Question

Derive Einstein's photoelectric equation.

Answer 1

Einstein's photoelectric equation.

According to Einstein, the emission of photoelectron is the result of the interaction between a single photon of the incident radiation and an electron in the metal.

When a photon of energy $hv$ is incident on a metal surface, its energy is used up in two ways.

(i) A part of the energy of the photon is used in extracting the electron from the surface of metal, since the electrons in the metal are bound to the nucleus. This energy $W$ spent in releasing the photoelectron is known as photoelectric work function of the metal. The work function of a photo metal is defined as the minimum amount of energy required to liberate an electron from the metal surface.

(ii) The remaining energy of the photon is used to impart kinetic energy to the liberated electron. If $m$ is the mass of an electron and $v$ its velocity, then

Energy of the incident photon $=$ Work function $+$ Kinetic energy of the electron

$h\gamma =W+\frac{1}{2}m{v}^2....\left(1\right)$

If the electron does not lose energy by internal collisions, as it escapes from the metal, the entire energy $(h-W)$ will be exhibited as the kinetic energy of the electron Thus, $(h-W)$ represents the maximum kinetic energy of the ejected photoelectron. If Vman is the maximum velocity with which the photoelectron can be ejected, then

$hv=W+\frac{1}{2}m{v}_{max}^2....\left(2\right)$

This equation is known as Einstein's photoelectric equation.

When the frequency $\left(\gamma \right)$ of the incident radiation is equal to the threshold frequency $\left({\gamma }_0\right)$ of the metal surface, kinetic energy of the electron is zero. Then equation $\left(2\right)$ becomes,

$h{\gamma }_0=W....\left(3\right)$

Substituting the value of $W$ in equation $\left(2\right)$ we get,

$h\gamma -h{\gamma }_0=m{v}_{max}^2$ (or) $h(\gamma -{\gamma }_0)=m{v}_{max}^2$

This is another form of Einstein's photoelectric equation.