Halliday Physics Fundamentals

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1. Measurement & Vectors SI Base Units: Length (m), Mass (kg), Time (s), Current (A), Temp (K), Amount (mol), Luminous Intensity (cd) Prefixes: G (10$^9$), M (10$^6$), k (10$^3$), c (10$^{-2}$), m (10$^{-3}$), $\mu$ (10$^{-6}$), n (10$^{-9}$), p (10$^{-12}$) Significant Figures: Non-zero digits, zeros between non-zeros, trailing zeros with decimal. Vector Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\tan \theta = A_y / A_x$ Dot Product: $\vec{A} \cdot \vec{B} = AB \cos \phi = A_x B_x + A_y B_y + A_z B_z$ Cross Product: $|\vec{A} \times \vec{B}| = AB \sin \phi$. Direction by right-hand rule. $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ 2. Kinematics (1D & 2D) 1D Motion Avg Velocity: $v_{avg} = \Delta x / \Delta t$ Instantaneous Velocity: $v = dx/dt$ Avg Acceleration: $a_{avg} = \Delta v / \Delta t$ Instantaneous Acceleration: $a = dv/dt = d^2x/dt^2$ Constant Acceleration Equations: $v = v_0 + at$ $x = x_0 + v_0 t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ 2D Motion (Projectile) Position: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$ Velocity: $\vec{v}(t) = v_x(t)\hat{i} + v_y(t)\hat{j}$ Acceleration: $\vec{a}(t) = a_x(t)\hat{i} + a_y(t)\hat{j}$ Projectile Motion ($a_x=0, a_y=-g$): $v_x = v_{0x}$ $x = x_0 + v_{0x}t$ $v_y = v_{0y} - gt$ $y = y_0 + v_{0y}t - \frac{1}{2}gt^2$ $v_y^2 = v_{0y}^2 - 2g(y - y_0)$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for $y_0=0$) Uniform Circular Motion Centripetal Acceleration: $a_c = v^2/r$ (directed towards center) Period: $T = 2\pi r / v$ 3. Newton's Laws of Motion 1st Law (Inertia): If $\vec{F}_{net}=0$, then $\vec{v}=$ constant. 2nd Law: $\vec{F}_{net} = m\vec{a}$ 3rd Law: $\vec{F}_{AB} = -\vec{F}_{BA}$ (action-reaction pairs) Weight: $W = mg$ (force due to gravity) Normal Force: $\vec{F}_N$ (perpendicular to surface) Friction: Static: $f_s \le \mu_s F_N$ Kinetic: $f_k = \mu_k F_N$ ($\mu_k Drag Force (high speed): $D = \frac{1}{2} C \rho A v^2$ Terminal Speed: $v_t = \sqrt{\frac{2mg}{C\rho A}}$ 4. Work & Energy Work by Constant Force: $W = \vec{F} \cdot \vec{d} = Fd \cos \phi$ Work by Variable Force (1D): $W = \int_{x_i}^{x_f} F(x) dx$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Power: $P = dW/dt = \vec{F} \cdot \vec{v}$ (instantaneous), $P_{avg} = W/\Delta t$ Gravitational Potential Energy: $U_g = mgy$ Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ Conservative Forces: Work independent of path, $W_c = -\Delta U$. (Gravity, Spring) Non-Conservative Forces: Work depends on path. (Friction, Drag) Conservation of Mechanical Energy: $E_{mech} = K + U$. If only conservative forces do work, $E_{mech}$ is conserved. $\Delta E_{mech} = 0$. General Conservation of Energy: $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ 5. Momentum & Collisions Linear Momentum: $\vec{p} = m\vec{v}$ Newton's 2nd Law (Momentum): $\vec{F}_{net} = d\vec{p}/dt$ Impulse: $\vec{J} = \int \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ Conservation of Momentum: If $\vec{F}_{net, ext} = 0$, then $\Delta \vec{P}_{sys} = 0$, so $\vec{P}_{sys,i} = \vec{P}_{sys,f}$. Collisions: Elastic: Both momentum and kinetic energy are conserved. Inelastic: Momentum conserved, kinetic energy not conserved. Perfectly Inelastic: Objects stick together. Momentum conserved, max KE loss. Center of Mass: Particles: $\vec{r}_{CM} = \frac{1}{M} \sum m_i \vec{r}_i$ Continuous: $\vec{r}_{CM} = \frac{1}{M} \int \vec{r} dm$ Velocity of CM: $\vec{v}_{CM} = \frac{1}{M} \sum m_i \vec{v}_i = \vec{P}_{sys}/M$ Newton's 2nd Law for System: $\vec{F}_{net, ext} = M\vec{a}_{CM}$ 6. Rotation Angular Position: $\theta$ (radians) Angular Velocity: $\omega = d\theta/dt$ Angular Acceleration: $\alpha = d\omega/dt$ 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 & Angular: $s = r\theta$ $v = r\omega$ $a_t = r\alpha$ (tangential acceleration) $a_c = v^2/r = r\omega^2$ (centripetal acceleration) Moment of Inertia: $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous) Parallel-Axis Theorem: $I = I_{CM} + Mh^2$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$, $|\vec{\tau}| = rF \sin \phi$ 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: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})$ Angular Momentum (rigid body): $L = I\omega$ Newton's 2nd Law (Angular): $\vec{\tau}_{net} = d\vec{L}/dt$ Conservation of Angular Momentum: If $\vec{\tau}_{net, ext} = 0$, then $\vec{L}_{sys}$ is conserved. $I_i\omega_i = I_f\omega_f$. 7. Gravity Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$ ($G = 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$) Gravitational Potential Energy: $U = -G \frac{m_1 m_2}{r}$ (zero at $r=\infty$) Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's Laws: Orbits are ellipses with Sun at one focus. Equal areas swept in equal times ($\Delta A/\Delta t = L/(2m)$). $T^2 \propto r^3$ (for circular orbits, $T^2 = (\frac{4\pi^2}{GM}) r^3$). 8. Oscillations Simple Harmonic Motion (SHM): $x(t) = x_m \cos(\omega t + \phi)$ $v(t) = -\omega x_m \sin(\omega t + \phi)$ $a(t) = -\omega^2 x_m \cos(\omega t + \phi) = -\omega^2 x(t)$ Angular Frequency: $\omega = \sqrt{k/m}$ (spring-mass), $\omega = \sqrt{g/L}$ (pendulum) Frequency: $f = \omega/(2\pi)$ Period: $T = 1/f = 2\pi/\omega$ Energy in SHM: $E = K+U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kx_m^2 = \frac{1}{2}m\omega^2x_m^2$ Torsion Pendulum: $T = 2\pi \sqrt{I/\kappa}$ ($\kappa$ is torsion constant) Physical Pendulum: $T = 2\pi \sqrt{I/(mgd)}$ ($d$ is distance from pivot to CM) Damped SHM: $x(t) = x_m e^{-bt/2m} \cos(\omega't + \phi)$ Forced Oscillations & Resonance: Max amplitude when driving frequency equals natural frequency. 9. Waves (Mechanical) Wave Function: $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Wave Number: $k = 2\pi/\lambda$ Angular Frequency: $\omega = 2\pi f$ Wave Speed: $v = \lambda f = \omega/k$ Speed on Stretched String: $v = \sqrt{\tau/\mu}$ ($\tau$ tension, $\mu$ linear density) Power Transmitted: $P_{avg} = \frac{1}{2}\mu v \omega^2 y_m^2$ Principle of Superposition: $y'(x,t) = y_1(x,t) + y_2(x,t)$ Interference: Constructive: $\Delta \phi = m(2\pi)$ or $\Delta L = m\lambda$ Destructive: $\Delta \phi = (m + \frac{1}{2})(2\pi)$ or $\Delta L = (m + \frac{1}{2})\lambda$ Standing Waves on String (fixed ends): $\lambda_n = 2L/n$ $f_n = n v/(2L) = n f_1$ ($n=1,2,3,...$) Beats: $f_{beat} = |f_1 - f_2|$ 10. Sound Waves Speed of Sound: $v = \sqrt{B/\rho}$ (fluids), $v = \sqrt{Y/\rho}$ (solids) Intensity: $I = P/A$ Intensity Level (Decibels): $\beta = (10 \text{ dB}) \log_{10}(I/I_0)$, $I_0 = 10^{-12} \text{ W/m}^2$ Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D = detector, S = source; choose top sign for approaching, bottom for receding) Standing Waves in Pipes: Open-Open: $\lambda_n = 2L/n$, $f_n = n(v/2L)$ ($n=1,2,3,...$) Open-Closed: $\lambda_n = 4L/n$, $f_n = n(v/4L)$ ($n=1,3,5,...$) 11. Temperature & Heat 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: $\Delta V = V \beta \Delta T$ ($\beta \approx 3\alpha$) Heat Capacity: $Q = C \Delta T$ Specific Heat: $Q = cm \Delta T$ Latent Heat (Phase Change): $Q = L_F m$ (fusion), $Q = L_V m$ (vaporization) Heat Transfer: Conduction: $P_{cond} = k A \frac{T_H - T_C}{L}$ Radiation: $P_{rad} = \sigma \epsilon A T^4$ ($\sigma = 5.67 \times 10^{-8} \text{ W/(m}^2\text{K}^4)$, $\epsilon$ emissivity) 12. Kinetic Theory & Thermodynamics Kinetic Theory of Gases Ideal Gas Law: $PV = nRT = NkT$ ($R=8.31 \text{ J/(mol K)}$, $k=1.38 \times 10^{-23} \text{ J/K}$) Average Kinetic Energy per Molecule: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{3RT/M} = \sqrt{3kT/m}$ Internal Energy (Ideal Monatomic Gas): $E_{int} = \frac{3}{2}nRT$ Thermodynamics 1st Law of Thermodynamics: $\Delta E_{int} = Q - W$ ($Q$ heat added, $W$ work done by system) Work done by gas: $W = \int P dV$ Molar Specific Heats: Constant Volume: $Q = n C_V \Delta T$ Constant Pressure: $Q = n C_P \Delta T$ $C_P = C_V + R$ Ratio: $\gamma = C_P/C_V$ Adiabatic Process: $Q=0$. $PV^\gamma = \text{constant}$ 2nd Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of an isolated system never decreases. Entropy: $\Delta S = \int dQ/T$ (reversible), $\Delta S \ge 0$ (isolated system) Heat Engines: $\epsilon = |W|/|Q_H| = 1 - |Q_C|/|Q_H|$ Carnot Engine (Ideal): $\epsilon_C = 1 - T_C/T_H$ Refrigerators/Heat Pumps: $K = |Q_C|/|W|$ (refrigerator), $K = |Q_H|/|W|$ (heat pump) Carnot Refrigerator: $K_C = T_C/(T_H - T_C)$ 13. Electric Charge & Field Charge Quantization: $q = ne$ ($e = 1.602 \times 10^{-19} \text{ C}$) Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ ($k = 1/(4\pi\epsilon_0) \approx 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2$) Electric Field: $\vec{E} = \vec{F}/q_0$ Field of Point Charge: $E = k |q|/r^2$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ (from - to +) Torque on Dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\Phi_E = \oint \vec{E} \cdot d\vec{A} = Q_{enc}/\epsilon_0$ Field for common geometries: Infinite line of charge: $E = \lambda / (2\pi\epsilon_0 r)$ Infinite non-conducting sheet: $E = \sigma / (2\epsilon_0)$ Conducting sheet (surface): $E = \sigma / \epsilon_0$ Spherical shell (outside): $E = kQ/r^2$ Spherical shell (inside): $E=0$ 14. Electric Potential Potential Energy: $\Delta U = -W = - \int \vec{F} \cdot d\vec{s}$ Electric Potential: $V = U/q_0$ Potential Difference: $\Delta V = V_f - V_i = - \int_i^f \vec{E} \cdot d\vec{s}$ Potential of Point Charge: $V = kq/r$ Potential of Dipole (far field): $V = \frac{kp \cos\theta}{r^2}$ Relating E-field & Potential: $\vec{E} = -\nabla V = -(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k})$ Equipotential Surfaces: Perpendicular to E-field lines. 15. Capacitance Capacitance: $C = Q/V$ Parallel Plate Capacitor: $C = \epsilon_0 A/d$ Cylindrical Capacitor: $C = 2\pi\epsilon_0 L / \ln(b/a)$ Spherical Capacitor: $C = 4\pi\epsilon_0 \frac{ab}{b-a}$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Capacitors in Series: $1/C_{eq} = \sum (1/C_i)$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{1}{2}Q^2/C = \frac{1}{2}QV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Dielectrics: $C = \kappa C_{air}$, $E = E_{air}/\kappa$, $V = V_{air}/\kappa$ 16. Current & Resistance Electric Current: $I = dq/dt$ Current Density: $\vec{J} = n e \vec{v}_d$ ($n$ charge carriers per unit volume, $\vec{v}_d$ drift velocity) Ohm's Law (Microscopic): $\vec{E} = \rho \vec{J}$ Resistance: $R = V/I$ Resistivity: $\rho = E/J = RA/L$ Resistance Temperature Dependence: $\rho - \rho_0 = \rho_0 \alpha (T - T_0)$ Power: $P = IV = I^2R = V^2/R$ 17. Circuits EMF: $\mathcal{E}$ (ideal battery) Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $1/R_{eq} = \sum (1/R_i)$ Kirchhoff's Rules: Junction Rule: $\sum I_{in} = \sum I_{out}$ Loop Rule: $\sum \Delta V = 0$ (around any closed loop) RC Circuits (Charging Capacitor): $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = (\mathcal{E}/R)e^{-t/RC}$ RC Circuits (Discharging Capacitor): $q(t) = Q_0 e^{-t/RC}$, $I(t) = -(Q_0/RC)e^{-t/RC}$ Time Constant: $\tau = RC$ 18. Magnetic Fields Lorentz Force: $\vec{F} = q\vec{v} \times \vec{B}$ (on charge), $\vec{F} = I\vec{L} \times \vec{B}$ (on current-carrying wire) Magnetic Force on Current Loop (Torque): $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NIA\hat{n}$ Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = I B / (ne L)$ Circular Motion in B-field: $r = mv/(|q|B)$, $T = 2\pi m/(|q|B)$ 19. Magnetic Fields from Currents Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ ($\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$) Field of Long Straight Wire: $B = \mu_0 I / (2\pi r)$ Force between Parallel Wires: $F/L = \mu_0 I_1 I_2 / (2\pi d)$ Field at Center of Arc: $B = \frac{\mu_0 I \phi}{4\pi R}$ ($\phi$ in radians) Field at Center of Loop: $B = \mu_0 I / (2R)$ Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Solenoid: $B = \mu_0 n I$ ($n$ turns per unit length) Toroid: $B = \mu_0 N I / (2\pi r)$ 20. Induction & Inductance Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law of Induction: $\mathcal{E} = -d\Phi_B/dt$ Lenz's Law: Induced current opposes the change in magnetic flux. Motional EMF: $\mathcal{E} = (BLv)\sin\theta$ Inductance: $L = N\Phi_B/I$ Solenoid Inductance: $L = \mu_0 n^2 A l$ Self-Induced EMF: $\mathcal{E}_L = -L dI/dt$ RL Circuits (Current Growth): $I(t) = (\mathcal{E}/R)(1 - e^{-t/\tau_L})$ 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: $\mathcal{E}_2 = -M dI_1/dt$ 21. Electromagnetic Oscillations & AC LC Oscillations: Angular Frequency: $\omega = 1/\sqrt{LC}$ Charge: $q(t) = Q \cos(\omega t + \phi)$ Current: $i(t) = - \omega Q \sin(\omega t + \phi)$ Damped RLC Circuit: $q(t) = Q e^{-Rt/2L} \cos(\omega't + \phi)$, $\omega' = \sqrt{\omega^2 - (R/2L)^2}$ AC Circuits (Series RLC): Inductive Reactance: $X_L = \omega L$ Capacitive Reactance: $X_C = 1/(\omega C)$ Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Current Amplitude: $I = V_m/Z$ Phase Angle: $\tan \phi = (X_L - X_C)/R$ ($V_L$ leads $I$, $V_C$ lags $I$) Power in AC Circuits: Average Power: $P_{avg} = I_{rms} V_{rms} \cos \phi = I_{rms}^2 R$ RMS Values: $V_{rms} = V_m/\sqrt{2}$, $I_{rms} = I_m/\sqrt{2}$ Power Factor: $\cos \phi$ Resonance: $\omega_R = 1/\sqrt{LC}$ (where $X_L = X_C$, $Z=R$, $\phi=0$) 22. Maxwell's Equations & EM Waves Gauss' Law for Electricity: $\oint \vec{E} \cdot d\vec{A} = Q_{enc}/\epsilon_0$ Gauss' Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$ (no magnetic monopoles) Faraday's Law: $\oint \vec{E} \cdot d\vec{s} = -d\Phi_B/dt$ Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 d\Phi_E/dt$ Displacement Current: $I_d = \epsilon_0 d\Phi_E/dt$ Speed of Light: $c = 1/\sqrt{\mu_0 \epsilon_0} \approx 3 \times 10^8 \text{ m/s}$ EM Waves: $E = E_m \sin(kx - \omega t)$, $B = B_m \sin(kx - \omega t)$ Relationship E and B: $E=cB$ Intensity of EM Wave: $I = P_{avg}/A = \frac{1}{c\mu_0} E_{rms}^2 = \frac{c}{\mu_0} B_{rms}^2 = \frac{E_{rms}B_{rms}}{\mu_0}$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 23. Optics: Reflection & 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: $n = c/v$ Total Internal Reflection: Occurs when $n_1 > n_2$ and $\theta_1 > \theta_c$, where $\sin\theta_c = n_2/n_1$ 24. Optics: Mirrors & Lenses Mirror Equation: $1/p + 1/i = 1/f$ Lens Equation: $1/p + 1/i = 1/f$ Magnification: $m = -i/p = h'/h$ Focal Length: Concave Mirror: $f = R/2$ (real focus) Convex Mirror: $f = -R/2$ (virtual focus) Converging Lens: $f > 0$ Diverging Lens: $f Sign Conventions (Standard): $p$: + if real object (in front of mirror/lens) $i$: + if real image (in front of mirror, behind lens) $f$: + for converging (concave mirror, converging lens) $h'$: + if upright Lensmaker's Equation: $1/f = (n-1)(1/r_1 - 1/r_2)$ Optical Instruments: Magnifying Glass: $m_\theta = (25 \text{ cm})/f$ (relaxed eye) Compound Microscope: $M = m_o m_e = (-L/f_o)((25 \text{ cm})/f_e)$ Refracting Telescope: $m_\theta = -f_o/f_e$ 25. Interference Young's Double-Slit Experiment: Path Difference: $\Delta L = d \sin\theta$ Bright Fringes (Max): $d \sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Dark Fringes (Min): $d \sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2, ...$) For small $\theta$: $y = L \tan\theta \approx L \sin\theta$ Approx. Bright Fringes: $y_m = m\frac{\lambda L}{d}$ Thin-Film Interference: Phase change upon reflection: $\pi$ rad (when reflecting from higher $n$) Optical Path Difference (OPD): $2n_2d$ Constructive Interference ($2n_2d = (m+\frac{1}{2})\lambda$ or $m\lambda$ depending on phase shifts) Destructive Interference ($2n_2d = m\lambda$ or $(m+\frac{1}{2})\lambda$ depending on phase shifts) Michelson Interferometer: $\Delta L = m\lambda/2$ for $m$ fringes (change in arm length) 26. Diffraction Single-Slit Diffraction: Minima: $a \sin\theta = m\lambda$ ($m=\pm 1, \pm 2, ...$) Central Max width: $2\lambda L/a$ Diffraction Grating: Maxima: $d \sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Rayleigh's Criterion (Resolution): $\theta_R = 1.22 \lambda/D$ (circular aperture) X-Ray Diffraction (Bragg's Law): $2d \sin\theta = m\lambda$ ($d$ is spacing between crystal planes) 27. Relativity Postulates: Principle of Relativity: Laws of physics are same in all inertial frames. Speed of Light: $c$ is same for all inertial observers. Lorentz Factor: $\gamma = 1/\sqrt{1 - (v/c)^2}$ Length Contraction: $L = L_0/\gamma$ Time Dilation: $\Delta t = \gamma \Delta t_0$ Relativistic Momentum: $\vec{p} = \gamma m\vec{v}$ Relativistic Energy: $E = \gamma mc^2 = K + mc^2$ Rest Energy: $E_0 = mc^2$ Kinetic Energy: $K = (\gamma - 1)mc^2$ Momentum-Energy Relation: $E^2 = (pc)^2 + (mc^2)^2$ 28. Quantum Physics Planck's Constant: $h = 6.626 \times 10^{-34} \text{ J s}$ Photon Energy: $E = hf = hc/\lambda$ Photoelectric Effect: $K_{max} = hf - \Phi$ ($\Phi$ work function) Compton Effect: $\Delta\lambda = \lambda' - \lambda = (h/mc)(1 - \cos\phi)$ De Broglie Wavelength: $\lambda = h/p$ Heisenberg Uncertainty Principle: $\Delta x \Delta p_x \ge \hbar/2$ $\Delta E \Delta t \ge \hbar/2$ ($\hbar = h/(2\pi)$) Schrödinger Equation (Time-Independent 1D): $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ Probability Density: $|\psi(x)|^2$ Normalization: $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$ Particle in a Box (1D, infinite well): Energy Levels: $E_n = \frac{h^2 n^2}{8mL^2}$ ($n=1,2,3...$) Wavefunctions: $\psi_n(x) = \sqrt{2/L} \sin(n\pi x/L)$ Quantum Numbers: $n$ (principal), $l$ (orbital), $m_l$ (magnetic), $m_s$ (spin) Pauli Exclusion Principle: No two electrons can have the same set of four quantum numbers.