Obtain Einstein's photoelectric equation with necessary explanation.

Explanation

When a photon of energy hv is incident on a metal surface, it is completely absorbed by a single electron and the electron is ejected.
In this process, a part of the photon energy is used for the ejection of the electrons from the metal surface (photoelectric work function `phi_(0)`) and the remaining energy as the kinetic energy of the ejected electron. From the law of conservation of energy.
`h upsilon = phi_(0) + (1)/(2) mv^(2)" "...(1)`
where m is the mass of the electron and `upsilon` its velocity

If we reduce the frequency of the incident light, the speed or kinetic energy of photo electrons is also reduced. At some frequency `v_(0)` of incident radiation, the photo electrons are ejected with almost zero kinetic energy. Then the equation (1) becomes
`h upsilon_(0) = phi_(0)`
where `v_(0)` is the threshold frequency. By rewriting the equation (1), we get
`h upsilon = h upsilon_(0) + (1)/(2) mv^(2)" "...(2)`
The equation (2) is known as Einstein.s Photoelectric equation.
If the electron does not lose energy by internal collisions, then it is emitted with maximum kinetic energy `K_(max)`. Then
`K_(max) = (1)/(2) mv_(max)^(2)`
where `upsilon_(max)` is the maximum velocity of the electron ejected. The equation (1) is rearranged as follows :
`K_(max) = h upsilon - phi_(0)`