Question

What is the photo-electric effect ? Write its laws. Establish the Einstein's equation of photoelectric effect \(\frac{1}{2} mv^2 = h(\nu - \nu_0)\).

Hint:

For the photoelectric effect, remember the energy balance: \(E_{photon} = W_{eject} + KE_{electron}\).

Answer 1

What is the Photoelectric Effect?
The photoelectric effect is the phenomenon of emission of electrons from a substance (usually a metal) when electromagnetic radiation, such as light, of a sufficiently high frequency is incident upon it. The emitted electrons are called photoelectrons.
Laws of Photoelectric Emission: Based on experimental observations, the laws are: \begin{enumerate} \item For a given material, there exists a certain minimum frequency of incident radiation, called the threshold frequency (\(\nu_0\)), below which no photoelectric emission occurs, no matter how intense the light is. \item The photoelectric emission is an instantaneous process. The time lag between the incidence of radiation and the emission of photoelectrons is very small (less than \(10^{-9}\) s). \item The number of photoelectrons emitted per second (which determines the photoelectric current) is directly proportional to the intensity of the incident radiation, provided the frequency is above the threshold frequency. \item The maximum kinetic energy of the emitted photoelectrons is independent of the intensity of the incident light but depends linearly on its frequency. \end{enumerate} Einstein's Photoelectric Equation:

Step 1: Understanding the Concept (Einstein's Postulate):
To explain the photoelectric effect, Einstein proposed that light is quantized and consists of discrete packets of energy called photons. The energy of each photon is \(E = h\nu\), where \(h\) is Planck's constant and \(\nu\) is the frequency of light.

Step 2: Derivation:
When a photon of energy \(h\nu\) strikes the metal surface, its energy is used in two parts: \begin{itemize} \item A portion of the energy is used to overcome the surface barrier and free the electron from the metal. This minimum energy required is called the work function (\(\Phi_0\)) of the metal. \item The rest of the photon's energy is imparted to the emitted electron as its maximum kinetic energy (\(K_{max}\)). \end{itemize} By the law of conservation of energy: \[ \text{Energy of Photon} = \text{Work Function} + \text{Max. Kinetic Energy of Electron} \] \[ h\nu = \Phi_0 + K_{max} \] The work function can be expressed in terms of the threshold frequency (\(\nu_0\)) as \(\Phi_0 = h\nu_0\). Substituting this: \[ h\nu = h\nu_0 + K_{max} \] Rearranging the equation to solve for the kinetic energy: \[ K_{max} = h\nu - h\nu_0 = h(\nu - \nu_0) \] Since \(K_{max} = \frac{1}{2}m v_{max}^2\), where \(m\) is the mass of the electron and \(v_{max}\) is its maximum velocity, the equation becomes: \[ \frac{1}{2}mv_{max}^2 = h(\nu - \nu_0) \] This is Einstein's photoelectric equation.