Description of the Light:
According to the Albert Einstein
The light is consist of small packets or bundles of energy. These packets are called photon. The energy of each photon is $h \nu$.
Theory of Photoelectric Effect:
When the photon of ultraviolet ray is incident on the metal then this energy is used in two parts:
1.) To emit the electron from a metal surface i.e. Work Function. The electron emitted from a metal surface is called the photoelectron.
2.) Remaining energy increases the kinetic energy of the emitted photoelectron. This is the maximum kinetic energy of the photoelectron.
Let us consider
The energy of incident Ultraviolet Photon $E=h\nu$
The work function $W= h \nu_{\circ}$
The maximum kinetic energy of photo electron $K_{max}= \frac{1}{2}mv^{2}_{max}$
According to Einstein's photoelectric effect theory
$E=W+K_{max}$
$h \nu = h \nu_{\circ} + \frac{1}{2}m v^{2}_{max}$
$h \nu - h \nu_{\circ} = \frac{1}{2}m v^{2}_{max}$
$h \left( \nu - \nu_{\circ} \right) = \frac{1}{2}m v^{2}_{max}$
Where
$\nu \rightarrow$ Frequency of incident Ultraviolet ray photon
$\nu_{\circ} \rightarrow$ Threshold frequency to emit the photoelectron from the metal surface
$ v_{max} \rightarrow$ Maximum velocity of emitted photoelectron from the metal surface
