Modern Physics: The Photoelectric Effect and Wave-Particle Duality Study Guide

Definition: The photoelectric effect is the phenomenon of the emission of electrons from a metal surface after absorbing photonic energy (light energy).
Photoelectrons: The electrons that are emitted from the metal surface during this process are specifically referred to as photoelectrons.
Photoelectric Current: The current that flows in a closed circuit as a result of these emitted photoelectrons is known as the photoelectric current.

Theoretical Basis: According to this theory, light energy is not continuous.
Quanta or Photons: Light energy is always available in integral multiples of small elementary energy packets called quanta or photons.
Energy of a Photon: Each photon carries a specific amount of energy defined by the formula: $E = h \times \nu$ - Where $h$ is Planck's constant ( $6.6 \times 10^{-34}\,J \cdot s$ ). - Where $\nu$ is the frequency of the light.

Work Function ( $W$ ): This is the minimum energy of incident light required to cause photoelectric emission from a surface. - Example: The work function of Tungsten can be increased by coating it with metal oxide.
Threshold Frequency ( $\nu_0$ ): This is the minimum frequency of incident light that can cause photoelectric emission.
Threshold Wavelength ( $\lambda_0$ ): This is the maximum wavelength of incident light that can cause photoelectric emission.
Material Dependence: All three of these physical quantities ( $W$ , $\nu_0$ , and $\lambda_0$ ) depend strictly on the material properties of the metal and will differ for different metals.

General Equation: The energy of an incident photon is split between the work function and the maximum kinetic energy of the emitted electron: $E = W + KE_{max}$
Frequency-based Equation: $h \times \nu = h \times \nu_0 + KE_{max}$
Wavelength-based Equation: $\frac{h \times c}{\lambda} = \frac{h \times c}{\lambda_0} + KE_{max}$
Conceptual Scenarios (Example with $W = 10\,eV$ ): - If incident energy $E = 15\,eV$ : Results in the "most disturbance" (significant emission). - If incident energy $E = 10\,eV$ : The electron will "just pop up" (threshold reached). - If incident energy $E = 7\,eV$ : Results in the "least disturbance" (no emission).

Definition: Stopping potential is the minimum reverse potential applied at the anode to seize or stop the photoelectric current in the circuit.
Mechanism: It is the point where the electron stops as it travels by depleting its kinetic energy ( $KE = 0$ ).
Mathematical Relation: $KE_{max} = h \times \nu - W$ $e \times V_{st} = h \times \nu - W$ $V_{st} = \frac{h \times \nu}{e} - \frac{W}{e}$
Dependencies: - Stopping potential depends on the frequency of the incident light. - Stopping potential is independent of the intensity of the incident light.

Methodology: Different frequencies of incident light are used to obtain different values of stopping potential. These observations are recorded in a table ( $\nu$ vs $V_{st}$ ).
Graphical Analysis: A graph is plotted with frequency on the x-axis and stopping potential on the y-axis, resulting in a straight line.
Linear Equation: Using $y = m \times x + c$ : $V_{st} = \left( \frac{h}{e} \right) \times \nu - \frac{W}{e}$
Slope ( $m$ ): The slope of the line is defined as: $m = \tan(\theta) = \frac{h}{e}$
Calculating $h$ : By multiplying the slope ( $m$ ) by the charge of an electron ( $e$ ), one obtains Planck's constant: $h = m \times e$ $h = m \times 1.6 \times 10^{-19} = 6.6 \times 10^{-34}\,J \cdot s$

Photoelectric Current Relationship: Photoelectric current is directly proportional to the intensity of incident light and the number of photoelectrons (or number of photons). $\text{Photoelectric Current} \propto \text{Intensity} \propto \text{Number of Photons}$
Independence: Both Stopping Potential ( $V_{st}$ ) and Maximum Kinetic Energy ( $KE_{max}$ ) are independent of the intensity of the incident light.

Classical Energy: According to classical mechanics, the energy of a particle of mass $m$ is $E = m \times c^2$ .
Quantum Energy: Referencing Planck's theory, the energy of a wave is $E = h \times \nu$ .
Equating Energy: $m \times c^2 = \frac{h \times c}{\lambda}$ , which leads to $m \times c = \frac{h}{\lambda}$ .
Momentum ( $p$ ): Since $p = m \times c$ , then $p = \frac{h}{\lambda}$ .
De Broglie Wavelength: $\lambda = \frac{h}{p}$ .
Relation to Kinetic Energy ( $K$ ): - Since $K = \frac{1}{2} \times m \times v^2$ and $p = m \times v$ , then $p = \sqrt{2 \times K \times m}$ . - $\lambda = \frac{h}{\sqrt{2 \times K \times m}}$ .
Relation to Potential Difference ( $V_m$ ): - For a charge $q$ accelerated through a potential $V_m$ , $W = q \times \Delta V = K$ . - $\lambda = \frac{h}{\sqrt{2 \times q \times V_m \times m}}$ .
Photon Mass Properties: - Mass of a photon: $m = \frac{h \times \nu}{c^2}$ . - The rest mass of a photon is $0$ .

Purpose: The Davisson-Germer experiment confirms the wave nature of electrons using the phenomenon of diffraction.
Crystallography: For the study of crystal structures (crystallography), X-rays are typically utilized.

Problem 1: Calculate the De Broglie wavelength of an electron with an energy of $100\,eV$ . - Formula: $\lambda = \frac{h}{\sqrt{2 \times K \times m}}$ - Parameters: $K = 100 \times 1.6 \times 10^{-19}\,J$ , $m = 9.1 \times 10^{-31}\,kg$ .
Problem 2: Light of wavelength $198\,nm$ is incident on a metallic cathode with a work function of $2.5\,eV$ . How much potential difference must be applied between the cathode and anode of a photocell to stop the photocurrent? Also, find the threshold wavelength for the metallic cathode. - Parameters: $\lambda = 198 \times 10^{-9}\,m$ , $W = 2.5\,eV$ .