Question

Write Einstein’s photoelectric equation and point out any two characteristic properties of photons on which this equation is based. Briefly explain the three observed features which can be explained by this caution.

Answer 1

Hint: Einstein explained the photoelectric effect as the emission of electrons by a metal surface when light of a minimum frequency irradiated it. He devised an equation to calculate the kinetic energy of the fastest of these emitted electrons. Einstein’s photoelectric equation is an extension of Planck’s quantum theory.

Formula Used:
Einstein planck relation $E = hv$
Where,
$h$ is the planck's constant
$v$ is the frequency of radiation emitted.
Kinetic energy ${E_k} = \dfrac{1}{2}m{\upsilon ^2}$
Where, $m$is the mass of an object
$\upsilon $ is the velocity of an object

Complete step by step answer:
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 in overcoming the potential barrier of 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,
$\dfrac{1}{2}m{\upsilon ^2} = hv - {\phi _0}$
$h\nu = {\phi _0} + \dfrac{1}{2}m{\upsilon ^2}$
Where $m$ is the mass of the electron and $\upsilon $ is velocity.
If we reduce the frequency of incident light, the speed or kinetic energy of photoelectrons is also reduced. At some frequency ${v_0}$ of the incident radiation, the photo electrons are just ejected with almost zero kinetic energy. Then the equation becomes,
$h{v_0} = {\phi _0}$
Where ${v_0}$ is the threshold frequency
By rewriting this we get,
$hv = h{v_0} + \dfrac{1}{2}m{\upsilon ^2}$, This equation is known as Einstein’s photoelectric equation.
If the electron does not lose energy by internal collision, then it is emitted with maximum kinetic energy ${K_{\max }}$. Then,
${K_{\max }} = \dfrac{1}{2}m{\upsilon ^2}_{\max }$
Where ${\upsilon _{\max }}$ is the maximum velocity of the electron ejected, the equation is rearranged as,${K_{\max }} = hv - {\phi _0}$
The two characteristic properties of photons: Photons have particle and wave characteristics. In wave characteristics, it travels in space of particular frequency. In particle characteristics, it will be emitted or absorbed in units called quanta of lights.
The three observed features: 1) Burglar alarm – In this there is ultraviolet radiation due to this it will stop the supply of the photons and work as off mode and so the bell will ring automatically. 2) Solar cells – It will convert solar radiation to electrical emf. It is also called photo-volatic cells. 3) Television Telecast – According to photoelectric emission principle, the dark and the light part of an image is interpreted in high and low electrical charges.

Note: Einstein's equation was experimentally verified by R.A.Millikan. He drew ${K_{\max }}$ versus $v$ graph for many metals ( cesium, potassium, sodium, and lithium) and found the slope is independent of the metals.