Halliday Physics Cheat Sheet

Cheatsheet Content

### Kinematics (1D & 2D) #### 1. One-Dimensional 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: - $v = v_0 + at$ - $x - x_0 = v_0 t + \frac{1}{2}at^2$ - $v^2 = v_0^2 + 2a(x - x_0)$ - $x - x_0 = \frac{1}{2}(v_0 + v)t$ #### 2. Two-Dimensional Motion (Projectile Motion) - **Position Vector:** $\vec{r} = x\hat{i} + y\hat{j}$ - **Velocity Vector:** $\vec{v} = v_x\hat{i} + v_y\hat{j}$ - **Acceleration Vector:** $\vec{a} = a_x\hat{i} + a_y\hat{j}$ ##### Projectile Motion (Horizontal $a_x=0$, Vertical $a_y=-g$): - $x = (v_0 \cos\theta_0)t$ - $y = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ - $v_x = v_0 \cos\theta_0$ - $v_y = v_0 \sin\theta_0 - gt$ - **Range:** $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ - **Max Height:** $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ ### Newton's Laws of Motion #### 1. First 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. #### 2. Second Law - $\sum \vec{F} = m\vec{a}$ - **Weight:** $W = mg$ (force due to gravity) #### 3. Third Law (Action-Reaction) - If object A exerts a force on object B, then object B must exert a force of equal magnitude and opposite direction back on object A. $\vec{F}_{AB} = -\vec{F}_{BA}$ ##### Common Forces: - **Normal Force ($F_N$):** Perpendicular to surface - **Friction Force ($f_s, f_k$):** Parallel to surface, opposes motion - **Static Friction:** $f_s \le \mu_s F_N$ - **Kinetic Friction:** $f_k = \mu_k F_N$ - **Tension ($T$):** Force transmitted through a rope/cable ### Work and Energy #### 1. Work - **Constant Force:** $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$ - **Variable Force (1D):** $W = \int_{x_i}^{x_f} F(x) dx$ - **Work-Kinetic Energy Theorem:** $W_{net} = \Delta K = K_f - K_i$ #### 2. Kinetic Energy - **Translational:** $K = \frac{1}{2}mv^2$ #### 3. Potential Energy - **Gravitational:** $U_g = mgh$ - **Elastic (Spring):** $U_s = \frac{1}{2}kx^2$ (Hooke's Law: $F_s = -kx$) #### 4. Conservation of Energy - **Conservative Forces Only:** $E_{mech} = K + U = \text{constant}$ - $K_i + U_i = K_f + U_f$ - **With Non-Conservative Forces:** $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ #### 5. Power - **Average Power:** $P_{avg} = \frac{W}{\Delta t}$ - **Instantaneous Power:** $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ ### Momentum and Collisions #### 1. Momentum - **Linear Momentum:** $\vec{p} = m\vec{v}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta \vec{p}$ - **Impulse-Momentum Theorem:** $\vec{J} = \vec{p}_f - \vec{p}_i$ #### 2. Conservation of Momentum - If net external force is zero: $\sum \vec{p}_i = \sum \vec{p}_f$ - **Center of Mass (CM):** - Position: $\vec{r}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$ - Velocity: $\vec{v}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i} = \frac{\vec{P}_{total}}{M_{total}}$ #### 3. Collisions - **Elastic Collision:** Both momentum and kinetic energy are conserved. - $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$ - $\frac{1}{2}m_1 v_{1i}^2 + \frac{1}{2}m_2 v_{2i}^2 = \frac{1}{2}m_1 v_{1f}^2 + \frac{1}{2}m_2 v_{2f}^2$ - **Inelastic Collision:** Momentum conserved, kinetic energy NOT conserved. - **Perfectly Inelastic Collision:** Objects stick together after collision. - $m_1 v_{1i} + m_2 v_{2i} = (m_1 + m_2)v_f$ ### Rotational Motion #### 1. Rotational Kinematics - **Angular Position:** $\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ ##### Constant Angular Acceleration Equations: - $\omega = \omega_0 + \alpha t$ - $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ #### 2. Relationship between Linear and Angular Variables - $s = r\theta$ - $v_t = r\omega$ (tangential speed) - $a_t = r\alpha$ (tangential acceleration) - $a_c = \frac{v_t^2}{r} = r\omega^2$ (centripetal acceleration) #### 3. Rotational Dynamics - **Torque:** $\tau = rF\sin\phi = r_{\perp}F = r F_{\perp}$ (where $\phi$ is angle between $\vec{r}$ and $\vec{F}$) - **Newton's Second Law for Rotation:** $\sum \tau = I\alpha$ - **Moment of Inertia ($I$):** $I = \sum m_i r_i^2$ (for point masses) or $I = \int r^2 dm$ - **Common I values:** - Hoop/Thin Ring: $MR^2$ - Solid Cylinder/Disc: $\frac{1}{2}MR^2$ - Solid Sphere: $\frac{2}{5}MR^2$ - Thin Rod (center): $\frac{1}{12}ML^2$ - Thin Rod (end): $\frac{1}{3}ML^2$ - **Parallel-Axis Theorem:** $I = I_{CM} + Md^2$ #### 4. Rotational Work & Energy - **Work done by Torque:** $W = \int_{\theta_i}^{\theta_f} \tau d\theta$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Total Kinetic Energy (rolling):** $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$ #### 5. Angular Momentum - **For a particle:** $\vec{l} = \vec{r} \times \vec{p} = r p \sin\phi$ - **For a rigid body:** $L = I\omega$ - **Newton's Second Law (Angular):** $\sum \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$ - **Conservation of Angular Momentum:** If $\sum \vec{\tau}_{ext} = 0$, then $\vec{L}_{total} = \text{constant}$ - $I_i \omega_i = I_f \omega_f$ ### Gravitation - **Newton's Law of Universal Gravitation:** $F = G\frac{m_1 m_2}{r^2}$ - $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ - **Gravitational Potential Energy:** $U = -G\frac{m_1 m_2}{r}$ - **Escape Speed:** $v_{esc} = \sqrt{\frac{2GM}{R}}$ - **Orbital Speed (circular orbit):** $v_{orbit} = \sqrt{\frac{GM}{r}}$ - **Kepler's Laws:** 1. Orbits are ellipses with the Sun at one focus. 2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3. $T^2 \propto a^3$ (Period squared is proportional to semi-major axis cubed). For circular orbits, $T^2 = (\frac{4\pi^2}{GM})r^3$. ### Oscillations #### 1. Simple Harmonic Motion (SHM) - **Condition for SHM:** Restoring force proportional to displacement ($F = -kx$) - **Displacement:** $x(t) = A\cos(\omega t + \phi)$ - **Velocity:** $v(t) = -\omega A\sin(\omega t + \phi)$ - **Acceleration:** $a(t) = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x(t)$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (spring-mass) - **Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Frequency:** $f = \frac{1}{T} = \frac{\omega}{2\pi}$ #### 2. Energy in SHM - **Total Mechanical Energy:** $E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ #### 3. Pendulum - **Simple Pendulum (small angles):** $T = 2\pi\sqrt{\frac{L}{g}}$ - **Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgd}}$ (where $d$ is distance from pivot to CM) ### Waves #### 1. Transverse and Longitudinal Waves - **Transverse:** Oscillation perpendicular to wave direction (e.g., light, waves on a string) - **Longitudinal:** Oscillation parallel to wave direction (e.g., sound) #### 2. Wave Characteristics - **Wave Speed:** $v = \lambda f$ - **Wave on a String:** $v = \sqrt{\frac{T}{\mu}}$ (where $\mu$ is linear mass density) - **Sound Wave Speed:** $v = \sqrt{\frac{B}{\rho}}$ (Bulk modulus $B$, density $\rho$) - **Intensity:** $I = \frac{P}{A}$ (Power $P$, Area $A$) - **Intensity Level (Decibels):** $\beta = (10 \text{ dB})\log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) #### 3. Superposition and Interference - **Principle of Superposition:** When waves overlap, resultant displacement is sum of individual displacements. - **Constructive Interference:** Path difference = $n\lambda$ (n=0, 1, 2...) - **Destructive Interference:** Path difference = $(n + \frac{1}{2})\lambda$ (n=0, 1, 2...) #### 4. Standing Waves - **On a String (fixed ends):** - Wavelengths: $\lambda_n = \frac{2L}{n}$ ($n=1, 2, 3...$) - Frequencies: $f_n = n\frac{v}{2L} = n f_1$ ($f_1$ is fundamental frequency) - **In an Open-Open Pipe:** Same as string. - **In a Closed-Open Pipe:** - Wavelengths: $\lambda_n = \frac{4L}{n}$ ($n=1, 3, 5...$) - Frequencies: $f_n = n\frac{v}{4L} = n f_1$ #### 5. Doppler Effect - **Approaching Source/Observer:** Higher frequency - **Receding Source/Observer:** Lower frequency - **General Formula:** $f' = f \left(\frac{v \pm v_D}{v \mp v_S}\right)$ - $v$: speed of sound - $v_D$: speed of detector (observer) - $v_S$: speed of source - Top signs (+ for $v_D$, - for $v_S$) for "towards," bottom signs (- for $v_D$, + for $v_S$) for "away." ### Thermodynamics #### 1. Temperature and Heat - **Temperature Scales:** - Celsius to Kelvin: $T_K = T_C + 273.15$ - Fahrenheit to Celsius: $T_C = \frac{5}{9}(T_F - 32)$ - **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:** $Q = mc\Delta T$ - **Latent Heat (Phase Change):** $Q = mL$ ($L_f$ for fusion, $L_v$ for vaporization) #### 2. Heat Transfer - **Conduction:** $P_{cond} = kA\frac{T_H - T_C}{L}$ - **Convection:** Heat transfer via fluid motion. - **Radiation:** $P_{rad} = \sigma A e T^4$ (Stefan-Boltzmann Law) - $\sigma = 5.67 \times 10^{-8} \text{ W/(m}^2\cdot\text{K}^4)$, $e$ is emissivity. #### 3. First Law of Thermodynamics - **Statement:** $\Delta E_{int} = Q - W$ - $\Delta E_{int}$: Change in internal energy - $Q$: Heat added to system - $W$: Work done BY system - **Work for a gas:** $W = \int P dV$ - **Processes:** - **Isobaric:** Constant pressure, $W = P\Delta V$ - **Isochoric:** Constant volume, $W = 0 \implies \Delta E_{int} = Q$ - **Isothermal:** Constant temperature, $\Delta E_{int} = 0 \implies Q = W$ (for ideal gas) - **Adiabatic:** No heat exchange, $Q = 0 \implies \Delta E_{int} = -W$ #### 4. Kinetic Theory of Gases - **Ideal Gas Law:** $PV = nRT = NkT$ - $R = 8.314 \text{ J/(mol}\cdot\text{K})$ (gas constant) - $k = 1.38 \times 10^{-23} \text{ J/K}$ (Boltzmann constant) - **Internal Energy of Monatomic Ideal Gas:** $E_{int} = \frac{3}{2}nRT = \frac{3}{2}NkT$ - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M}}$ (M is molar mass in kg/mol) #### 5. Second Law of Thermodynamics - **Statement (Entropy):** The entropy of an isolated system never decreases. $\Delta S \ge 0$ - **Entropy Change:** - Reversible: $\Delta S = \int \frac{dQ}{T}$ - Isothermal: $\Delta S = \frac{Q}{T}$ - **Heat Engines:** - Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ - Carnot Efficiency: $\epsilon_C = 1 - \frac{T_C}{T_H}$ - **Refrigerators/Heat Pumps:** - Coefficient of Performance (COP): $K = \frac{|Q_C|}{|W|}$ (refrigerator), $K = \frac{|Q_H|}{|W|}$ (heat pump) - Carnot COP: $K_C = \frac{T_C}{T_H - T_C}$ (refrigerator), $K_C = \frac{T_H}{T_H - T_C}$ (heat pump) ### Electricity #### 1. Electric Charge & Force - **Quantization of Charge:** $q = ne$ ($e = 1.602 \times 10^{-19} \text{ C}$) - **Coulomb's Law:** $F = k\frac{|q_1 q_2|}{r^2}$ - $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ - $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ #### 2. Electric Field - **Definition:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Point Charge:** $E = k\frac{|q|}{r^2}$ - **Electric Dipole Moment:** $\vec{p} = q\vec{d}$ - **Torque on Dipole:** $\vec{\tau} = \vec{p} \times \vec{E}$ - **Potential Energy of Dipole:** $U = -\vec{p} \cdot \vec{E}$ #### 3. Gauss's Law - **Statement:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - **Applications:** - **Infinite Line of Charge:** $E = \frac{\lambda}{2\pi\epsilon_0 r}$ - **Infinite Nonconducting Sheet:** $E = \frac{\sigma}{2\epsilon_0}$ - **Conducting Sheet/Surface:** $E = \frac{\sigma}{\epsilon_0}$ - **Spherical Shell (outside):** $E = k\frac{Q}{r^2}$ - **Spherical Shell (inside):** $E = 0$ #### 4. Electric Potential - **Definition:** $V = \frac{U}{q_0}$ - **Relationship E and V:** $E_x = -\frac{\partial V}{\partial x}$, $\vec{E} = -\nabla V$ - **Potential Difference:** $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ - **Point Charge:** $V = k\frac{q}{r}$ - **Potential Energy of Two Point Charges:** $U = k\frac{q_1 q_2}{r}$ #### 5. Capacitance - **Definition:** $C = \frac{Q}{V}$ - **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:** $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ - **Dielectrics:** $C = \kappa C_0$ (where $\kappa$ is dielectric constant) ### DC Circuits #### 1. Current and Resistance - **Current:** $I = \frac{dQ}{dt}$ - **Current Density:** $J = \frac{I}{A}$ - **Drift Velocity:** $I = n e A v_d$ - **Resistance:** $R = \rho \frac{L}{A}$ ($\rho$ is resistivity) - **Ohm's Law:** $V = IR$ #### 2. DC Circuits - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Power Dissipation:** $P = IV = I^2 R = \frac{V^2}{R}$ #### 3. Kirchhoff's Rules - **Junction Rule:** Sum of currents entering a junction equals sum of currents leaving it. ($\sum I_{in} = \sum I_{out}$) - **Loop Rule:** Sum of potential differences around any closed loop is zero. ($\sum \Delta V = 0$) #### 4. RC Circuits - **Charging Capacitor:** $Q(t) = Q_{max}(1 - e^{-t/RC})$, $I(t) = I_{max} e^{-t/RC}$ - **Discharging Capacitor:** $Q(t) = Q_0 e^{-t/RC}$, $I(t) = I_0 e^{-t/RC}$ - **Time Constant:** $\tau = RC$ ### Magnetism #### 1. Magnetic Fields & Forces - **Magnetic Force on a Moving Charge:** $\vec{F}_B = q\vec{v} \times \vec{B}$ - Magnitude: $F_B = |q|vB\sin\phi$ - **Magnetic Force on a Current-Carrying Wire:** $\vec{F}_B = I\vec{L} \times \vec{B}$ - Magnitude: $F_B = ILB\sin\phi$ - **Torque on a Current Loop:** $\vec{\tau} = \vec{\mu} \times \vec{B}$ - Magnetic Dipole Moment: $\mu = NIA$ #### 2. Sources of Magnetic Field - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{s} \times \hat{r}}{r^2}$ - $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ (permeability of free space) - **Long Straight Wire:** $B = \frac{\mu_0 I}{2\pi r}$ - **Current Loop (center):** $B = \frac{\mu_0 I}{2R}$ - **Solenoid (inside):** $B = \mu_0 n I$ ($n$ is turns per unit length) #### 3. Ampere's Law - **Statement:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ - **Applications:** Used for highly symmetric current distributions. #### 4. Faraday's Law of Induction - **Statement:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ - Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ - **Motional EMF:** $\mathcal{E} = BLv$ (for a conductor moving perpendicular to B) #### 5. Lenz's Law - The direction of an induced current is such that it opposes the change in magnetic flux that created it. #### 6. Inductance - **Self-Inductance:** $L = \frac{N\Phi_B}{I}$ - **Solenoid Inductance:** $L = \mu_0 n^2 A l$ - **Induced EMF:** $\mathcal{E}_L = -L\frac{dI}{dt}$ - **Energy Stored in Inductor:** $U_B = \frac{1}{2}LI^2$ - **Energy Density of Magnetic Field:** $u_B = \frac{B^2}{2\mu_0}$ #### 7. RL Circuits - **Current increase:** $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ - **Current decrease:** $I(t) = I_0 e^{-t/\tau_L}$ - **Time Constant:** $\tau_L = \frac{L}{R}$ ### AC Circuits #### 1. RLC Series Circuits - **Impedance:** $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - **Inductive Reactance:** $X_L = \omega L$ - **Capacitive Reactance:** $X_C = \frac{1}{\omega C}$ - **Phase Angle:** $\tan\phi = \frac{X_L - X_C}{R}$ (voltage leads current if positive) - **RMS Values:** $V_{rms} = \frac{V_{max}}{\sqrt{2}}$, $I_{rms} = \frac{I_{max}}{\sqrt{2}}$ - **Ohm's Law for AC:** $V_{rms} = I_{rms}Z$ - **Power Factor:** $\cos\phi = \frac{R}{Z}$ - **Average Power:** $P_{avg} = I_{rms}^2 R = I_{rms} V_{rms} \cos\phi$ #### 2. Resonance - **Resonance Condition:** $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ - At resonance, $Z=R$ and $\phi=0$. ### Electromagnetic Waves #### 1. Maxwell's Equations (Integral Form) 1. **Gauss's Law for Electricity:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ 2. **Gauss's Law for Magnetism:** $\oint \vec{B} \cdot d\vec{A} = 0$ 3. **Faraday's Law:** $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ 4. **Ampere-Maxwell Law:** $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ - Displacement Current: $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$ #### 2. Properties of EM Waves - **Speed of Light:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3.00 \times 10^8 \text{ m/s}$ - **Relationship E and B:** $E = cB$ - **Wave Equation:** $\frac{\partial^2 E}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 E}{\partial t^2}$ - **Poynting Vector:** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (direction of energy flow) - **Intensity:** $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{E_{max}B_{max}}{2\mu_0}$ - **Radiation Pressure:** $P_{rad} = \frac{I}{c}$ (absorbed), $P_{rad} = \frac{2I}{c}$ (reflected) ### Optics #### 1. Reflection - **Law of Reflection:** $\theta_i = \theta_r$ - **Plane Mirrors:** Image is virtual, upright, same size, same distance behind mirror. #### 2. Refraction (Snell's Law) - $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Index of Refraction:** $n = \frac{c}{v}$ - **Critical Angle:** $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) #### 3. Thin Lenses & Spherical Mirrors - **Mirror/Lens Equation:** $\frac{1}{f} = \frac{1}{p} + \frac{1}{i}$ - $f$: focal length (+ concave mirror, converging lens; - convex mirror, diverging lens) - $p$: object distance (+ real object, - virtual object) - $i$: image distance (+ real image, - virtual image) - **Magnification:** $m = -\frac{i}{p} = \frac{h'}{h}$ - $|m| > 1$: magnified; $|m| 0$: upright; $m ### Modern Physics #### 1. Special Relativity - **Postulates:** 1. The laws of physics are the same for all inertial reference frames. 2. The speed of light in vacuum has the same value in all inertial reference frames. - **Lorentz Factor:** $\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$ - **Length Contraction:** $L = L_0/\gamma$ - **Time Dilation:** $\Delta t = \gamma \Delta t_0$ - **Relativistic Momentum:** $p = \gamma mv$ - **Relativistic Energy:** $E = \gamma mc^2 = K + mc^2$ - Rest Energy: $E_0 = mc^2$ - Kinetic Energy: $K = (\gamma - 1)mc^2$ - Energy-Momentum Relation: $E^2 = (pc)^2 + (mc^2)^2$ #### 2. Quantum Physics - **Planck's Quantum Hypothesis:** $E = hf$ - $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$ (Planck's constant) - **Photoelectric Effect:** $K_{max} = hf - \Phi$ ($\Phi$ is work function) - **Photon Momentum:** $p = \frac{h}{\lambda} = \frac{E}{c}$ - **De Broglie Wavelength:** $\lambda = \frac{h}{p}$ - **Heisenberg Uncertainty Principle:** - Position/Momentum: $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - Energy/Time: $\Delta E \Delta t \ge \frac{\hbar}{2}$ ($\hbar = h/2\pi$) - **Schrödinger Equation (1D, time-independent):** $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi = E\psi$ #### 3. Atomic Physics - **Bohr Model (Hydrogen-like atoms):** - Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}Z^2$ - Radius: $r_n = a_0 \frac{n^2}{Z}$ ($a_0 = 0.0529 \text{ nm}$ Bohr radius) - **Quantum Numbers:** - Principal (n): $1, 2, 3, ...$ (energy, size) - Orbital (l): $0, 1, ..., n-1$ (shape, angular momentum) - Magnetic ($m_l$): $-l, ..., 0, ..., +l$ (orientation) - Spin ($m_s$): $\pm \frac{1}{2}$ - **Pauli Exclusion Principle:** No two electrons in an atom can have the same set of four quantum numbers. #### 4. Nuclear Physics - **Nuclear Notation:** $_Z^A X$ (A: mass number, Z: atomic number) - **Mass Defect & Binding Energy:** $E_B = \Delta m c^2$ - **Radioactive Decay:** - Activity: $R = -\frac{dN}{dt} = \lambda N$ - Decay Law: $N(t) = N_0 e^{-\lambda t}$ - Half-life: $T_{1/2} = \frac{\ln 2}{\lambda}$