Physics B.Tech I - Study Notes

Course Information

• Course Code: PH24101

• Course Name: Physics (B.Tech. I)

• Instructors: Dr. Rajyavardhan Ray and Dr. Anupam Roy

• Department: Dept of Physics, BIT Mesra

• Contact: Email at royanupam [AT] bitmesra [DOT] ac [DOT] in

• Sections: J & K

Syllabus Overview

• Module-1: Physical Optics

  • Polarization

  • Malus’ Law

  • Brewster’s Law

  • Double Refraction

  • Interference in Thin Films (Parallel Films)

  • Interference in Wedge-Shaped Layers

  • Newton’s Rings

  • Fraunhofer Diffraction by Single Slit & Double Slit

  • Elementary Ideas of Fibre Optics and Application of Fibre Optic Cables

• Module-2: Electromagnetic Theory

  • Gradient, Divergence, and Curl

  • Statement of Gauss's Theorem & Stokes' Theorem

  • Gauss’s Law and Applications

  • Concept of Electric Potential

  • Relationship between E and V

  • Polarization of Dielectrics, Dielectric Constant

  • Boundary Conditions for E & D

  • Gauss’s Law in Magnetostatics

  • Ampere’s Circuital Law

  • Boundary Conditions for B & H

  • Equation of Continuity

  • Displacement Current

  • Maxwell’s Equations

• Module-3: Special Theory of Relativity

  • Introduction to Relativity

  • Inertial Frame of Reference

  • Galilean Transformations

  • Postulates of Special Relativity

  • Lorentz Transformations and Its Conclusions

  • Length Contraction

  • Time Dilation

  • Velocity Addition

  • Mass Change

  • Einstein's Mass-Energy Relation

• Module-4: Quantum Mechanics

  • Planck's Theory of Black-Body Radiation

  • Compton Effect

  • Wave-Particle Duality

  • De Broglie Waves

  • Davisson and Germer's Experiment

  • Uncertainty Principle

  • Brief Idea of Wave Packet, Wave Function and Its Physical Interpretation

  • Schrodinger Equation in One Dimension

  • For Free Particle

  • For Particle in an Infinite Square Well

• Module-5: Modern Physics

  • Laser: Spontaneous and Stimulated Emission

  • Einstein's A and B Coefficients

  • Population Inversion

  • Light Amplification

  • Basic Laser Action

  • Ruby and He-Ne Lasers

  • Properties and Applications of Laser Radiation

  • Nuclear Physics: Binding Energy Curve, Nuclear Force, Liquid Drop Model, Introduction to Shell Model

  • Applications of Nuclear Physics

  • Concept of Plasma Physics and Its Applications

Textbooks

1. A. Ghatak, Optics, 4th Edition, Tata McGraw Hill, 2009

2. Mathew N.O. Sadiku, Elements of Electromagnetics, Oxford University Press (2001)

3. Arthur Beiser, Concept of Modern Physics, 6th edition 2009, Tata McGraw-Hill

4. F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, Springer, Edition 2016

Reference Books

1. Fundamentals of Physics, Halliday, Walker, and Resnick

Module 2 Details: Electromagnetic Theory

• Topics Include:

  • Gradient, Divergence and Curl

  • Statement of Gauss's Theorem & Stokes' Theorem

  • Applications of Gauss’s Law

  • Concept of Electric Potential, Relationship between E and V

  • Polarization of Dielectrics, Dielectric Constant

  • Boundary Conditions for E & D

  • Gauss’s Law in Magnetostatics, Ampere’s Circuital Law

  • Displacement Current, Maxwell's Equations

• Class Structure:

  • 4 Lectures including 1 Tutorial per week (8 hours over ~2 weeks for this module)

Introductory Concepts in Module 2: Electromagnetic Theory

• Mathematical Preliminaries: Gradient, Divergence, Curl and their Applications

• Scalar and Vector Fields

  • Example of a Scalar Field: Temperature distribution in a room, denoted as $T(x, y, z)$, where temperature is a scalar quantity.

  • Example of a Vector Field: Average flow of air particles in the room, represented by velocity $v(x, y, z)$, which is a vector quantity.

• Computing Derivatives/Integrals of a Vector Field:

  • Requires multivariate calculus for functions with multiple variables.

Gradient (Grad)

• Definition: Gradient is a vector operator that represents the rate of change of a scalar field.

• Mathematical Expression:
  \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k}

• Operation: Operates on a scalar field to produce a vector function.

• Interpretation:

  • The gradient points in the direction of the steepest increase of the scalar field.

  • Example with Temperature Mapping: For $T(x, y, z)$, the temperature gradient indicates the direction of maximum temperature increase, useful in optimization, machine learning, AI (gradient descent algorithm).

Mathematical Derivation

• Variation Expression of Temperature:
  dT = \frac{\partial T}{\partial x} dx + \frac{\partial T}{\partial y} dy + \frac{\partial T}{\partial z} dz = \nabla T \cdot d\mathbf{l}

• Geometric Interpretation:
  $dT = \nabla T \, d\mathbf{l}$ gives the maximum temperature variation when the gradient direction aligns with the displacement direction ($\theta = 0$, $\cos \theta = 1$).

Divergence (Div)

• Definition: Divergence is a scalar operator that measures the spread or dispersion of a vector field from a point.

• Mathematical Expression:
  \nabla \cdot \mathbf{F} = \frac{\partial Fx}{\partial x} + \frac{\partial Fy}{\partial y} + \frac{\partial F_z}{\partial z}

• Operation: Produces a scalar function through the dot product on a vector field.

• Interpretation: Indicates the rate of outward/inward flow of the vector field at the specified point; positive divergence indicates a source, while negative indicates a sink.

Examples of Divergence

• For $ extbf{F}(x, y) = 6x^2 \hat{i} + 4y \hat{j}$,
  \text{div} \textbf{F} = 12x + 4

• For $ extbf{F}(x, y, z) = x^2 \hat{i} + 2z \hat{j} - y \hat{k}$,
  \text{div} \textbf{F} = 2x

Curl

• Definition: Curl is a vector operator measuring the rotation or circulation of a vector field at a point.

• Mathematical Expression:
  \nabla \times \mathbf{F} = \frac{\partial Fz}{\partial y} - \frac{\partial Fy}{\partial z} \hat{i} + \frac{\partial Fx}{\partial z} - \frac{\partial Fz}{\partial x} \hat{j} + \frac{\partial Fy}{\partial x} - \frac{\partial Fx}{\partial y} \hat{k}

• Operation: Produces a vector field through the cross product on a vector field.

• Interpretation: Determines the circulation's direction and strength in the vector field.

Application Examples:

1. For $ extbf{F}(x, y, z) = y^3 \hat{i} + xy \hat{j} - z \hat{k}$,
   \text{curl } \textbf{F} = y - 3y^2 \hat{k}

2. For $ extbf{F}(x, y, z) = x \hat{i} + y \hat{j} + z \hat{k}$,
   \text{curl } \textbf{F} = 0

3. For $ extbf{v} = -y \hat{i} + x \hat{j}$,
   \text{curl } extbf{v} = 2 \hat{k}

4. For $ extbf{v} = x \hat{j}$,
   \text{curl } extbf{v} = \hat{k}

Application of Divergence and Curl

• Divergence of Curl:
  \nabla \cdot (\nabla \times \textbf{F}) = 0

• Example:
  \nabla \cdot (\nabla \times \textbf{v}) = 0
  Laplacian Operator

• Operates on a scalar field and produces a scalar function.

• Mathematical Expression:
  \nabla^2 f = \nabla \cdot \nabla f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

Gauss’s Theorem (Divergence Theorem)

• Statement:
  \intV \nabla \cdot \mathbf{v} \, d\tau = \ointS \mathbf{v} \cdot d\mathbf{a}

• Physical Meaning:

  • Relates the flux of a vector field through a closed surface to the divergence within the volume enclosed by that surface.

  • Represents the total flux entering/leaving a closed surface equal to the net source/sink of the vector field inside the volume.

Stokes’ Theorem

• Statement:
  \intS \nabla \times \mathbf{v} \, d\mathbf{a} = \ointC \mathbf{v} \, d\mathbf{l}

• Physical Meaning:

  • Connects the circulation around a closed curve to the curl of a vector field over the surface enclosed by the curve.

  • Fundamental in fluid dynamics, relating circulation to the rotation of the vector field.

Important Concepts in Electrostatics

• Coulomb’s Law:

  • Force on a test charge $Q$ due to another point charge $q$:
    F = \frac{1}{4\pi \epsilon_0} \frac{Qq}{r^2} \hat{r}

  • Positive/negative signs indicate attractive/repulsive nature based on charge sign.

• Electric Field (E):

  • Force per unit charge experienced by a test charge $Q$:
    E = \frac{F}{Q} = \frac{1}{4\pi \epsilon0} \sum{i=1}^{n} \frac{qi}{ri^2} \hat{r_i}

• Electric Flux (Φ_E):

  • Total flux $ΦE = \intS \mathbf{E} \cdot d extbf{a}$,

• Gauss’s Law:

  • For a closed surface $S$, the electric flux through the surface is proportional to the enclosed charge $Q{enc}$: \ointS \mathbf{E} \cdot d extbf{a} = \frac{Q{enc}}{\epsilon0}

Electromagnetic Applications and Theories

• Modules related to electrostatics, currents, fields in matter, magnetostatics

• Theoretical contributions from Maxwell’s equations, their applications in electromagnetic fields