanta (photons), each carrying energy $h\nu$. Applying this to the photoelectric effect: when a single photon of energy $h\nu$ strikes a metal surface, it is completely absorbed by one single electron; part of this energy, equal to the metal's work function $\phi_0$, is spent freeing the electron from the surface, and the remainder becomes the ejected electron's kinetic energy. Conservation of energy for this one-photon-one-electron process gives Einstein's photoelectric equation, $h\nu=\phi_0+\tfrac12mv^2$; if the electron loses no energy internally on its way out, it emerges with the maximum possible kinetic energy, giving the most commonly used form $K_{max}=h\nu-\phi_0$. At the threshold frequency $\nu_0$, the photoelectron is emitted with essentially zero kinetic energy, so $h\nu_0=\phi_0$, meaning the threshold frequency is simply the photon frequency whose energy exactly equals the work function. …