Organizer
MAKAUT Physics-I (Gr-A) Organizer | BS-PH101
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 (F = ma = (dp)/(dt)). · Third Law (Action and Reaction): When two bodies interact, they exert equal and opposite forces on each other (F12 = -F21). · 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 (W12 = T2 - T1). · 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 (d²x)/(dt²) + 2K(dx)/(dt) + ω²x = 0, where K is the damping constant and ω is the natural angular frequency. · Three Regimes of Damping: Motion is classified as Overdamped (K > ω), Critically Damped (K = ω), or Underdamped (K < ω). · Energy Decay and Relaxation Time: The energy of a damped oscillator decays exponentially with time (E = E_0e⁻2Kt); the relaxation time (τ = (1)/(2K)) is the time required for the energy to decrease to (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 (λ = KT). · Forced Vibrations and Resonance · Forced Oscillator Equation: A damped oscillator subjected to an external periodic force F0cos(qt) obeys m(d²x)/(dt²) + L(dx)/(dt) + kx = F0cos(qt). · Steady-State Response: In the steady state, the particle oscillates at the driving frequency q, with displacement amplitude A = fracf_0√(ω²-q²)²+4K²q². · Amplitude and Velocity Resonance: Amplitude resonance occurs when the driving frequency is q = √ω²-2K², whereas velocity resonance occurs at the natural angular frequency q = ω. · Quality Factor (Q): Defined as 2π times the ratio of the average energy stored to the energy dissipated per cycle (Q = (ω)/(2K) = ωτ). · 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 (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 ∇ · vecA = 0, while an irrotational vector field satisfies ∇ × vecA = 0. · Maxwell's Integral Equations: Include Gauss's Law for Electricity (oint vecE· dvecA = (Q)/(ε0)), Gauss's Law for Magnetism (oint vecB· dvecA = 0), Faraday's Law (oint vecE· dvecl = -∫ frac∂ vecB∂ t· dvecA), and the Ampère-Maxwell Law (oint vecB· dvecl = μ0∫(vecJ+frac∂vecD∂ t)· dvecA). · Maxwell's Differential Equations: Expressed as ∇ · vecD = ρ, ∇ · vecB = 0, ∇ × vecE = -frac∂ vecB∂ t, and ∇ × vecH = vecJ + frac∂ vecD∂ t. · Electromagnetic Waves: Electromagnetic waves propagate through vacuum at the speed c = frac1√μ0ε0, with the electric field vecE, magnetic field vecB, 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 δ = √(2)/(ωσμ). · 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(sin²β)/(β²); 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_0cos²θ. · 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, vE < vO) or Negative (e.g., Calcite, vE > vO). · Laser Fundamentals: Laser action requires an active medium, population inversion (N2 > N1), 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 (B12), spontaneous emission (A21), and stimulated emission (B21), leading to the relations B12 = B21 and fracA21B21 = (8π hν³)/(c³).
