MAKAUT Physics-I (Gr-A) Organizer | BS-PH101

Here is a structured, two-level outline summarizing the key concepts, formulas, and principles from the provided physics document:

• Newton's Laws of Motion and Foundations of Mechanics

  • First Law (Law of Inertia): A body remains at rest or continues to move with uniform velocity in a straight line unless acted upon by a net external force.
  • Second Law (Law of Force): The rate of change of linear momentum of a particle is proportional to the applied net force and occurs in the direction of that force $\left(F = ma = \frac{dp}{dt}\right)$.
  • Third Law (Action and Reaction): When two bodies interact, they exert equal and opposite forces on each other $\left(F_{12} = -F_{21}\right)$.
  • Conservation Laws: Linear momentum is conserved when the net external force on a system is zero, while angular momentum is conserved when the net external torque is zero.
  • Work-Energy Theorem: The net work done on a particle by all external forces is equal to the change in its kinetic energy $\left(W_{12} = T_2 - T_1\right)$.
  • System Constraints: Constraints on mechanical systems are classified as Rheonomic/Scleronomic (depending on time dependence), Holonomic/Non-holonomic (depending on the form of the constraint), Conservative/Dissipative (based on energy conservation), and Bilateral/Unilateral (based on the allowed direction of motion).

• Simple and Damped Harmonic Motion (SHM)

  • SHM Fundamentals: Periodic motion in which the restoring acceleration is directly proportional to the displacement from the mean position and is directed toward the mean position.
  • Damped Motion Differential Equation: Damped motion is described by $\frac{d^2x}{dt^2} + 2K\frac{dx}{dt} + \omega^2x = 0$, where $K$ is the damping constant and $\omega$ is the natural angular frequency.
  • Three Regimes of Damping: Motion is classified as Overdamped $(K > \omega)$, Critically Damped $(K = \omega)$, or Underdamped $(K < \omega)$.
  • Energy Decay and Relaxation Time: The energy of a damped oscillator decays exponentially with time $\left(E = E_0e^{-2Kt}\right)$; the relaxation time $\left(\tau = \frac{1}{2K}\right)$ is the time required for the energy to decrease to $\frac{1}{e}$ (approximately 37%) of its initial value.
  • Logarithmic Decrement: The natural logarithm of the ratio of two successive amplitudes measured on the same side of the mean position $\left(\lambda = KT\right)$.

• Forced Vibrations and Resonance

  • Forced Oscillator Equation: A damped oscillator subjected to an external periodic force $F_0\cos(qt)$ obeys $m\frac{d^2x}{dt^2} + L\frac{dx}{dt} + kx = F_0\cos(qt)$.
  • Steady-State Response: In the steady state, the particle oscillates at the driving frequency $q$, with displacement amplitude $A = \frac{f_0}{\sqrt{(\omega^2-q^2)^2+4K^2q^2}}$.
  • Amplitude and Velocity Resonance: Amplitude resonance occurs when the driving frequency is $q = \sqrt{\omega^2-2K^2}$, whereas velocity resonance occurs at the natural angular frequency $q = \omega$.
  • Quality Factor ($Q$): Defined as $2\pi$ times the ratio of the average energy stored to the energy dissipated per cycle $\left(Q = \frac{\omega}{2K} = \omega\tau\right)$.
  • Electrical Analogy: A mechanical oscillator can be represented by an analogous series $LCR$ circuit, where inductance $L$ corresponds to mass $m$, resistance $R$ corresponds to the mechanical damping coefficient, and capacitance $C$ corresponds to $\frac{1}{k}$.

• Vector Calculus and Electromagnetism

  • Vector Theorems and Definitions: Includes Gauss's Divergence Theorem, which relates a volume integral to a closed-surface integral, and Stokes' Theorem, which relates the surface integral of the curl of a vector field to a closed line integral.
  • Field Characteristics: A solenoidal vector field satisfies $\nabla \cdot \vec{A} = 0$, while an irrotational vector field satisfies $\nabla \times \vec{A} = 0$.
  • Maxwell's Integral Equations: Include Gauss's Law for Electricity $\left(\oint \vec{E}\cdot d\vec{A} = \frac{Q}{\epsilon_0}\right)$, Gauss's Law for Magnetism $\left(\oint \vec{B}\cdot d\vec{A} = 0\right)$, Faraday's Law $\left(\oint \vec{E}\cdot d\vec{l} = -\int \frac{\partial \vec{B}}{\partial t}\cdot d\vec{A}\right)$, and the Ampère-Maxwell Law $\left(\oint \vec{B}\cdot d\vec{l} = \mu_0\int\left(\vec{J}+\frac{\partial\vec{D}}{\partial t}\right)\cdot d\vec{A}\right)$.
  • Maxwell's Differential Equations: Expressed as $\nabla \cdot \vec{D} = \rho$, $\nabla \cdot \vec{B} = 0$, $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$, and $\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$.
  • Electromagnetic Waves: Electromagnetic waves propagate through vacuum at the speed $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$, with the electric field $\vec{E}$, magnetic field $\vec{B}$, and direction of propagation mutually perpendicular to one another.
  • Skin Depth: The characteristic depth to which an electromagnetic wave penetrates into a conducting medium, given by $\delta = \sqrt{\frac{2}{\omega\sigma\mu}}$.

• Optics, Polarization, and Lasers

  • Diffraction Phenomena: Single-slit diffraction produces a central maximum and secondary maxima, with the intensity distribution given by $I = I_0\frac{\sin^2\beta}{\beta^2}$; the intensity of the secondary maxima decreases rapidly with increasing order.
  • Grating Spectra: The resolving power of a plane transmission grating is given by $nN$, where $n$ is the spectral order and $N$ is the total number of illuminated rulings.
  • Polarization Laws: Polarization demonstrates the transverse nature of light; Malus's Law states that the transmitted intensity varies as $I = I_0\cos^2\theta$.
  • Double Refraction: A uniaxial crystal splits an incident light wave into an Ordinary ($O$-ray) and an Extraordinary ($E$-ray) wave. The $O$-ray obeys Snell's law, whereas the $E$-ray generally does not. Uniaxial crystals are classified as Positive (e.g., Quartz, $v_E < v_O$) or Negative (e.g., Calcite, $v_E > v_O$).
  • Laser Fundamentals: Laser action requires an active medium, population inversion $(N_2 > N_1)$, optical or electrical pumping, and an optical resonant cavity to produce coherent and monochromatic amplification through stimulated emission.
  • Einstein Coefficients: Describe the probabilities of absorption $(B_{12})$, spontaneous emission $(A_{21})$, and stimulated emission $(B_{21})$, leading to the relations $B_{12} = B_{21}$ and $\frac{A_{21}}{B_{21}} = \frac{8\pi h\nu^3}{c^3}$.