Halliday

Cheatsheet Content

1. Measurement SI Base Units: Length: meter (m) Mass: kilogram (kg) Time: second (s) Current: ampere (A) Temperature: kelvin (K) Amount: mole (mol) Luminous Intensity: candela (cd) Prefixes: Giga (G): $10^9$ Mega (M): $10^6$ Kilo (k): $10^3$ Centi (c): $10^{-2}$ Milli (m): $10^{-3}$ Micro ($\mu$): $10^{-6}$ Nano (n): $10^{-9}$ Pico (p): $10^{-12}$ Significant Figures: Non-zero digits are significant. Zeros between non-zeros are significant. Leading zeros are not. Trailing zeros are significant if a decimal point is present. 2. Kinematics (1D & 2D) 2.1. 1D Motion Displacement: $\Delta x = x_f - x_i$ Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Instantaneous Velocity: $v = \frac{dx}{dt}$ Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$ Instantaneous Acceleration: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ Constant Acceleration Equations: $v = v_0 + at$ $x - x_0 = v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ Free Fall: $a = -g = -9.8 \, \text{m/s}^2$ (downwards) 2.2. 2D & 3D Motion Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity Vector: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ Acceleration Vector: $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ Projectile Motion: $x = (v_0 \cos\theta_0)t$ $y = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ $v_x = v_0 \cos\theta_0$ $v_y = v_0 \sin\theta_0 - gt$ Max Height: $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Uniform Circular Motion: Speed: $v = \frac{2\pi r}{T}$ Centripetal Acceleration: $a_c = \frac{v^2}{r}$ (directed towards center) 3. Newton's Laws 1st Law (Inertia): An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. 2nd Law: $\vec{F}_{net} = m\vec{a}$ 3rd Law: If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) Weight: $W = mg$ (force of gravity) Normal Force: $\vec{F}_N$ (perpendicular to surface) Tension: $\vec{T}$ (force through a rope/cable) Friction: Static Friction: $f_s \le \mu_s F_N$ Kinetic Friction: $f_k = \mu_k F_N$ Usually, $\mu_s > \mu_k$ Drag Force: $D = \frac{1}{2} C \rho A v^2$ (for high speeds) Terminal Speed: $v_t = \sqrt{\frac{2mg}{C\rho A}}$ 4. Work and Energy Work done by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$ Work done by Variable Force: $W = \int_{x_i}^{x_f} F(x) dx$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ Conservative Forces: Work done is path-independent ($W_c = -\Delta U$) Non-Conservative Forces: Work done is path-dependent (e.g., friction) Conservation of Mechanical Energy: $E_{mech} = K + U = \text{constant}$ (if only conservative forces do work) General Conservation of Energy: $W_{ext} = \Delta E_{mech} + \Delta E_{th} + \Delta E_{int}$ Thermal Energy (Friction): $\Delta E_{th} = f_k d$ Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ (instantaneous) Average Power: $P_{avg} = \frac{W}{\Delta t}$ 5. Momentum and Collisions Momentum: $\vec{p} = m\vec{v}$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ Impulse-Momentum Theorem: $\vec{J} = \vec{p}_f - \vec{p}_i$ Average Force from Impulse: $\vec{F}_{avg} = \frac{\Delta \vec{p}}{\Delta t}$ Conservation of Momentum: If $\vec{F}_{net,ext} = 0$, then $\vec{P}_{total} = \text{constant}$ Collisions: Elastic: $K_{total}$ is conserved. ($m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$) Inelastic: $K_{total}$ is NOT conserved. Perfectly Inelastic: Objects stick together after collision. ($m_1v_{1i} + m_2v_{2i} = (m_1+m_2)V_f$) Center of Mass: Discrete Particles: $\vec{r}_{com} = \frac{1}{M} \sum m_i \vec{r}_i$ Continuous Body: $\vec{r}_{com} = \frac{1}{M} \int \vec{r} dm$ Velocity of COM: $\vec{v}_{com} = \frac{1}{M} \sum m_i \vec{v}_i$ Newton's 2nd Law for System: $\vec{F}_{net,ext} = M\vec{a}_{com}$ 6. Rotation Angular Position: $\theta$ (radians) Angular Displacement: $\Delta\theta = \theta_f - \theta_i$ Average Angular Velocity: $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$ Instantaneous Angular Velocity: $\omega = \frac{d\theta}{dt}$ Average Angular Acceleration: $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$ Instantaneous Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Constant Angular Acceleration Equations: $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relating Linear and Angular: Arc Length: $s = r\theta$ Tangential Speed: $v_t = r\omega$ Tangential Acceleration: $a_t = r\alpha$ Centripetal Acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Moment of Inertia: $I = \sum m_i r_i^2$ (discrete) or $I = \int r^2 dm$ (continuous) Parallel-Axis Theorem: $I = I_{com} + Md^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ or $\tau = rF\sin\phi$ Newton's 2nd Law for Rotation: $\tau_{net} = I\alpha$ Work in Rotation: $W = \int_{\theta_i}^{\theta_f} \tau d\theta$ Power in Rotation: $P = \tau\omega$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p}$ Angular Momentum for Rigid Body: $L = I\omega$ Newton's 2nd Law (Angular Form): $\vec{\tau}_{net} = \frac{d\vec{L}}{dt}$ Conservation of Angular Momentum: If $\vec{\tau}_{net,ext} = 0$, then $\vec{L}_{total} = \text{constant}$ Rolling without Slipping: $v_{com} = R\omega$, $a_{com} = R\alpha$ Total Kinetic Energy (Rolling): $K = \frac{1}{2}Mv_{com}^2 + \frac{1}{2}I_{com}\omega^2$ 7. Oscillations Simple Harmonic Motion (SHM): Position: $x(t) = x_m \cos(\omega t + \phi)$ Velocity: $v(t) = -\omega x_m \sin(\omega t + \phi)$ Acceleration: $a(t) = -\omega^2 x_m \cos(\omega t + \phi) = -\omega^2 x(t)$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass system) Period: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$ Frequency: $f = \frac{1}{T} = \frac{\omega}{2\pi}$ Simple Pendulum: $T = 2\pi \sqrt{\frac{L}{g}}$ (for small angles) Physical Pendulum: $T = 2\pi \sqrt{\frac{I}{mgd}}$ (I = moment of inertia about pivot, d = distance from pivot to COM) Damped SHM: $x(t) = x_m e^{-bt/2m} \cos(\omega' t + \phi)$ Angular Frequency (Damped): $\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$ Forced Oscillations & Resonance: When driving frequency equals natural frequency, amplitude is maximum. 8. Waves 8.1. Transverse and Longitudinal Waves Wave Function: $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Wave Number: $k = \frac{2\pi}{\lambda}$ Angular Frequency: $\omega = 2\pi f$ Wave Speed: $v = \frac{\omega}{k} = \lambda f$ Speed on a String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear mass density) Power Transmitted: $P_{avg} = \frac{1}{2}\mu v \omega^2 y_m^2$ 8.2. Sound Waves Speed of Sound: $v = \sqrt{\frac{B}{\rho}}$ (B = bulk modulus, $\rho$ = density) Speed in Air: $v \approx 343 \, \text{m/s}$ at $20^\circ \text{C}$ Displacement: $s(x,t) = s_m \cos(kx - \omega t)$ Pressure Variation: $\Delta P(x,t) = \Delta P_m \sin(kx - \omega t)$ where $\Delta P_m = (v\rho\omega)s_m$ Intensity: $I = \frac{P_{avg}}{A} = \frac{1}{2}\rho v \omega^2 s_m^2 = \frac{\Delta P_m^2}{2\rho v}$ Sound Level (Decibels): $\beta = (10 \, \text{dB}) \log_{10}\left(\frac{I}{I_0}\right)$ where $I_0 = 10^{-12} \, \text{W/m}^2$ Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D = detector, S = source; use top signs for "towards", bottom for "away") 8.3. Standing Waves Resonance (String fixed at both ends): $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ for $n=1,2,3...$ Resonance (Open-Open or Closed-Closed Pipe): Same as string. Resonance (Open-Closed Pipe): $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ for $n=1,3,5...$ (only odd harmonics) Beats: $f_{beat} = |f_1 - f_2|$ 9. Thermodynamics Temperature Scales: $T_C = T_K - 273.15$ $T_F = \frac{9}{5}T_C + 32^\circ$ Thermal Expansion: Linear: $\Delta L = L\alpha\Delta T$ Volume: $\Delta V = V\beta\Delta T$, where $\beta = 3\alpha$ Heat: $Q$ (energy transferred due to temperature difference) Heat Capacity: $Q = C\Delta T$ Specific Heat: $Q = cm\Delta T$ Latent Heat (Phase Change): $Q = Lm$ ($L_f$ for fusion, $L_v$ for vaporization) First Law of Thermodynamics: $\Delta E_{int} = Q - W$ $E_{int}$ = internal energy of system $Q$ = heat added to system $W$ = work done BY system Work done by Gas: $W = \int P dV$ Ideal Gas Law: $PV = nRT = NkT$ $R = 8.31 \, \text{J/(mol} \cdot \text{K)}$ (gas constant) $k = 1.38 \times 10^{-23} \, \text{J/K}$ (Boltzmann constant) Kinetic Theory of Gases: Average Kinetic Energy per molecule: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ (M = molar mass in kg/mol) Molar Specific Heat: Constant Volume: $C_V = \frac{Q}{n\Delta T}$ Constant Pressure: $C_P = \frac{Q}{n\Delta T}$ For ideal gas: $C_P = C_V + R$ Monatomic ideal gas: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$ Adiabatic Process: $PV^\gamma = \text{constant}$ or $TV^{\gamma-1} = \text{constant}$ where $\gamma = C_P/C_V$ Heat Transfer: Conduction: $P_{cond} = \frac{Q}{t} = kA\frac{T_H - T_L}{L}$ Radiation: $P_{rad} = \sigma \epsilon A T^4$ ($\sigma$ = Stefan-Boltzmann constant, $\epsilon$ = emissivity) Second Law of Thermodynamics: Heat flows spontaneously from hotter to colder. Entropy of isolated system never decreases. Heat Engines: Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_L|}{|Q_H|}$ Carnot Engine (Ideal): $\epsilon_C = 1 - \frac{T_L}{T_H}$ Refrigerators/Heat Pumps: Coefficient of Performance (Refrigerator): $K = \frac{|Q_L|}{|W|}$ Coefficient of Performance (Heat Pump): $K_{HP} = \frac{|Q_H|}{|W|} = K + 1$ Carnot Refrigerator: $K_C = \frac{T_L}{T_H - T_L}$ Entropy: $\Delta S = \int \frac{dQ}{T}$ Entropy Change (Reversible): $\Delta S = \frac{Q}{T}$ Ideal Gas Entropy Change: $\Delta S = nR \ln\left(\frac{V_f}{V_i}\right) + nC_V \ln\left(\frac{T_f}{T_i}\right)$ 10. Electric Charge & Field Charge Quantization: $q = ne$ ($e = 1.602 \times 10^{-19} \, \text{C}$) Coulomb's Law: $F = k \frac{|q_1q_2|}{r^2}$ where $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2$ Electric Field (Point Charge): $E = k \frac{|q|}{r^2}$ (direction away from positive, towards negative) Force on Charge in E-field: $\vec{F} = q\vec{E}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ (from -q to +q) Torque on Dipole in E-field: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ Electric Flux: $\Phi = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ Electric Field (Infinite Line of Charge): $E = \frac{\lambda}{2\pi\epsilon_0 r}$ Electric Field (Infinite Non-conducting Sheet): $E = \frac{\sigma}{2\epsilon_0}$ Electric Field (Conducting Sheet): $E = \frac{\sigma}{\epsilon_0}$ 11. Electric Potential Potential Difference: $\Delta V = V_f - V_i = -\int_{i}^{f} \vec{E} \cdot d\vec{s}$ Potential Energy: $\Delta U = q\Delta V$ Potential from Point Charge: $V = k \frac{q}{r}$ (relative to $V=0$ at infinity) Potential from Dipole (far field): $V = \frac{kp\cos\theta}{r^2}$ Electric Field from Potential: $E_x = -\frac{\partial V}{\partial x}$, $\vec{E} = -\vec{\nabla}V$ Potential Energy of System of Charges: $U = \sum_{pairs} k \frac{q_i q_j}{r_{ij}}$ 12. Capacitance Capacitance: $C = \frac{Q}{V}$ (Unit: Farad, F) Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Capacitors in Parallel: $C_{eq} = C_1 + C_2 + ...$ Capacitors in Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Dielectrics: $C = \kappa C_{air}$, $E = E_{air}/\kappa$, $V = V_{air}/\kappa$ 13. Current and Resistance Electric Current: $i = \frac{dQ}{dt}$ (Unit: Ampere, A) Current Density: $\vec{J} = n e \vec{v}_d$ ($n$ = charge carrier density, $\vec{v}_d$ = drift velocity) Current and Current Density: $i = \int \vec{J} \cdot d\vec{A}$ Resistance: $R = \frac{V}{i}$ (Unit: Ohm, $\Omega$) Resistivity: $\rho = \frac{RA}{L}$ Resistance and Resistivity: $R = \rho \frac{L}{A}$ Temperature Dependence of Resistivity: $\rho - \rho_0 = \rho_0 \alpha (T - T_0)$ Power in Circuits: $P = iV = i^2R = \frac{V^2}{R}$ 14. DC Circuits EMF: $\mathcal{E}$ (ideal battery voltage) Ohm's Law: $V = iR$ Resistors in Series: $R_{eq} = R_1 + R_2 + ...$ Resistors in Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ Kirchhoff's Rules: Loop Rule: Sum of voltage changes around any closed loop is zero. Junction Rule: Sum of currents entering a junction equals sum of currents leaving. RC Circuits: Charging Capacitor: $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $i(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ Discharging Capacitor: $q(t) = Q_0 e^{-t/RC}$, $i(t) = -\frac{Q_0}{RC}e^{-t/RC}$ Time Constant: $\tau = RC$ 15. Magnetic Field & Forces Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ ($F_B = |q|vB\sin\phi$) Circular Motion in B-field: $r = \frac{mv}{|q|B}$ (cyclotron frequency $f = \frac{|q|B}{2\pi m}$) Magnetic Force on Current Wire: $\vec{F}_B = i\vec{L} \times \vec{B}$ ($F_B = iLB\sin\phi$) Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = Ni\vec{A}$ (N = turns, A = area, direction by right-hand rule) Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = \frac{iB}{neL}$ (n = charge carrier density) 16. Sources of Magnetic Field Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{i d\vec{s} \times \hat{r}}{r^2}$ ($\mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A}$) Magnetic Field (Long Straight Wire): $B = \frac{\mu_0 i}{2\pi r}$ Force between Parallel Wires: $\frac{F}{L} = \frac{\mu_0 i_1 i_2}{2\pi d}$ (attractive if currents same direction) Magnetic Field (Center of Circular Arc): $B = \frac{\mu_0 i \phi}{4\pi r}$ ($\phi$ in radians) Magnetic Field (Center of Loop): $B = \frac{\mu_0 i}{2r}$ Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 i_{enc}$ Magnetic Field (Solenoid): $B = \mu_0 n i$ ($n$ = turns per unit length) Magnetic Field (Toroid): $B = \frac{\mu_0 i N}{2\pi r}$ 17. Induction and Inductance Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Lenz's Law: Induced current/EMF opposes the change in magnetic flux that created it. Motional EMF: $\mathcal{E} = BLv$ (for conductor moving perpendicular to B-field) Induced Electric Field: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ Inductance: $L = \frac{N\Phi_B}{i}$ (Unit: Henry, H) Solenoid Inductance: $L = \mu_0 n^2 A l$ RL Circuits (Current Growth): $i(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$, where $\tau_L = L/R$ RL Circuits (Current Decay): $i(t) = \frac{i_0}{R}e^{-t/\tau_L}$ Energy Stored in Inductor: $U_B = \frac{1}{2}Li^2$ Magnetic Energy Density: $u_B = \frac{B^2}{2\mu_0}$ Mutual Inductance: $M_{21} = \frac{N_2\Phi_{B21}}{i_1}$, $\mathcal{E}_2 = -M \frac{di_1}{dt}$ 18. AC Circuits AC Current/Voltage: $i = I \sin(\omega t)$, $v = V \sin(\omega t + \phi)$ Reactance: Inductive Reactance: $X_L = \omega L$ Capacitive Reactance: $X_C = \frac{1}{\omega C}$ Ohm's Law for AC Components: Resistor: $V_R = IR$ (in phase) Inductor: $V_L = IX_L$ ($V_L$ leads $I$ by $90^\circ$) Capacitor: $V_C = IX_C$ ($I$ leads $V_C$ by $90^\circ$) Impedance (RLC Series Circuit): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ Resonance (Series RLC): Occurs when $X_L = X_C$, so $\omega_0 = \frac{1}{\sqrt{LC}}$ Power in AC Circuits: $P_{avg} = I_{rms} V_{rms} \cos\phi = I_{rms}^2 R$ RMS Values: $V_{rms} = V_{peak}/\sqrt{2}$, $I_{rms} = I_{peak}/\sqrt{2}$ Transformers: $\frac{V_S}{V_P} = \frac{N_S}{N_P} = \frac{I_P}{I_S}$ 19. Electromagnetic Waves Speed of Light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \, \text{m/s}$ Wave Speed in Medium: $v = \frac{c}{\sqrt{\kappa_m \kappa_e}}$ Wave Equation: $\frac{\partial^2 E}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}$, $\frac{\partial^2 B}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 B}{\partial t^2}$ Relationship E and B: $E = cB$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (direction of energy flow) Intensity (Average): $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{1}{2c\mu_0}E_m^2$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 20. Optics 20.1. Reflection and Refraction Law of Reflection: $\theta_i = \theta_r$ Snell's Law (Refraction): $n_1 \sin\theta_1 = n_2 \sin\theta_2$ Index of Refraction: $n = c/v$ Critical Angle: $\sin\theta_c = \frac{n_2}{n_1}$ ($n_1 > n_2$) 20.2. Mirrors and Lenses Mirror/Lens Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ $p$: object distance (positive if real, negative if virtual) $i$: image distance (positive if real, negative if virtual) $f$: focal length (+ for concave mirror/converging lens, - for convex mirror/diverging lens) Magnification: $m = -\frac{i}{p} = \frac{h_i}{h_p}$ (positive for upright, negative for inverted) Lensmaker's Equation: $\frac{1}{f} = (n-1)\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$ 20.3. Interference Young's Double-Slit: Bright Fringes (Max): $d \sin\theta = m\lambda$, $y = \frac{m\lambda L}{d}$ ($m=0, \pm 1, \pm 2, ...$) Dark Fringes (Min): $d \sin\theta = (m + \frac{1}{2})\lambda$, $y = \frac{(m + \frac{1}{2})\lambda L}{d}$ Thin Films: (consider phase changes at interfaces) Path difference $= 2Lt$ Phase change upon reflection at $n_1 \to n_2$ interface: $180^\circ$ if $n_2 > n_1$, $0^\circ$ if $n_2 20.4. Diffraction Single-Slit Diffraction: Minima (Dark Fringes): $a \sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Central Max: Width $2y = \frac{2\lambda L}{a}$ Diffraction Grating: Max: $d \sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh's Criterion (Resolution): $\theta_R = 1.22\frac{\lambda}{D}$ (circular aperture) 21. Special Relativity Postulates: The laws of physics are the same for all inertial reference frames. The speed of light in vacuum ($c$) has the same value in all inertial reference frames. Time Dilation: $\Delta t = \gamma \Delta t_0$ ($\Delta t_0$ = proper time, $\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$) Length Contraction: $L = L_0/\gamma$ ($L_0$ = proper length) Relativistic Momentum: $\vec{p} = \gamma m\vec{v}$ Relativistic Kinetic Energy: $K = (\gamma - 1)mc^2$ Total Energy: $E = \gamma mc^2 = K + mc^2$ Mass-Energy Equivalence: $E_0 = mc^2$ (rest energy) Energy-Momentum Relation: $E^2 = (pc)^2 + (mc^2)^2$ 22. Quantum Physics Planck's Quantum Hypothesis: $E = hf$ ($h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s}$) Photoelectric Effect: $K_{max} = hf - \Phi$ ($\Phi$ = work function) Photon Momentum: $p = h/\lambda$ Compton Effect: $\Delta\lambda = \lambda' - \lambda = \frac{h}{mc}(1 - \cos\phi)$ De Broglie Wavelength: $\lambda = h/p = h/(mv)$ Heisenberg Uncertainty Principle: $\Delta x \Delta p_x \ge \hbar/2$ $\Delta E \Delta t \ge \hbar/2$ ($\hbar = h/2\pi$) Schrödinger Equation (1D, Time-Independent): $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ 23. Atomic Physics Bohr Model (Hydrogen): Quantized Energy Levels: $E_n = -\frac{13.6 \, \text{eV}}{n^2}$ Quantized Radii: $r_n = a_0 n^2$ ($a_0 = 0.0529 \, \text{nm}$ Bohr radius) Photon Energy: $hf = E_i - E_f$ Quantum Numbers: Principal (n): $1, 2, 3, ...$ (energy, size) Orbital (l): $0, 1, ..., n-1$ (shape, angular momentum, $s, p, d, f...$) Magnetic ($m_l$): $-l, ..., 0, ..., +l$ (orientation of orbital) Spin ($m_s$): $\pm 1/2$ (electron spin) Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers. X-rays: Short wavelength EM radiation. 24. Nuclear Physics Nucleus Composition: Protons (Z), Neutrons (N), Mass Number (A=Z+N) Nuclear Radius: $R = R_0 A^{1/3}$ ($R_0 \approx 1.2 \, \text{fm}$) Mass Defect: $\Delta m = (Z m_p + N m_n) - m_{nucleus}$ Binding Energy: $E_B = \Delta m c^2$ Radioactive Decay Law: $N(t) = N_0 e^{-\lambda t}$ Decay Constant: $\lambda = \frac{\ln 2}{T_{1/2}}$ ($T_{1/2}$ = half-life) Activity: $R = |\frac{dN}{dt}| = \lambda N$ (Unit: Becquerel, Bq or Curie, Ci) Alpha Decay: $(Z, A) \to (Z-2, A-4) + {}_2^4\text{He}$ Beta Decay ($e^-$): $(Z, A) \to (Z+1, A) + e^- + \bar{\nu}$ Beta Decay ($e^+$): $(Z, A) \to (Z-1, A) + e^+ + \nu$ Gamma Decay: Excited nucleus emits photon. Nuclear Fission: Heavy nucleus splits into lighter ones. Nuclear Fusion: Light nuclei combine to form heavier ones.