Physics Essentials (GTU BE01000021)

Cheatsheet Content

1. Properties of Matter Stress, Strain, and Hooke's Law Stress ($\sigma$): Force per unit area. Formula: $\sigma = \frac{F}{A}$ Terms: $F$ (force), $A$ (cross-sectional area) Strain ($\epsilon$): Deformation per unit length. Formula: $\epsilon = \frac{\Delta L}{L_0}$ Terms: $\Delta L$ (change in length), $L_0$ (original length) Hooke's Law: Within the elastic limit, stress is proportional to strain. Formula: $\sigma = E \epsilon$ Terms: $E$ (Young's Modulus) Young's Modulus ($E$): Measure of stiffness of an elastic material. Formula: $E = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L_0}$ Torsional Pendulum & Rigidity Modulus Couple per unit twist ($C$): Torque required to produce a unit twist in a wire. Formula for cylindrical wire: $C = \frac{\pi \eta r^4}{2L}$ Terms: $\eta$ (Rigidity Modulus), $r$ (radius of wire), $L$ (length of wire) Period of Torsional Pendulum ($T$): Formula: $T = 2\pi \sqrt{\frac{I}{C}}$ Terms: $I$ (moment of inertia of the suspended mass), $C$ (couple per unit twist) Rigidity Modulus ($\eta$): Derived from torsional pendulum experiment. Formula: $\eta = \frac{8\pi I L}{r^4 T^2}$ (from combining the above two) 2. Waves, Motion and Acoustics Simple Harmonic Motion (SHM) Displacement ($x$): $x(t) = A \cos(\omega t + \phi)$ (or sine) Velocity ($v$): $v(t) = -A\omega \sin(\omega t + \phi)$ Acceleration ($a$): $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$ Angular Frequency ($\omega$): $\omega = \sqrt{\frac{k}{m}}$ (for spring-mass system) or $\omega = \frac{2\pi}{T} = 2\pi f$ Terms: $A$ (amplitude), $\omega$ (angular frequency), $t$ (time), $\phi$ (phase constant), $k$ (spring constant), $m$ (mass), $T$ (period), $f$ (frequency) Damped Harmonic Motion Equation of motion: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$ Displacement (underdamped): $x(t) = A e^{-\gamma t} \cos(\omega' t + \phi)$ Terms: $b$ (damping coefficient), $\gamma = \frac{b}{2m}$ (damping constant), $\omega' = \sqrt{\omega_0^2 - \gamma^2}$ (damped angular frequency), $\omega_0 = \sqrt{\frac{k}{m}}$ (natural angular frequency) Forced Vibration and Resonance Equation of motion: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \cos(\omega t)$ Amplitude at resonance: $A_{res} = \frac{F_0}{b\omega_0}$ (when $\omega = \omega_0$) Terms: $F_0$ (amplitude of driving force), $\omega$ (driving frequency) Wave Motion Wave Equation (general): $y(x,t) = A \sin(kx - \omega t + \phi)$ Wave Speed ($v$): $v = f\lambda = \frac{\omega}{k}$ Wave Number ($k$): $k = \frac{2\pi}{\lambda}$ Terms: $y$ (displacement), $A$ (amplitude), $k$ (wave number), $\omega$ (angular frequency), $\lambda$ (wavelength), $f$ (frequency) Acoustics - Reverberation Time (Sabine's Formula) Reverberation Time ($T_R$): Time taken for sound intensity to fall by 60 dB. Formula: $T_R = \frac{0.161 V}{\sum A_i S_i}$ (or $T_R = \frac{0.161 V}{A_{total}}$) Terms: $V$ (volume of the room in $m^3$), $A_i$ (absorption coefficient of surface $i$), $S_i$ (area of surface $i$ in $m^2$) 3. Optics Interference (Young's Double Slit Experiment) Path Difference ($\Delta x$): $\Delta x = d \sin\theta = \frac{yd}{D}$ Condition for Constructive Interference (Bright Fringes): $\Delta x = m\lambda$ Condition for Destructive Interference (Dark Fringes): $\Delta x = (m + \frac{1}{2})\lambda$ Fringe Width ($\beta$): $\beta = \frac{\lambda D}{d}$ Terms: $d$ (slit separation), $D$ (distance to screen), $y$ (distance from center of screen), $\lambda$ (wavelength), $m$ (order of fringe) Thin Film Interference Path difference in thin film: $2\mu t \cos r$ Condition for Constructive Interference (reflected light): $2\mu t \cos r = (m + \frac{1}{2})\lambda$ (for thin film of higher refractive index than surrounding) Condition for Destructive Interference (reflected light): $2\mu t \cos r = m\lambda$ (for thin film of higher refractive index than surrounding) Terms: $\mu$ (refractive index of film), $t$ (thickness of film), $r$ (angle of refraction), $\lambda$ (wavelength in vacuum) Note: Phase change of $\pi$ (or $\lambda/2$) occurs upon reflection from a denser medium. These conditions swap if reflection occurs from a less dense medium or if considering transmitted light. Newton's Rings Radius of m-th Bright Ring ($R_m$): $R_m^2 = (m + \frac{1}{2})\lambda R$ Radius of m-th Dark Ring ($R_m$): $R_m^2 = m\lambda R$ Terms: $\lambda$ (wavelength of light), $R$ (radius of curvature of plano-convex lens), $m=0, 1, 2, ...$ Michelson Interferometer Path difference: $2(d_2 - d_1)$ (where $d_1, d_2$ are path lengths) Number of fringes shifted ($N$): $N = \frac{2\Delta d}{\lambda}$ (for measuring wavelength or displacement) Terms: $\Delta d$ (change in path length of one arm), $\lambda$ (wavelength) Diffraction (Grating) Grating Equation: $(a+b)\sin\theta = m\lambda$ (or $d\sin\theta = m\lambda$) Terms: $(a+b)$ or $d$ (grating element, distance between centers of adjacent slits), $\theta$ (angle of diffraction), $m$ (order of diffraction), $\lambda$ (wavelength) Rayleigh Criterion for Resolution Angular limit of resolution ($\theta_{min}$): For circular aperture: $\theta_{min} = \frac{1.22\lambda}{D}$ For slit: $\theta_{min} = \frac{\lambda}{a}$ Terms: $\lambda$ (wavelength), $D$ (diameter of circular aperture), $a$ (width of slit) 4. Quantum Physics Black Body Radiation (Planck's Law) Planck's Radiation Law (Energy density per unit wavelength): $u(\lambda, T) = \frac{8\pi hc}{\lambda^5} \frac{1}{e^{hc/(\lambda k_B T)} - 1}$ Terms: $h$ (Planck's constant), $c$ (speed of light), $\lambda$ (wavelength), $k_B$ (Boltzmann constant), $T$ (temperature) Wave-Particle Duality (de Broglie Wavelength) de Broglie Wavelength ($\lambda$): $\lambda = \frac{h}{p} = \frac{h}{mv}$ Terms: $h$ (Planck's constant), $p$ (momentum), $m$ (mass), $v$ (velocity) Heisenberg's Uncertainty Principle Position and Momentum: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ Energy and Time: $\Delta E \Delta t \ge \frac{\hbar}{2}$ Terms: $\Delta x$ (uncertainty in position), $\Delta p_x$ (uncertainty in momentum), $\Delta E$ (uncertainty in energy), $\Delta t$ (uncertainty in time), $\hbar = \frac{h}{2\pi}$ (reduced Planck's constant) Schrödinger's Wave Equation Time-Independent (1D): $-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$ Time-Dependent (1D): $i\hbar\frac{\partial\Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t)$ Terms: $\psi(x)$ (time-independent wave function), $\Psi(x,t)$ (time-dependent wave function), $V(x)$ (potential energy), $E$ (total energy), $m$ (mass) Probability Density: $|\Psi(x,t)|^2$ gives the probability of finding the particle at position $x$ at time $t$. 5. Lasers Einstein's Theory of Matter Radiation (A & B Coefficients) Rate of Stimulated Absorption ($R_{12}$): $R_{12} = B_{12} N_1 u(\nu)$ Rate of Spontaneous Emission ($R_{21,spont}$): $R_{21,spont} = A_{21} N_2$ Rate of Stimulated Emission ($R_{21,stim}$): $R_{21,stim} = B_{21} N_2 u(\nu)$ Ratio of Coefficients: $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$ (and $B_{12}=B_{21}$) Terms: $A_{21}$ (Einstein coefficient for spontaneous emission), $B_{12}, B_{21}$ (Einstein coefficients for stimulated absorption/emission), $N_1, N_2$ (population of lower and upper energy levels), $u(\nu)$ (radiation energy density at frequency $\nu$) Population Inversion: Condition $N_2 > N_1$ required for laser action. 6. New Engineering Materials Semiconductor Materials Intrinsic Carrier Concentration ($n_i$): $n_i = \sqrt{N_c N_v} \exp\left(-\frac{E_g}{2k_B T}\right)$ Conductivity ($\sigma$): $\sigma = (n\mu_e + p\mu_h)e$ Terms: $N_c, N_v$ (effective density of states in conduction/valence band), $E_g$ (band gap energy), $k_B$ (Boltzmann constant), $T$ (temperature), $n, p$ (electron/hole concentration), $\mu_e, \mu_h$ (electron/hole mobility), $e$ (elementary charge) P-N Junction Diode (Ideal Diode Equation): $I = I_0 \left(e^{\frac{eV}{k_B T}} - 1\right)$ Terms: $I$ (diode current), $I_0$ (reverse saturation current), $V$ (applied voltage), $e$ (electronic charge) Superconducting Materials Critical Temperature ($T_c$): Temperature below which a material becomes superconducting. Critical Magnetic Field ($H_c(T)$): $H_c(T) = H_c(0)\left[1 - \left(\frac{T}{T_c}\right)^2\right]$ Terms: $H_c(0)$ (critical field at 0 K) Meissner Effect: Expulsion of magnetic field from a superconductor. Nanomaterials Quantum Confinement: Effects observed when dimensions of material are comparable to de Broglie wavelength of electrons, leading to quantized energy levels. No specific formulas generally provided as this is a conceptual area in this context. Focus on properties due to small size.