Halliday

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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 ($10^9$), Mega ($10^6$), Kilo ($10^3$), Centi ($10^{-2}$), Milli ($10^{-3}$), Micro ($10^{-6}$), Nano ($10^{-9}$), Pico ($10^{-12}$) Significant Figures: Rules for addition/subtraction and multiplication/division. 2. Kinematics in 1D Displacement: $\Delta x = x_f - x_i$ Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Average Speed: $\frac{\text{total distance}}{\Delta t}$ 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 Acceleration: $g = 9.8 \text{ m/s}^2$ (downwards) 3. Vectors Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\theta = \arctan\left(\frac{A_y}{A_x}\right)$ Vector Addition: $\vec{C} = \vec{A} + \vec{B} \implies C_x = A_x + B_x, C_y = A_y + B_y$ Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = AB \cos \theta = A_x B_x + A_y B_y + A_z B_z$ Vector (Cross) Product: $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ Magnitude of Cross Product: $|\vec{A} \times \vec{B}| = AB \sin \theta$ (direction by right-hand rule) 4. Kinematics in 2D & 3D Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Displacement: $\Delta \vec{r} = \vec{r}_f - \vec{r}_i$ Average Velocity: $\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$ Instantaneous Velocity: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ Average Acceleration: $\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}$ Instantaneous Acceleration: $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ Projectile Motion: $v_{0x} = v_0 \cos \theta_0$, $v_{0y} = v_0 \sin \theta_0$ $x = (v_0 \cos \theta_0)t$ $y = (v_0 \sin \theta_0)t - \frac{1}{2}gt^2$ $v_y = v_0 \sin \theta_0 - gt$ $v_y^2 = (v_0 \sin \theta_0)^2 - 2gy$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for level ground) Uniform Circular Motion: Speed: $v = \frac{2\pi r}{T}$ Centripetal Acceleration: $a_c = \frac{v^2}{r}$ (directed towards center) 5. Newton's Laws of Motion First Law: 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. Second Law: $\sum \vec{F} = m\vec{a}$ Third Law: If object A exerts a force on object B, then object B exerts a force of equal magnitude and opposite direction on object A. ($\vec{F}_{AB} = -\vec{F}_{BA}$) Weight: $W = mg$ (force of gravity) Normal Force: $\vec{F}_N$ (perpendicular to surface) Tension: $\vec{T}$ (pulling force via rope/string) Friction: Static Friction: $f_s \le \mu_s F_N$ Kinetic Friction: $f_k = \mu_k F_N$ 6. Work, Energy, and Power Work done by a constant force: $W = \vec{F} \cdot \vec{d} = Fd \cos \theta$ Work done by a variable force: $W = \int \vec{F} \cdot d\vec{r}$ 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): $U_s = \frac{1}{2}kx^2$ Conservation of Mechanical Energy: $E_{mech} = K + U = \text{constant}$ (if only conservative forces do work) Work done by non-conservative forces: $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ (instantaneous), $P_{avg} = \frac{W}{\Delta t}$ 7. Center of Mass and Linear Momentum Center of Mass: For particles: $x_{com} = \frac{\sum m_i x_i}{\sum m_i}$, $y_{com} = \frac{\sum m_i y_i}{\sum m_i}$, $z_{com} = \frac{\sum m_i z_i}{\sum m_i}$ For solid objects: $\vec{r}_{com} = \frac{1}{M}\int \vec{r} dm$ Linear Momentum: $\vec{p} = m\vec{v}$ Newton's Second Law (momentum form): $\sum \vec{F} = \frac{d\vec{p}}{dt}$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum: If $\sum \vec{F}_{ext} = 0$, then $\Delta \vec{P}_{sys} = 0 \implies \vec{P}_{initial} = \vec{P}_{final}$ Collisions: Elastic: Momentum and Kinetic Energy conserved. Inelastic: Momentum conserved, Kinetic Energy NOT conserved. Perfectly Inelastic: Objects stick together, momentum conserved, max KE loss. 8. Rotation Angular Position: $\theta$ (radians) Angular Displacement: $\Delta \theta$ Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Rotational Kinematics (constant $\alpha$): $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relating Linear and Angular: $s = r\theta$ $v_t = r\omega$ (tangential speed) $a_t = r\alpha$ (tangential acceleration) $a_c = r\omega^2 = \frac{v_t^2}{r}$ (centripetal acceleration) Moment of Inertia: $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous) Parallel-Axis Theorem: $I = I_{com} + Md^2$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$, magnitude $\tau = rF \sin \phi$ Newton's Second Law for Rotation: $\sum \tau = I\alpha$ Work done by Torque: $W = \int \tau d\theta$ Power: $P = \tau \omega$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) Conservation of Angular Momentum: If $\sum \vec{\tau}_{ext} = 0$, then $\Delta \vec{L}_{sys} = 0 \implies \vec{L}_{initial} = \vec{L}_{final}$ 9. Gravity Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$, where $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ Gravitational Acceleration: $g = \frac{GM}{r^2}$ Gravitational Potential Energy: $U = -\frac{GMm}{r}$ (zero at infinity) Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's Laws: Orbits are ellipses with the Sun at one focus. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. The square of the orbital period $T$ is proportional to the cube of the semi-major axis $a$: $T^2 = \left(\frac{4\pi^2}{GM}\right)a^3$ 10. 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), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) Frequency: $f = \frac{\omega}{2\pi}$ Period: $T = \frac{1}{f} = \frac{2\pi}{\omega}$ Energy in SHM: $E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kx_m^2 = \frac{1}{2}m\omega^2 x_m^2$ Physical Pendulum: $T = 2\pi \sqrt{\frac{I}{mgd}}$ (I = moment of inertia, d = dist from pivot to COM) 11. Waves (Mechanical) Wave Function: $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Wave Number: $k = \frac{2\pi}{\lambda}$ Wave Speed: $v = \frac{\omega}{k} = \lambda f$ Speed on a String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ = tension, $\mu$ = linear mass density) Power of a wave: $P_{avg} = \frac{1}{2}\mu v \omega^2 y_m^2$ Principle of Superposition: $y'(x,t) = y_1(x,t) + y_2(x,t)$ Standing Waves on a String (fixed ends): Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1, 2, 3, ...$) Frequencies: $f_n = \frac{nv}{2L} = n f_1$ 12. 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 $20^\circ \text{C}$ Intensity: $I = \frac{P}{A}$ Sound Level (Decibels): $\beta = (10 \text{ dB}) \log_{10}\left(\frac{I}{I_0}\right)$, $I_0 = 10^{-12} \text{ W/m}^2$ Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D = detector, S = source; choose sign for relative motion) Standing Waves in Pipes: Open at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1, 2, 3, ...$) Closed at one end: $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ ($n=1, 3, 5, ...$) Beats: $f_{beat} = |f_1 - f_2|$ 13. Temperature, Heat, and First Law of Thermodynamics Temperature Scales: $T_F = \frac{9}{5}T_C + 32^\circ$ $T_K = T_C + 273.15$ Thermal Expansion: Linear: $\Delta L = L\alpha\Delta T$ Volume: $\Delta V = V\beta\Delta T$, where $\beta = 3\alpha$ Heat Capacity: $Q = C\Delta T = mc\Delta T$ ($c$ = specific heat) Phase Changes (Latent Heat): $Q = mL$ ($L_f$ for fusion, $L_v$ for vaporization) Heat Transfer: Conduction: $P_{cond} = \frac{kA(T_H - T_C)}{L}$ ($k$ = thermal conductivity) Radiation: $P_{rad} = \sigma \epsilon A T^4$ ($\sigma$ = Stefan-Boltzmann constant, $\epsilon$ = emissivity) First Law of Thermodynamics: $\Delta E_{int} = Q - W$ ($\Delta E_{int}$ = change in internal energy, $Q$ = heat added to system, $W$ = work done BY system) Work done by gas: $W = \int P dV$ 14. Kinetic Theory of Gases Ideal Gas Law: $PV = nRT = NkT$ ($R = 8.31 \text{ J/(mol}\cdot\text{K})$, $k = 1.38 \times 10^{-23} \text{ J/K}$) Average Kinetic Energy (translational): $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ Internal Energy of Ideal Gas: Monatomic: $E_{int} = \frac{3}{2}nRT$ Diatomic: $E_{int} = \frac{5}{2}nRT$ (at moderate T) Molar Specific Heats: Constant Volume: $C_V = \frac{1}{n}\frac{dE_{int}}{dT}$ ($Q = nC_V\Delta T$) Constant Pressure: $C_P = C_V + R$ ($Q = nC_P\Delta T$) Adiabatic Process: $PV^\gamma = \text{constant}$, where $\gamma = C_P/C_V$ 15. Entropy and Second Law of Thermodynamics Second Law: Heat flows spontaneously from hot to cold. Entropy of an isolated system never decreases: $\Delta S \ge 0$. Entropy Change: Reversible: $\Delta S = \int \frac{dQ}{T}$ Isothermal: $\Delta S = \frac{Q}{T}$ Carnot Engine (ideal): Efficiency: $\epsilon = 1 - \frac{T_C}{T_H}$ Coefficient of Performance (Refrigerator): $K = \frac{|Q_C|}{|W|} = \frac{T_C}{T_H - T_C}$ 16. Electric Charge and Field Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$, where $k = \frac{1}{4\pi\epsilon_0} \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ Field of a Point Charge: $E = k \frac{|q|}{r^2}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on a Dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of a Dipole: $U = -\vec{p} \cdot \vec{E}$ 17. Gauss' Law Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\epsilon_0 \Phi_E = q_{enc}$ Symmetry Applications: Sphere: $E = k \frac{Q}{r^2}$ (outside), $E = k \frac{Qr}{R^3}$ (inside, uniform charge) Infinite Line: $E = \frac{\lambda}{2\pi\epsilon_0 r}$ Infinite Sheet: $E = \frac{\sigma}{2\epsilon_0}$ Parallel Plates (capacitor): $E = \frac{\sigma}{\epsilon_0}$ (between plates) 18. Electric Potential Potential Difference: $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ Potential due to Point Charge: $V = k \frac{q}{r}$ Potential due to Dipole: $V = \frac{kp \cos \theta}{r^2}$ Relating E and V: $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ Potential Energy of System of Charges: $U = \sum_{i Potential Energy in Electric Field: $U = qV$ 19. Capacitance Capacitance: $C = \frac{q}{V}$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Cylindrical Capacitor: $C = \frac{2\pi\epsilon_0 L}{\ln(b/a)}$ Spherical Capacitor: $C = 4\pi\epsilon_0 \frac{ab}{b-a}$ Series Capacitors: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Parallel Capacitors: $C_{eq} = \sum C_i$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{q^2}{2C} = \frac{1}{2}qV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Dielectrics: $C = \kappa C_{air}$, $E = E_{air}/\kappa$ 20. Current and Resistance Electric Current: $I = \frac{dq}{dt}$ Current Density: $J = \frac{I}{A} = nqv_d$ ($v_d$ = drift speed) Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ ($\rho$ = resistivity) Resistivity and Temperature: $\rho - \rho_0 = \rho_0 \alpha (T - T_0)$ Power in Circuits: $P = IV = I^2R = \frac{V^2}{R}$ 21. Circuits EMF: $\mathcal{E}$ (ideal battery voltage) Series Resistors: $R_{eq} = \sum R_i$ Parallel Resistors: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ Kirchhoff's Laws: Junction Rule: $\sum I_{in} = \sum I_{out}$ Loop Rule: $\sum \Delta V = 0$ around any closed loop. RC Circuits (Charging): $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ RC Circuits (Discharging): $q(t) = Q_0 e^{-t/RC}$, $I(t) = -\frac{Q_0}{RC}e^{-t/RC}$ Time Constant: $\tau = RC$ 22. Magnetic Fields Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ ($F_B = |q|vB \sin \phi$) Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ ($F_B = ILB \sin \phi$) Magnetic Field Units: Tesla (T), $1 \text{ T} = 1 \text{ N/(A}\cdot\text{m})$ Hall Effect: $V_H = \frac{IB}{net}$ ($n$ = charge carrier density, $t$ = thickness) Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NIA\hat{n}$ ($N$ = turns, $I$ = current, $A$ = area) Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ 23. Magnetic Fields from Currents Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ Magnetic Permeability of Free Space: $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ Magnetic Field of Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Force between Parallel Wires: $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$ Magnetic Field at Center of Circular Loop: $B = \frac{\mu_0 I}{2R}$ Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Magnetic Field of Solenoid: $B = \mu_0 n I$ ($n$ = turns per unit length) Magnetic Field of Toroid: $B = \frac{\mu_0 N I}{2\pi r}$ 24. Induction and Inductance Faraday's Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Lenz's Law: Induced current's magnetic field opposes the change in magnetic flux. Motional EMF: $\mathcal{E} = BLv$ Self-Inductance: $L = \frac{N\Phi_B}{I}$ Inductor EMF: $\mathcal{E}_L = -L \frac{dI}{dt}$ RL Circuits (Current Growth): $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ RL Circuits (Current Decay): $I(t) = I_0 e^{-t/\tau_L}$ Time Constant: $\tau_L = L/R$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Magnetic Energy Density: $u_B = \frac{B^2}{2\mu_0}$ Mutual Inductance: $M_{21} = M_{12} = M$ Mutual EMF: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ 25. Electromagnetic Oscillations and AC Circuits LC Oscillations: Angular Frequency: $\omega = \frac{1}{\sqrt{LC}}$ Charge: $q(t) = Q \cos(\omega t + \phi)$ Current: $I(t) = - \omega Q \sin(\omega t + \phi)$ Damped RLC Oscillations: $\omega' = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2}$ Forced Oscillations (AC): $I = I_m \sin(\omega_d t - \phi)$ Capacitive Reactance: $X_C = \frac{1}{\omega_d C}$ Inductive Reactance: $X_L = \omega_d L$ Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan \phi = \frac{X_L - X_C}{R}$ Resonance: $X_L = X_C \implies \omega_d = \frac{1}{\sqrt{LC}}$ Power (AC Circuit): $P_{avg} = I_{rms}^2 R = I_{rms}V_{rms}\cos\phi$ ($\cos\phi$ = power factor) Transformer: $\frac{V_S}{V_P} = \frac{N_S}{N_P}$ (ideal) 26. Maxwell's Equations and EM Waves Maxwell's Equations (Integral Form): $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ (Gauss' Law for Electricity) $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss' Law for Magnetism) $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (Faraday's Law) $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ (Ampere-Maxwell Law) Displacement Current: $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$ Speed of Light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Properties of EM Waves: $c = E/B$ $\vec{E}$ and $\vec{B}$ are perpendicular to each other and to direction of propagation. $\frac{E}{B} = c$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (direction of energy flow) Intensity of EM Wave: $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{c}{\mu_0}B_{rms}^2 = \frac{E_{rms}B_{rms}}{\mu_0}$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 27. Light: Reflection and Refraction Law of Reflection: $\theta_1 = \theta_1'$ Law of Refraction (Snell's Law): $n_1 \sin \theta_1 = n_2 \sin \theta_2$ Index of Refraction: $n = c/v$ Critical Angle: $\sin \theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) Polarization by Reflection (Brewster's Angle): $\tan \theta_B = \frac{n_2}{n_1}$ 28. Lenses and Mirrors Spherical Mirrors: Mirror Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Focal Length: $f = R/2$ (concave), $f = -R/2$ (convex) Thin Lenses: Lensmaker's Equation: $\frac{1}{f} = (n-1)\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$ Thin Lens Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -\frac{i}{p} = \frac{h'}{h}$ Sign Conventions: $p$: + (real object), - (virtual object) $i$: + (real image, opposite side of mirror/lens), - (virtual image, same side) $f$: + (concave mirror, converging lens), - (convex mirror, diverging lens) $R$: + (center on same side as outgoing light), - (center on opposite side) $h, h'$: + (upright), - (inverted) 29. Interference Young's Double-Slit Experiment: Constructive Interference (bright fringes): $d \sin \theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Destructive Interference (dark fringes): $d \sin \theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) Fringe Spacing: $\Delta y = L \frac{\lambda}{d}$ Thin Films: (Reflected light) Phase change upon reflection: $180^\circ$ if $n_2 > n_1$ Path length difference: $2Lt$ (for normal incidence) Conditions depend on phase changes. 30. Diffraction Single-Slit Diffraction: Minima (dark fringes): $a \sin \theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Central maximum width: $2\theta_1 \approx \frac{2\lambda}{a}$ Diffraction Grating: Maxima (bright fringes): $d \sin \theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh's Criterion: $\theta_R = 1.22 \frac{\lambda}{D}$ (circular aperture)