Halliday Physics Fundamentals Cheat Shee

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1. Measurement & Units 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, $10^{-12}$) Dimensional Analysis: Check equation consistency using units. 2. Kinematics (1D & 2D) 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) 2D Motion (Projectile Motion) Components: $v_x = v_0 \cos \theta_0$, $v_y = v_0 \sin \theta_0$ Horizontal Motion (Constant Velocity): $x = x_0 + (v_0 \cos \theta_0)t$ Vertical Motion (Constant Acceleration, $a_y = -g$): $y = y_0 + (v_0 \sin \theta_0)t - \frac{1}{2}gt^2$ $v_y = v_0 \sin \theta_0 - gt$ $v_y^2 = (v_0 \sin \theta_0)^2 - 2g(y - y_0)$ Range: $R = \frac{v_0^2 \sin(2\theta_0)}{g}$ (for $y_0 = 0$) 3. Vectors Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$ Unit Vector Notation: $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ Vector Addition: $\vec{C} = \vec{A} + \vec{B} \implies C_x = A_x + B_x$, etc. Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = AB \cos \phi = A_x B_x + A_y B_y + A_z B_z$ 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: $|\vec{A} \times \vec{B}| = AB \sin \phi$ Direction: Right-Hand Rule 4. 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: $\vec{F}_{net} = m\vec{a}$ 3rd Law: For every action, there is an equal and opposite reaction. $\vec{F}_{AB} = -\vec{F}_{BA}$ Weight: $W = mg$ (force of gravity) Normal Force: $F_N$ (perpendicular to surface) Friction: Static: $f_s \le \mu_s F_N$ Kinetic: $f_k = \mu_k F_N$ Usually $\mu_s > \mu_k$ Tension: Force transmitted through a string/rope. Drag Force: $D = \frac{1}{2} C \rho A v^2$ (high speeds), $D = bv$ (low speeds) 5. Work, Energy & Power Work (Constant Force): $W = \vec{F} \cdot \vec{d} = Fd \cos \phi$ Work (Variable Force 1D): $W = \int_{x_i}^{x_f} F(x) dx$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ Conservative Forces: Work done is path-independent ($W_c = -\Delta U$). Examples: Gravity, spring force. 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: $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ Power: $P_{avg} = \frac{\Delta W}{\Delta t}$, $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ 6. Center of Mass & Momentum Center of Mass (Discrete Particles): $x_{com} = \frac{1}{M} \sum m_i x_i$ $\vec{r}_{com} = \frac{1}{M} \sum m_i \vec{r}_i$ Center of Mass (Continuous Body): $\vec{r}_{com} = \frac{1}{M} \int \vec{r} dm$ Momentum: $\vec{p} = m\vec{v}$ Newton's 2nd Law (Momentum Form): $\vec{F}_{net} = \frac{d\vec{p}}{dt}$ Impulse: $\vec{J} = \int_{t_i}^{t_f} \vec{F}(t) dt = \vec{F}_{avg} \Delta t = \Delta \vec{p}$ Conservation of Linear Momentum: $\vec{P}_{net} = \text{constant}$ if $\vec{F}_{net,ext} = 0$. $\sum \vec{p}_i = \sum \vec{p}_f$ Collisions: Elastic: Momentum and Kinetic Energy conserved. Inelastic: Momentum conserved, KE not conserved. Perfectly Inelastic: Momentum conserved, objects stick together. 7. 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: (analogous to 1D 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: ($r$ = distance from axis) 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 towards center) 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: $K_{rot} = \frac{1}{2}I\omega^2$ Total Kinetic Energy (Rolling): $K = K_{trans} + K_{rot} = \frac{1}{2}Mv_{com}^2 + \frac{1}{2}I_{com}\omega^2$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ Magnitude: $\tau = rF \sin \phi = r F_t = dF$ (d = lever arm) Newton's 2nd Law for Rotation: $\vec{\tau}_{net} = I\vec{\alpha}$ Work in Rotation: $W = \int_{\theta_i}^{\theta_f} \tau d\theta$ (constant $\tau$: $W = \tau \Delta \theta$) Power in Rotation: $P = \tau \omega$ Angular Momentum: Particle: $\vec{l} = \vec{r} \times \vec{p} = \vec{r} \times m\vec{v}$ Rigid Body: $\vec{L} = I\vec{\omega}$ Newton's 2nd Law (Angular Momentum Form): $\vec{\tau}_{net} = \frac{d\vec{L}}{dt}$ Conservation of Angular Momentum: $\vec{L}_{net} = \text{constant}$ if $\vec{\tau}_{net,ext} = 0$. 8. Equilibrium & Elasticity Static Equilibrium Conditions: Translational: $\sum \vec{F}_{net} = 0 \Rightarrow \sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ Rotational: $\sum \vec{\tau}_{net} = 0$ (about any axis) Stress: Force per unit area ($P = F/A$) Strain: Fractional change in dimension ($\Delta L/L$, $\Delta V/V$) Hooke's Law (Elastic Moduli): Stress = Modulus $\times$ Strain Young's Modulus (tension/compression): $E = \frac{F/A}{\Delta L/L}$ Shear Modulus (shearing): $G = \frac{F/A}{\Delta x/h}$ Bulk Modulus (volume compression): $B = -\frac{\Delta P}{\Delta V/V}$ 9. Gravitation 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 Acceleration: $g = G \frac{M_E}{R_E^2}$ (at surface), $g(r) = G \frac{M_E}{r^2}$ (at distance $r$ from center) 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 Sun at one focus. 2nd: A line joining a planet and the Sun sweeps out equal areas in equal times. 3rd: $T^2 \propto a^3$ (for circular orbit $T^2 = \left(\frac{4\pi^2}{GM}\right)r^3$) 10. Fluids Density: $\rho = \frac{m}{V}$ Pressure: $P = \frac{F}{A}$ Pressure in a Fluid at Depth: $P = P_0 + \rho gh$ Pascal's Principle: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. ($\frac{F_1}{A_1} = \frac{F_2}{A_2}$) Archimedes' Principle (Buoyancy): $F_B = \rho_{fluid} g V_{disp}$ (Buoyant force equals weight of displaced fluid) Equation of Continuity: $A_1 v_1 = A_2 v_2$ (for incompressible fluid) Bernoulli's Equation: $P_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2$ 11. Oscillations Simple Harmonic Motion (SHM): Displacement: $x(t) = x_m \cos(\omega t + \phi)$ Velocity: $v(t) = -x_m \omega \sin(\omega t + \phi)$ Acceleration: $a(t) = -x_m \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (mass-spring), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) Period: $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$ (mass-spring), $T = 2\pi \sqrt{\frac{L}{g}}$ (simple pendulum) Frequency: $f = \frac{1}{T}$ Energy: $E = \frac{1}{2}kx_m^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ Damped SHM: $x(t) = x_m e^{-bt/2m} \cos(\omega' t + \phi)$ Forced Oscillations & Resonance: Amplitude peaks when driving frequency matches natural frequency. 12. Waves Wave Speed: $v = \lambda f$ Transverse Wave on String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$=tension, $\mu$=linear density) Sound Wave Speed: $v = \sqrt{\frac{B}{\rho}}$ (fluids), $v = \sqrt{\frac{E}{\rho}}$ (solids) Wave Function: $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Wave number: $k = \frac{2\pi}{\lambda}$ Angular frequency: $\omega = 2\pi f$ Intensity: $I = \frac{P}{A}$ (Power per unit area) Sound Intensity Level: $\beta = (10 \text{ dB}) \log_{10}\left(\frac{I}{I_0}\right)$ ($I_0 = 10^{-12} \text{ W/m}^2$) Standing Waves: Occur when waves interfere constructively after reflection. On a string fixed at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1,2,3,...$) In an open-open pipe: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ ($n=1,2,3,...$) In an open-closed pipe: $\lambda_n = \frac{4L}{n}$, $f_n = \frac{nv}{4L}$ ($n=1,3,5,...$) Beats: $f_{beat} = |f_1 - f_2|$ Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (top signs for detector/source moving towards each other) 13. Temperature, Heat & Thermodynamics Temperature Scales: $T_F = \frac{9}{5}T_C + 32^\circ$ $T_K = T_C + 273.15$ Thermal Expansion: Linear: $\Delta L = L \alpha \Delta T$ Area: $\Delta A = A \beta \Delta T$ ($\beta \approx 2\alpha$) Volume: $\Delta V = V \gamma \Delta T$ ($\gamma \approx 3\alpha$) Heat Capacity: $Q = C \Delta T$ Specific Heat: $Q = cm \Delta T$ Latent Heat (Phase Change): $Q = L m$ ($L_f$ for fusion, $L_v$ for vaporization) Heat Transfer: Conduction: $P_{cond} = kA \frac{T_H - T_C}{L}$ Convection: Involves fluid motion. Radiation: $P_{rad} = \sigma \epsilon A T^4$ (Stefan-Boltzmann Law) Ideal Gas Law: $PV = nRT = NkT$ ($R = 8.31 \text{ J/mol}\cdot\text{K}$, $k = 1.38 \times 10^{-23} \text{ J/K}$) Kinetic Theory of Gases: Average KE: $K_{avg} = \frac{3}{2} kT$ RMS speed: $v_{rms} = \sqrt{\frac{3RT}{M}}$ Internal Energy of Ideal Gas: $E_{int} = \frac{3}{2}nRT$ (monatomic) 1st Law of Thermodynamics: $\Delta E_{int} = Q - W$ ($Q$ added to system, $W$ done by system) Work Done by Gas: $W = \int P dV$ Isobaric ($P=$ const): $W = P \Delta V$ Isothermal ($T=$ const): $W = nRT \ln\left(\frac{V_f}{V_i}\right)$ Adiabatic ($Q=0$): $PV^\gamma = \text{const}$ Isochoric ($V=$ const): $W = 0$ Specific Heats of Ideal Gas: $C_P = C_V + R$ Monatomic: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$, $\gamma = \frac{5}{3}$ Diatomic: $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$, $\gamma = \frac{7}{5}$ 2nd Law of Thermodynamics: Heat flows spontaneously from hot to cold. Entropy of isolated system never decreases. $\Delta S \ge 0$. Entropy Change: $\Delta S = \int \frac{dQ}{T}$ (reversible process) Engines: Efficiency $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|}$ Carnot Engine (Ideal): $\epsilon_C = 1 - \frac{T_C}{T_H}$ Refrigerators/Heat Pumps: Coefficient of Performance (COP) Refrigerator: $K = \frac{|Q_C|}{|W|} = \frac{|Q_C|}{|Q_H| - |Q_C|}$ Heat Pump: $K = \frac{|Q_H|}{|W|} = \frac{|Q_H|}{|Q_H| - |Q_C|}$ Carnot COP: $K_C = \frac{T_C}{T_H - T_C}$ (refrig), $K_C = \frac{T_H}{T_H - T_C}$ (HP) 14. Electric Charge & Field Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{|q_1 q_2|}{r^2}$ ($\epsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$) Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ Electric Field (Point Charge): $E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2}$ Electric Field (Continuous Distribution): $d\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{dq}{r^2}\hat{r}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on Dipole in E-field: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ 15. Gauss' Law Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\epsilon_0 \Phi_E = q_{enc}$ Applications: Infinite line of charge: $E = \frac{\lambda}{2\pi\epsilon_0 r}$ Infinite non-conducting sheet: $E = \frac{\sigma}{2\epsilon_0}$ Conducting sheet: $E = \frac{\sigma}{\epsilon_0}$ Spherical shell (outside): $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ Spherical shell (inside): $E = 0$ Solid non-conducting sphere (outside): $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ Solid non-conducting sphere (inside): $E = \frac{1}{4\pi\epsilon_0} \frac{qr}{R^3}$ 16. Electric Potential Potential Difference: $\Delta V = V_f - V_i = -\frac{W_{ab}}{q_0} = -\int_i^f \vec{E} \cdot d\vec{s}$ Potential (Point Charge): $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ Potential (Continuous Distribution): $V = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{r}$ Relating E-field and Potential: $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ Potential Energy (System of Charges): $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$ (for two charges) Potential Energy (Charge in Potential): $U = qV$ 17. Capacitance Capacitance Definition: $C = \frac{q}{V}$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Cylindrical Capacitor: $C = 2\pi\epsilon_0 \frac{L}{\ln(b/a)}$ Spherical Capacitor: $C = 4\pi\epsilon_0 \frac{ab}{b-a}$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Capacitors in Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{q^2}{2C} = \frac{1}{2}qV$ Energy Density: $u_E = \frac{1}{2}\epsilon_0 E^2$ Dielectrics: $C = \kappa C_{air}$ ($\kappa$ = dielectric constant) 18. Current & Resistance Electric Current: $I = \frac{dq}{dt}$ Current Density: $\vec{J} = n e \vec{v}_d$ ($n$=charge carrier density, $e$=charge, $v_d$=drift speed) Ohm's Law (Microscopic): $\vec{E} = \rho \vec{J}$ ($\rho$=resistivity) Resistance: $R = \rho \frac{L}{A}$ Ohm's Law (Macroscopic): $V = IR$ Power: $P = IV = I^2R = \frac{V^2}{R}$ Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ 19. DC Circuits EMF: $\mathcal{E}$ (ideal battery voltage) Kirchhoff's Rules: Junction Rule: $\sum I_{in} = \sum I_{out}$ (Conservation of Charge) Loop Rule: $\sum \Delta V = 0$ (Conservation of Energy) RC Circuits: Charging: $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ Discharging: $q(t) = q_0 e^{-t/RC}$, $I(t) = -\frac{q_0}{RC}e^{-t/RC}$ Time Constant: $\tau = RC$ 20. Magnetic Fields Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ (Right-Hand Rule) Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ ($\vec{\mu} = NI\vec{A}$ = magnetic dipole moment) Potential Energy of Dipole: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = \frac{I B}{ne A}$ Motion in B-field: $r = \frac{mv}{|q|B}$ (circular motion) 21. Ampere's Law 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: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ 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): $B = \frac{\mu_0 N I}{2\pi r}$ 22. Faraday's Law & Induction 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 opposes the change in magnetic flux that created it. Motional EMF: $\mathcal{E} = B L v$ Induced Electric Field: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ 23. Inductance Self-Inductance: $L = \frac{N\Phi_B}{I}$ Solenoid Inductance: $L = \mu_0 n^2 A l$ Inductors in Series: $L_{eq} = \sum L_i$ Inductors in Parallel: $\frac{1}{L_{eq}} = \sum \frac{1}{L_i}$ 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: $u_B = \frac{B^2}{2\mu_0}$ 24. Electromagnetic Oscillations & AC Circuits LC Oscillations: Angular Frequency: $\omega = \frac{1}{\sqrt{LC}}$ Charge: $q(t) = Q \cos(\omega t + \phi)$ Current: $I(t) = -Q\omega \sin(\omega t + \phi)$ Energy: $U = U_E + U_B = \frac{q^2}{2C} + \frac{1}{2}LI^2 = \text{constant}$ Driven RLC Circuit: Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Inductive Reactance: $X_L = \omega L$ Capacitive Reactance: $X_C = \frac{1}{\omega C}$ RMS Current: $I_{rms} = \frac{V_{rms}}{Z}$ Phase Angle: $\tan \phi = \frac{X_L - X_C}{R}$ Power Factor: $\cos \phi$ Average Power: $P_{avg} = I_{rms} V_{rms} \cos \phi = I_{rms}^2 R$ Resonance: $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ Transformers: $\frac{V_S}{V_P} = \frac{N_S}{N_P}$ (voltage), $\frac{I_S}{I_P} = \frac{N_P}{N_S}$ (current) 25. Electromagnetic Waves Maxwell's Equations (Integral Form): Gauss' Law for E: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0}$ Gauss' Law for B: $\oint \vec{B} \cdot d\vec{A} = 0$ Faraday's Law: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ Speed of Light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Wave Equation: $\frac{\partial^2 E_x}{\partial z^2} = \frac{1}{c^2} \frac{\partial^2 E_x}{\partial t^2}$ Relationship E and B: $E=cB$ Poynting Vector (Energy Flow): $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ Intensity (Average Poynting Vector): $I = S_{avg} = \frac{1}{c\mu_0}E_{rms}^2 = \frac{c}{\mu_0}B_{rms}^2 = \frac{E_{rms}B_{rms}}{\mu_0}$ Radiation Pressure: $P_r = I/c$ (absorbed), $P_r = 2I/c$ (reflected) 26. Light: Reflection & Refraction Law of Reflection: $\theta_i = \theta_r$ Law of Refraction (Snell's Law): $n_1 \sin \theta_1 = n_2 \sin \theta_2$ Index of Refraction: $n = c/v$ Critical Angle: $\sin \theta_c = n_2/n_1$ (for $n_1 > n_2$) 27. Lenses & Mirrors Mirror/Lens Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -\frac{i}{p} = \frac{h_i}{h_p}$ Focal Length: $f = R/2$ (spherical mirrors) Sign Conventions: $p$: + real object, - virtual object $i$: + real image, - virtual image $f$: + concave mirror/converging lens, - convex mirror/diverging lens $R$: + center on same side as outgoing light (concave mirror/convex surface), - opposite side (convex mirror/concave surface) $m$: + upright, - inverted Power of Lens: $P = 1/f$ (diopters) Thin Lenses in Contact: $1/f_{eq} = 1/f_1 + 1/f_2$ 28. Interference Young's Double Slit: Constructive: $d \sin \theta = m\lambda$ ($m=0, \pm 1, \pm 2, ...$) Destructive: $d \sin \theta = (m + 1/2)\lambda$ ($m=0, \pm 1, \pm 2, ...$) Fringe separation: $\Delta y = \frac{\lambda L}{d}$ Thin Film Interference: Path length difference + phase shifts upon reflection. Phase shift of $\pi$ (or $1/2 \lambda$) when light reflects from higher index medium. 29. 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): $\theta_R = 1.22 \frac{\lambda}{D}$ (circular aperture) 30. Relativity Postulates: 1. The laws of physics are the same for all inertial observers. 2. The speed of light in vacuum is the same for all inertial observers. Time Dilation: $\Delta t = \gamma \Delta t_0$ ($\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$) Length Contraction: $L = L_0/\gamma$ 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) Momentum-Energy Relation: $E^2 = (pc)^2 + (mc^2)^2$ 31. Photons & Matter Waves Photon Energy: $E = hf = \frac{hc}{\lambda}$ ($h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$) Photoelectric Effect: $K_{max} = hf - \Phi$ ($\Phi$=work function) Compton Effect: $\Delta \lambda = \lambda' - \lambda = \frac{h}{mc}(1 - \cos \phi)$ De Broglie Wavelength: $\lambda = \frac{h}{p}$ Heisenberg Uncertainty Principle: $\Delta x \Delta p_x \ge \hbar/2$ $\Delta E \Delta t \ge \hbar/2$ ($\hbar = h/2\pi$) 32. Atomic Structure Bohr Model: Quantized Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ (Hydrogen) Quantized Radii: $r_n = n^2 a_0$ ($a_0 = 0.0529 \text{ nm}$ = Bohr radius) Energy of Photon: $\Delta E = |E_f - E_i| = hf$ Quantum Numbers: $n$: principal (energy, size) $1, 2, 3, ...$ $l$: orbital angular momentum (shape) $0, 1, ..., n-1$ $m_l$: magnetic (orientation) $-l, ..., 0, ..., +l$ $m_s$: spin magnetic ($\pm 1/2$) Pauli Exclusion Principle: No two electrons can have the same set of four quantum numbers. X-Rays: Continuous (Bremsstrahlung) and Characteristic (electron transitions). Lasers: Light Amplification by Stimulated Emission of Radiation. 33. Nuclear Physics Nucleus: Protons (Z) + Neutrons (N) = Mass Number (A) Isotopes: Same Z, different 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: $N(t) = N_0 e^{-\lambda t}$ ($\lambda$ = decay constant) Half-Life: $T_{1/2} = \frac{\ln 2}{\lambda}$ Activity: $R = |\frac{dN}{dt}| = \lambda N$ Types of Decay: Alpha ($\alpha$): Emission of $^4_2$He nucleus. Beta ($\beta^-$): Neutron converts to proton, emits $e^-$ and $\bar{\nu}$. Beta ($\beta^+$): Proton converts to neutron, emits $e^+$ and $\nu$. Gamma ($\gamma$): Emission of high-energy photon from excited nucleus. Fission: Large nucleus splits into smaller ones. Fusion: Small nuclei combine to form larger one.