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) Luminosity: 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): $10^{-12}$ 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: $v = v_0 + at$ $x = x_0 + v_0t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ Free Fall: $a = -g \approx -9.8 \text{ m/s}^2$ (downwards) 2.2. 2D Motion (Projectile Motion) Position Vector: $\vec{r} = x\hat{i} + y\hat{j}$ Velocity Vector: $\vec{v} = v_x\hat{i} + v_y\hat{j}$ where $v_x = \frac{dx}{dt}$, $v_y = \frac{dy}{dt}$ Acceleration Vector: $\vec{a} = a_x\hat{i} + a_y\hat{j}$ where $a_x = \frac{dv_x}{dt}$, $a_y = \frac{dv_y}{dt}$ Projectile Motion (No Air Resistance): $a_x = 0$, $a_y = -g$ $v_x = v_{0x} = v_0 \cos\theta_0$ (constant) $x = (v_0 \cos\theta_0)t$ $v_y = v_{0y} - gt = v_0 \sin\theta_0 - gt$ $y = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ Max Height: $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ 3. Newton's Laws of Motion 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: $\Sigma \vec{F} = m\vec{a}$ Force is a vector: $F_x = ma_x$, $F_y = ma_y$ Unit of Force: Newton (N), $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$ 3rd 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. $\vec{F}_{AB} = -\vec{F}_{BA}$ Weight: $W = mg$ (force due to gravity) Normal Force ($F_N$): Perpendicular to surface, prevents penetration. Tension ($T$): Force transmitted through a rope/cable. Friction: Static Friction: $f_s \le \mu_s F_N$ (prevents motion) Kinetic Friction: $f_k = \mu_k F_N$ (opposes motion) $\mu_s \ge \mu_k$ 4. Work, Energy, & Power Work done by a constant force: $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ Unit: Joule (J), $1 \text{ J} = 1 \text{ N} \cdot \text{m}$ Work done by a variable force: $W = \int_{x_i}^{x_f} F_x(x) dx$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Potential Energy: Gravitational: $U_g = mgy$ Elastic (Spring): $U_s = \frac{1}{2}kx^2$ (Hooke's Law: $F_s = -kx$) Conservative Force: Work done is independent of path, depends only on initial/final positions. Associated with potential energy. (e.g., gravity, spring force) Non-Conservative Force: Work done depends on path. (e.g., friction, air resistance) Conservation of Mechanical Energy: $E_{mech} = K + U$ (if only conservative forces do work) $K_i + U_i = K_f + U_f$ Conservation of Energy (General): $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ If $W_{nc} = 0$, then $E_{mech}$ is conserved. Power: Rate at which work is done. Average: $P_{avg} = \frac{\Delta W}{\Delta t}$ Instantaneous: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ Unit: Watt (W), $1 \text{ W} = 1 \text{ J/s}$ 5. Systems of Particles, Momentum, & Collisions Center of Mass (CM): 1D: $x_{CM} = \frac{1}{M} \sum m_i x_i$ 2D/3D: $\vec{r}_{CM} = \frac{1}{M} \sum m_i \vec{r}_i$ Continuous: $x_{CM} = \frac{1}{M} \int x \, dm$ Velocity of CM: $\vec{v}_{CM} = \frac{1}{M} \sum m_i \vec{v}_i = \frac{d\vec{r}_{CM}}{dt}$ Momentum: $\vec{p} = m\vec{v}$ Unit: $\text{kg} \cdot \text{m/s}$ Newton's 2nd Law (Momentum Form): $\Sigma \vec{F}_{ext} = \frac{d\vec{P}}{dt}$, where $\vec{P} = \sum \vec{p}_i = M\vec{v}_{CM}$ Impulse: $\vec{J} = \int_{t_i}^{t_f} \vec{F} dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ Unit: $\text{N} \cdot \text{s}$ Conservation of Momentum: If $\Sigma \vec{F}_{ext} = 0$, then $\vec{P}$ is conserved ($\vec{P}_{initial} = \vec{P}_{final}$) Collisions: Elastic: Both momentum and kinetic energy are conserved. Inelastic: Momentum is conserved, but kinetic energy is NOT conserved ($K_{final} Perfectly Inelastic: Objects stick together after collision. Momentum is conserved, and $K_{final}$ is minimized. 6. Rotation Angular Position: $\theta$ (radians) Angular Displacement: $\Delta\theta = \theta_f - \theta_i$ Angular Velocity: $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$, $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$, $\alpha = \frac{d\omega}{dt}$ Constant Angular Acceleration Equations: (analogous to linear) $\omega = \omega_0 + \alpha t$ $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relating Linear and Angular Variables (for a point on a rotating object): Arc length: $s = r\theta$ Tangential speed: $v_t = r\omega$ Tangential acceleration: $a_t = r\alpha$ Centripetal acceleration: $a_c = \frac{v^2}{r} = r\omega^2$ (always points to center) Moment of Inertia ($I$): Rotational inertia. Point mass: $I = mr^2$ Discrete system: $I = \sum m_i r_i^2$ Continuous object: $I = \int r^2 \, dm$ Parallel-Axis Theorem: $I = I_{CM} + Mh^2$ Torque ($\tau$): Rotational equivalent of force. $\vec{\tau} = \vec{r} \times \vec{F}$ Magnitude: $\tau = rF\sin\phi = rF_t = r_{\perp}F$ Newton's 2nd Law for Rotation: $\Sigma \tau = I\alpha$ Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Total Kinetic Energy (rolling): $K_{tot} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$ Angular Momentum ($\vec{L}$): Point particle: $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m\vec{v}$ Rigid body: $L = I\omega$ Newton's 2nd Law for Rotation (Angular Momentum Form): $\Sigma \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$ Conservation of Angular Momentum: If $\Sigma \vec{\tau}_{ext} = 0$, then $\vec{L}$ is conserved ($L_{initial} = L_{final}$). 7. Gravity 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 Acceleration ($g$): $g = G\frac{M}{r^2}$ (at surface of Earth, $r=R_E$) Gravitational Potential Energy: $U = -G\frac{m_1 m_2}{r}$ (zero at infinity) Escape Speed: $v_{esc} = \sqrt{\frac{2GM}{R}}$ Kepler's Laws: 1st: Orbits are ellipses with the Sun at one focus. 2nd: A line joining a planet and the Sun sweeps out equal areas in equal times. ($\frac{dL}{dt} = 0$, conservation of angular momentum) 3rd: $T^2 \propto a^3$ (for circular orbits, $T^2 = (\frac{4\pi^2}{GM})r^3$) 8. 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: $T = 2\pi\sqrt{\frac{I}{mgd}}$ ($d$ is distance from pivot to CM) Energy in SHM: $E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kx_m^2$ (conserved) Damped SHM: $x(t) = x_m e^{-bt/(2m)} \cos(\omega' t + \phi)$ Forced Oscillations & Resonance: When driving frequency matches natural frequency, amplitude is maximum. 9. Waves (Mechanical) General Wave Equation (1D): $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Wave Speed: $v = \lambda f = \frac{\omega}{k}$ Angular Wave Number: $k = \frac{2\pi}{\lambda}$ Speed of Wave on a String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ is tension, $\mu$ is linear mass density) Power of a Traveling Wave: $P = \frac{1}{2}\mu v \omega^2 y_m^2$ Interference: Constructive: $\Delta L = n\lambda$ ($\Delta\phi = 2\pi n$) Destructive: $\Delta L = (n + \frac{1}{2})\lambda$ ($\Delta\phi = (2n+1)\pi$) Standing Waves (on a string fixed at both ends): $\lambda_n = \frac{2L}{n}$ $f_n = n\frac{v}{2L} = nf_1$ (harmonics) Nodes at ends, antinodes in middle. Sound Waves: Longitudinal mechanical waves. Speed of Sound: $v = \sqrt{\frac{B}{\rho}}$ (fluids), $v = \sqrt{\frac{Y}{\rho}}$ (solids) Intensity: $I = \frac{P}{A} = \frac{P}{4\pi r^2}$ (for spherical wave) Sound Level: $\beta = (10 \text{ dB}) \log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (+ for detector moving towards, - for source moving towards) Standing Waves in Pipes: Open at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = n\frac{v}{2L}$ (all harmonics present) Closed at one end: $\lambda_n = \frac{4L}{n}$ ($n=1,3,5,...$), $f_n = n\frac{v}{4L}$ (only odd harmonics) 10. 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 $\beta = 3\alpha$ 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) $N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$ (Avogadro's number) Kinetic Theory of Gases: Average Kinetic Energy: $K_{avg} = \frac{3}{2}kT$ (for monatomic gas) RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ ($M$ is molar mass in kg/mol) Heat ($Q$): Energy transferred due to temperature difference. Specific Heat: $Q = mc\Delta T$ Latent Heat (Phase Change): $Q = mL$ ($L_f$ for fusion, $L_v$ for vaporization) Mechanisms of Heat Transfer: Conduction: $P_{cond} = kA\frac{dT}{dx}$ (Fourier's Law) Convection: Transfer by fluid motion. Radiation: $P_{rad} = \sigma \epsilon A T^4$ (Stefan-Boltzmann Law) $\sigma = 5.67 \times 10^{-8} \text{ W/(m}^2\cdot\text{K}^4)$ (Stefan-Boltzmann constant) First Law of Thermodynamics: $\Delta E_{int} = Q - W$ $\Delta E_{int}$: Change in internal energy (state function) $Q$: Heat added to system $W$: Work done BY system ($W = \int P \, dV$) For ideal gas: $\Delta E_{int} = nC_V\Delta T$ Thermodynamic Processes: Isobaric: $P = \text{constant}$, $W = P\Delta V$ Isochoric: $V = \text{constant}$, $W = 0$, $\Delta E_{int} = Q$ Isothermal: $T = \text{constant}$, $\Delta E_{int} = 0$, $Q = W = nRT \ln(\frac{V_f}{V_i})$ Adiabatic: $Q = 0$, $\Delta E_{int} = -W$, $PV^\gamma = \text{constant}$ $\gamma = C_P/C_V$ (adiabatic index) Molar Specific Heats of Ideal Gases: $C_P - C_V = R$ Monatomic: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$ Diatomic: $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$ (at room temp) Second Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of an isolated system never decreases. Heat Engines: Efficiency: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ Carnot Efficiency: $\epsilon_C = 1 - \frac{T_C}{T_H}$ (max possible) Refrigerators/Heat Pumps: Coefficient of Performance (Refrigerator): $K = \frac{|Q_C|}{|W|} = \frac{|Q_C|}{|Q_H| - |Q_C|}$ Carnot COP: $K_C = \frac{T_C}{T_H - T_C}$ Entropy: $\Delta S = \int \frac{dQ_{rev}}{T}$ For reversible process: $\Delta S = \frac{Q}{T}$ For isolated system: $\Delta S \ge 0$ 11. Electric Fields Coulomb's Law: $F = k\frac{|q_1 q_2|}{r^2}$ $k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/(\text{N}\cdot\text{m}^2)$ (permittivity of free space) Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ Point Charge: $E = k\frac{|q|}{r^2}$ (radially outward for +q, inward for -q) Superposition Principle: $\vec{E}_{net} = \sum \vec{E}_i$ Electric Dipole: Dipole Moment: $\vec{p} = q\vec{d}$ (from -q to +q) Torque on Dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy: $U = -\vec{p} \cdot \vec{E}$ Electric Field Lines: Originate on + charges, terminate on - charges. Density proportional to field strength. 12. Gauss' Law Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ Conductors: $E=0$ inside. Any excess charge resides on surface. E-field just outside is perpendicular to surface. Applications: Infinite Line of Charge: $E = \frac{\lambda}{2\pi\epsilon_0 r}$ Infinite Non-Conducting Sheet: $E = \frac{\sigma}{2\epsilon_0}$ Parallel Plate Capacitor: $E = \frac{\sigma}{\epsilon_0}$ (between plates) 13. Electric Potential Electric Potential Energy: $\Delta U = -W = - \int \vec{F} \cdot d\vec{s} = -q_0 \int \vec{E} \cdot d\vec{s}$ Electric Potential: $V = \frac{U}{q_0}$ $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ Unit: Volt (V), $1 \text{ V} = 1 \text{ J/C}$ Potential due to Point Charge: $V = k\frac{q}{r}$ Potential due to System of Charges: $V = \sum V_i = \sum k\frac{q_i}{r_i}$ Relating E-field and Potential: $\vec{E} = -\nabla V = -(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k})$ Potential Energy of a System of Point Charges: $U = \sum_{pairs} k\frac{q_i q_j}{r_{ij}}$ Equipotential Lines/Surfaces: Perpendicular to E-field lines. No work done moving charge along them. 14. Capacitance Capacitance: $C = \frac{q}{V}$ Unit: Farad (F), $1 \text{ F} = 1 \text{ C/V}$ Parallel-Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Energy Stored in Capacitor: $U = \frac{1}{2}CV^2 = \frac{q^2}{2C} = \frac{1}{2}qV$ Energy Density of Electric Field: $u_E = \frac{1}{2}\epsilon_0 E^2$ Capacitors in Parallel: $C_{eq} = C_1 + C_2 + ...$ Capacitors in Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ Dielectrics: Material inserted between plates. $C = \kappa C_0$ ($\kappa$ is dielectric constant) $E = E_0/\kappa$ 15. Current & Resistance Electric Current: $I = \frac{dq}{dt}$ Unit: Ampere (A), $1 \text{ A} = 1 \text{ C/s}$ Current Density: $\vec{J} = nq\vec{v}_d$ ($n$ is charge carrier density, $v_d$ is drift speed) $I = \int \vec{J} \cdot d\vec{A}$ Resistance: $R = \frac{V}{I}$ (Ohm's Law) Unit: Ohm ($\Omega$), $1 \Omega = 1 \text{ V/A}$ Resistivity: $\rho = \frac{E}{J}$ Resistance from Resistivity: $R = \rho \frac{L}{A}$ Temperature Dependence: $\rho - \rho_0 = \rho_0 \alpha(T - T_0)$ Power in Electric Circuits: $P = IV = I^2R = \frac{V^2}{R}$ 16. DC Circuits EMF ($\mathcal{E}$): "Electromotive force" (voltage source) 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: Junction Rule: $\Sigma I_{in} = \Sigma I_{out}$ (conservation of charge) Loop Rule: $\Sigma \Delta V = 0$ (conservation of energy) RC Circuits: Charging a Capacitor: $q(t) = C\mathcal{E}(1 - e^{-t/RC})$ $I(t) = \frac{\mathcal{E}}{R} e^{-t/RC}$ Discharging a Capacitor: $q(t) = Q_0 e^{-t/RC}$ $I(t) = -\frac{Q_0}{RC} e^{-t/RC}$ Time Constant: $\tau = RC$ 17. Magnetic Fields Magnetic Force on a Moving Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ Magnitude: $F_B = |q|vB\sin\theta$ Direction by right-hand rule. Unit: Tesla (T), $1 \text{ T} = 1 \text{ N}/(\text{A}\cdot\text{m})$ Magnetic force does no work. Motion of Charged Particle in Uniform B-field: Circular path (if $\vec{v} \perp \vec{B}$): $r = \frac{mv}{|q|B}$ Angular frequency (cyclotron freq.): $\omega = \frac{|q|B}{m}$ Magnetic Force on a Current-Carrying Wire: $\vec{F}_B = I\vec{L} \times \vec{B}$ Magnitude: $F_B = ILB\sin\theta$ Torque on a Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NIA\hat{n}$ (for N turns, area A) Potential Energy: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = \frac{IB}{net}$ (measures charge carrier density $n$) 18. Magnetic Fields from Currents Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ (permeability of free space) Magnetic Field of a Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Force Between Two Parallel Currents: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ (attractive if same direction) Magnetic Field at Center of Circular Arc: $B = \frac{\mu_0 I \phi}{4\pi R}$ ($\phi$ in radians) Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ Applications of Ampere's Law: Long Solenoid: $B = \mu_0 nI$ ($n$ is turns per unit length) (inside) Toroid: $B = \frac{\mu_0 N I}{2\pi r}$ (inside) 19. Induction & Inductance Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Unit: Weber (Wb), $1 \text{ Wb} = 1 \text{ T}\cdot\text{m}^2$ Faraday's Law of Induction: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Lenz's Law: The induced current/EMF opposes the change in magnetic flux. Motional EMF: $\mathcal{E} = BLv$ (for a conductor of length L moving with speed v perpendicular to B) Induced Electric Field: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (non-conservative E-field) Inductance: $L = \frac{N\Phi_B}{I}$ Unit: Henry (H), $1 \text{ H} = 1 \text{ Wb/A}$ Solenoid Inductance: $L = \mu_0 n^2 A l$ Self-Induced EMF: $\mathcal{E}_L = -L\frac{dI}{dt}$ RL Circuits: Current build-up: $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-t/\tau_L})$ 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: $M = \frac{N_2 \Phi_{B2}}{I_1}$ and $\mathcal{E}_2 = -M\frac{dI_1}{dt}$ 20. Electromagnetic Oscillations & AC Circuits LC Oscillations: Angular Frequency: $\omega = \frac{1}{\sqrt{LC}}$ Charge: $q(t) = Q \cos(\omega t + \phi)$ Current: $I(t) = -\omega Q \sin(\omega t + \phi)$ Energy: $U = U_E + U_B = \frac{q^2}{2C} + \frac{1}{2}LI^2 = \frac{Q^2}{2C}$ (conserved) Damped RLC Oscillations: Angular Frequency: $\omega' = \sqrt{\frac{1}{LC} - (\frac{R}{2L})^2}$ AC Circuits (Series RLC): Reactance: $X_L = \omega L$ (inductive), $X_C = \frac{1}{\omega C}$ (capacitive) Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Peak Current: $I_m = \frac{\mathcal{E}_m}{Z}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ ($\mathcal{E}$ leads $I$ if $\phi>0$) Resonance: $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ ($Z=R$, $I_m$ max) Power Factor: $\cos\phi = R/Z$ Average Power: $P_{avg} = I_{rms}^2 R = \mathcal{E}_{rms} I_{rms} \cos\phi$ RMS Values: $X_{rms} = X_m/\sqrt{2}$ Transformers: $\frac{V_S}{V_P} = \frac{N_S}{N_P}$ (ideal) 21. Electromagnetic Waves Maxwell's Equations (integral form): $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ (Gauss' Law for E-fields) $\oint \vec{B} \cdot d\vec{A} = 0$ (Gauss' Law for B-fields) $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ (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) Speed of Light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Properties of EM Waves: Transverse waves: $\vec{E} \perp \vec{B} \perp \vec{v}$ $E = cB$ Wave Equation: $E(x,t) = E_m \sin(kx - \omega t)$, $B(x,t) = B_m \sin(kx - \omega t)$ Intensity: $I = \frac{P_{avg}}{A} = \frac{1}{c\mu_0} E_{rms}^2 = \frac{1}{2c\mu_0} E_m^2$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 22. Light & Optics Reflection: Angle of incidence = Angle of reflection ($\theta_i = \theta_r$) Refraction (Snell's Law): $n_1 \sin\theta_1 = n_2 \sin\theta_2$ Index of Refraction: $n = c/v$ Total Internal Reflection: Occurs when $n_1 > n_2$ and $\theta_1 > \theta_c$, where $\sin\theta_c = n_2/n_1$ Mirrors & Lenses (Thin Lens Equation): $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -\frac{i}{p}$ $f = R/2$ (spherical mirrors) Real image: $i>0$, virtual image: $i Upright image: $m>0$, inverted image: $m Converging lens/concave mirror: $f>0$ Diverging lens/convex mirror: $f Diffraction (Single Slit): Minima: $a \sin\theta = m\lambda$ ($m=1, 2, 3, ...$) Central maximum width: $\Delta y = \frac{2L\lambda}{a}$ Diffraction Gratings: Maxima: $d \sin\theta = m\lambda$ ($m=0, 1, 2, ...$)