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) Scientific Notation: $N \times 10^n$ Unit Conversions: Use conversion factors (e.g., $1 \text{ inch} = 2.54 \text{ cm}$) Significant Figures: Rules for addition/subtraction and multiplication/division. 2. Motion in 1D Displacement: $\Delta x = x_f - x_i$ Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$ Average Speed: $s_{avg} = \frac{\text{total distance}}{\Delta t}$ 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: $v = v_0 + at$ $x - x_0 = v_0t + \frac{1}{2}at^2$ $v^2 = v_0^2 + 2a(x - x_0)$ $x - x_0 = \frac{1}{2}(v_0 + v)t$ Free-Fall Acceleration: $a = -g = -9.8 \text{ m/s}^2$ (near Earth's surface) 3. Vectors Components: $A_x = A \cos\theta$, $A_y = A \sin\theta$ Magnitude: $A = \sqrt{A_x^2 + A_y^2}$ Direction: $\tan\theta = \frac{A_y}{A_x}$ (check quadrant) Unit Vectors: $\hat{i}, \hat{j}, \hat{k}$ for x, y, z axes Vector Addition: $\vec{A} + \vec{B} = (A_x+B_x)\hat{i} + (A_y+B_y)\hat{j}$ Scalar (Dot) Product: $\vec{A} \cdot \vec{B} = AB \cos\phi = A_xB_x + A_yB_y + A_zB_z$ Vector (Cross) Product: $\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$ Magnitude of Cross Product: $|\vec{A} \times \vec{B}| = AB \sin\phi$ 4. Motion in 2D & 3D Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ Velocity Vector: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$ Acceleration Vector: $\vec{a} = \frac{d\vec{v}}{dt} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$ Projectile Motion: $x - x_0 = (v_0 \cos\theta_0)t$ $y - y_0 = (v_0 \sin\theta_0)t - \frac{1}{2}gt^2$ $v_x = v_0 \cos\theta_0$ (constant) $v_y = v_0 \sin\theta_0 - gt$ Range $R = \frac{v_0^2}{g} \sin(2\theta_0)$ (for level ground) Max Height $H = \frac{(v_0 \sin\theta_0)^2}{2g}$ Uniform Circular Motion: Speed: $v = \frac{2\pi r}{T}$ Centripetal Acceleration: $a_c = \frac{v^2}{r}$ (directed towards center) 5. Force & Motion (Newton's Laws) Newton's 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. Newton's Second Law: $\vec{F}_{net} = m\vec{a}$ Newton's Third Law: 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: $W = mg$ Normal Force: $\vec{F}_N$ (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: $\vec{T}$ (force through a cord/rope) 6. Kinetic Energy & Work Kinetic Energy (KE): $K = \frac{1}{2}mv^2$ Work done by constant force: $W = Fd \cos\phi = \vec{F} \cdot \vec{d}$ Work done by varying force: $W = \int_{x_i}^{x_f} F(x) dx$ Work-Kinetic Energy Theorem: $W_{net} = \Delta K = K_f - K_i$ Power: $P_{avg} = \frac{W}{\Delta t}$, $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ 7. Potential Energy & Conservation of Energy Gravitational Potential Energy: $U_g = mgh$ Elastic Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ Conservative Forces: Work done is path-independent (e.g., gravity, spring) Nonconservative Forces: Work done is path-dependent (e.g., friction) Mechanical Energy: $E_{mech} = K + U$ Conservation of Mechanical Energy: If only conservative forces do work, $E_{mech,i} = E_{mech,f}$ or $K_i + U_i = K_f + U_f$ Conservation of Energy (General): $W_{nc} = \Delta E_{mech} = \Delta K + \Delta U$ Total Energy: $E_{total} = E_{mech} + E_{thermal} + E_{internal} + ...$ (always conserved) 8. Center of Mass & Linear Momentum Center of Mass (CM): For particles: $x_{CM} = \frac{1}{M} \sum m_ix_i$, $y_{CM} = \frac{1}{M} \sum m_iy_i$ For continuous body: $x_{CM} = \frac{1}{M} \int x \,dm$ Linear Momentum: $\vec{p} = m\vec{v}$ Newton's Second Law (momentum form): $\vec{F}_{net} = \frac{d\vec{p}}{dt}$ Impulse: $\vec{J} = \int_{t_i}^{t_f} \vec{F}(t) dt = \Delta \vec{p} = \vec{p}_f - \vec{p}_i$ Conservation of Linear Momentum: If $\vec{F}_{net,ext} = 0$, then $\vec{P}_{total,i} = \vec{P}_{total,f}$ Collisions: Elastic: KE is conserved. Both momentum and KE are conserved. Inelastic: KE is NOT conserved. Momentum is conserved. Perfectly Inelastic: Objects stick together. Momentum is conserved. 9. 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: $\omega = \omega_0 + \alpha t$ $\theta - \theta_0 = \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ Relating Linear and Angular Variables: $s = r\theta$ $v_t = r\omega$ (tangential speed) $a_t = r\alpha$ (tangential acceleration) $a_c = \frac{v_t^2}{r} = r\omega^2$ (centripetal acceleration) Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Moment of Inertia: $I = \sum m_ir_i^2$ (discrete), $I = \int r^2 dm$ (continuous) Parallel-Axis Theorem: $I = I_{CM} + Mh^2$ 10. Torque & Angular Momentum Torque: $\vec{\tau} = \vec{r} \times \vec{F}$, magnitude $\tau = rF\sin\phi = rF_t = r_{\perp}F$ Newton's Second Law for Rotation: $\vec{\tau}_{net} = I\vec{\alpha}$ Work done by Torque: $W = \int_{\theta_i}^{\theta_f} \tau \,d\theta$ Power for Rotation: $P = \tau\omega$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ (for rigid body) Newton's Second Law (angular momentum form): $\vec{\tau}_{net} = \frac{d\vec{L}}{dt}$ Conservation of Angular Momentum: If $\vec{\tau}_{net,ext} = 0$, then $\vec{L}_{total,i} = \vec{L}_{total,f}$ 11. Equilibrium & Elasticity Conditions for Equilibrium: Translational: $\sum \vec{F} = 0$ Rotational: $\sum \vec{\tau} = 0$ (about any axis) Stress: $\text{Stress} = \frac{\text{Force}}{\text{Area}}$ Strain: $\text{Strain} = \frac{\Delta L}{L_0}$ (Tensile/Compressive), $\text{Strain} = \frac{\Delta x}{L_0}$ (Shear) Young's Modulus: $E = \frac{\text{Tensile Stress}}{\text{Tensile Strain}}$ Shear Modulus: $G = \frac{\text{Shear Stress}}{\text{Shear Strain}}$ Bulk Modulus: $B = -\frac{\Delta P}{\Delta V/V}$ 12. Gravitation Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$ (G is gravitational constant) Gravitational Acceleration: $g = G \frac{M}{r^2}$ Gravitational Potential Energy: $U = -G \frac{m_1 m_2}{r}$ 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. $T^2 \propto a^3$ (period squared proportional to semi-major axis cubed) For circular orbits: $T^2 = (\frac{4\pi^2}{GM})r^3$ 13. Fluids Density: $\rho = \frac{m}{V}$ Pressure: $P = \frac{F}{A}$ Pressure in a Fluid at Rest: $P = P_0 + \rho gh$ 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. ($F_1/A_1 = F_2/A_2$) Archimedes' Principle: Buoyant force $F_b = \rho_{fluid} V_{disp} g$ Equation of Continuity: $A_1v_1 = A_2v_2$ (for incompressible fluid) Bernoulli's Equation: $P_1 + \frac{1}{2}\rho v_1^2 + \rho gy_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gy_2$ 14. 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) Period: $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ Frequency: $f = \frac{1}{T}$ Energy in SHM: $E = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kx_m^2$ (constant) Pendulums: Simple Pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$ (for small angles) Physical Pendulum: $T = 2\pi\sqrt{\frac{I}{mgd}}$ Torsion Pendulum: $T = 2\pi\sqrt{\frac{I}{\kappa}}$ ($\kappa$ is torsion constant) 15. Waves Wave Speed: $v = \lambda f$ Transverse Wave in String: $v = \sqrt{\frac{\tau}{\mu}}$ ($\tau$ tension, $\mu$ linear density) Wave Equation: $y(x,t) = y_m \sin(kx - \omega t + \phi)$ Angular Wave Number: $k = \frac{2\pi}{\lambda}$ Angular Frequency: $\omega = 2\pi f$ Power of a Wave: $P_{avg} = \frac{1}{2}\mu v \omega^2 y_m^2$ Principle of Superposition: $y'(x,t) = y_1(x,t) + y_2(x,t)$ Interference: Constructive: $\Delta\phi = m(2\pi)$ Destructive: $\Delta\phi = (m + \frac{1}{2})(2\pi)$ Standing Waves: Formed by two waves traveling in opposite directions. Resonance in Strings: $f_n = n \frac{v}{2L}$, $\lambda_n = \frac{2L}{n}$ ($n=1,2,3...$) 16. Sound Waves Sound Speed: $v = \sqrt{\frac{B}{\rho}}$ (B is bulk modulus) Speed in Air: $v \approx 343 \text{ m/s}$ at $20^\circ \text{C}$ Intensity: $I = \frac{P}{A}$ Intensity Level (Decibels): $\beta = (10 \text{ dB}) \log_{10}\frac{I}{I_0}$ ($I_0 = 10^{-12} \text{ W/m}^2$) Doppler Effect: $f' = f \frac{v \pm v_D}{v \mp v_S}$ (D for detector, S for source; top sign for approach, bottom for recession) Beats: $f_{beat} = |f_1 - f_2|$ Resonance in Open Pipe: $f_n = n \frac{v}{2L}$ ($n=1,2,3...$) Resonance in Closed Pipe: $f_n = n \frac{v}{4L}$ ($n=1,3,5...$) 17. Temperature, Heat & First Law of Thermodynamics Temperature Scales: $T_C = T_K - 273.15$ $T_F = \frac{9}{5}T_C + 32^\circ$ 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): $Q = Lm$ (L is latent heat of fusion or vaporization) Heat Transfer: Conduction: $P_{cond} = kA\frac{dT}{dx}$ Radiation: $P_{rad} = \sigma \epsilon A T^4$ (Stefan-Boltzmann Law) First Law of Thermodynamics: $\Delta E_{int} = Q - W$ ($Q$ is heat added to system, $W$ is work done BY system) Work done by gas: $W = \int P dV$ 18. Kinetic Theory of Gases Ideal Gas Law: $PV = nRT = NkT$ ($R$ gas constant, $k$ Boltzmann constant) Avogadro's Number: $N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$ Molar Mass: $M = mN_A$ Average Translational KE per molecule: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ Internal Energy of Ideal Monatomic Gas: $E_{int} = \frac{3}{2}nRT$ Specific Heats of Ideal Gas: Constant Volume: $C_V = \frac{f}{2}R$ (f = degrees of freedom) Constant Pressure: $C_P = C_V + R$ Ratio: $\gamma = \frac{C_P}{C_V}$ Adiabatic Process: $PV^\gamma = \text{constant}$, $TV^{\gamma-1} = \text{constant}$ 19. Entropy & Second Law of Thermodynamics Second Law of Thermodynamics: Entropy of an isolated system never decreases. Entropy Change (reversible): $\Delta S = \int \frac{dQ_{rev}}{T}$ Entropy Change (constant T): $\Delta S = \frac{Q}{T}$ Efficiency of Heat Engine: $\epsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_L|}{|Q_H|}$ Coefficient of Performance (Refrigerator): $K = \frac{|Q_L|}{|W|}$ Carnot Cycle (Ideal Engine/Refrigerator): $\epsilon_C = 1 - \frac{T_L}{T_H}$ $K_C = \frac{T_L}{T_H - T_L}$ 20. Electric Charge & Field Coulomb's Law: $F = k \frac{|q_1q_2|}{r^2}$ ($k = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$) Electric Field: $\vec{E} = \frac{\vec{F}}{q_0}$ Electric Field of Point Charge: $E = k \frac{|q|}{r^2}$ 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: $U = -\vec{p} \cdot \vec{E}$ 21. Gauss' Law Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss' Law: $\epsilon_0 \Phi_E = q_{enc}$ Applications: Finding E-field for symmetric charge distributions (e.g., infinite line, plane, sphere) 22. Electric Potential Electric Potential Energy: $\Delta U = -W = - \int \vec{F} \cdot d\vec{s}$ Electric Potential: $V = \frac{U}{q_0}$ Potential Difference: $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$ Relation between E and V: $\vec{E} = -\nabla V = -(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k})$ Potential due to Point Charge: $V = k \frac{q}{r}$ Potential due to Dipole: $V = k \frac{p \cos\theta}{r^2}$ 23. Capacitance Capacitance: $C = \frac{q}{V}$ (Unit: Farad, F) Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Capacitors in Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Energy Stored in Capacitor: $U = \frac{1}{2}CV^2 = \frac{q^2}{2C} = \frac{1}{2}qV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ Capacitor with Dielectric: $C = \kappa C_{air}$, $E = E_{air}/\kappa$ 24. Current & Resistance Electric Current: $I = \frac{dq}{dt}$ (Unit: Ampere, A) Current Density: $\vec{J} = n e \vec{v}_d$ ($n$ charge carriers, $e$ charge, $v_d$ drift speed) Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ ($\rho$ resistivity) Resistivity (Temperature Dependence): $\rho - \rho_0 = \rho_0 \alpha (T - T_0)$ Power in Electric Circuits: $P = IV = I^2R = \frac{V^2}{R}$ 25. Circuits Resistors in Series: $R_{eq} = \sum R_i$ Resistors in Parallel: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ Kirchhoff's Laws: Junction Rule: $\sum I_{in} = \sum I_{out}$ Loop Rule: $\sum \Delta V = 0$ around any closed loop RC Circuits (Charging Capacitor): $q(t) = C\mathcal{E}(1 - e^{-t/RC})$, $I(t) = \frac{\mathcal{E}}{R}e^{-t/RC}$ RC Circuits (Discharging Capacitor): $q(t) = q_0 e^{-t/RC}$, $I(t) = -\frac{q_0}{RC}e^{-t/RC}$ Time Constant: $\tau = RC$ 26. Magnetic Fields Magnetic Force on Charge: $\vec{F}_B = q\vec{v} \times \vec{B}$ (magnitude $F_B = |q|vB\sin\phi$) Magnetic Force on Current: $\vec{F}_B = I\vec{L} \times \vec{B}$ (magnitude $F_B = ILB\sin\phi$) Torque on Current Loop: $\vec{\tau} = \vec{\mu} \times \vec{B}$ Magnetic Dipole Moment: $\vec{\mu} = NIA\hat{n}$ Potential Energy of Dipole in B-field: $U = -\vec{\mu} \cdot \vec{B}$ Hall Effect: $V_H = \frac{IB}{net}$ ($n$ charge carrier density, $t$ thickness) 27. Sources of Magnetic Fields Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ Magnetic Field of Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Force between Parallel Wires: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ Magnetic Field at Center of Loop: $B = \frac{\mu_0 I}{2R}$ Magnetic Field of Solenoid: $B = \mu_0 n I$ ($n$ turns per unit length) Ampere's Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ 28. 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/EMF opposes the change in magnetic flux that produced it. Motional EMF: $\mathcal{E} = BLv$ Induced Electric Field: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$ Generators: $\mathcal{E} = NBA\omega \sin(\omega t)$ 29. Inductance Self-Inductance: $L = \frac{N\Phi_B}{I}$ (Unit: Henry, H) Solenoid Inductance: $L = \mu_0 n^2 A l$ 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): $I(t) = I_0 e^{-t/\tau_L}$ Inductive Time Constant: $\tau_L = L/R$ Energy Stored in Inductor: $U_B = \frac{1}{2}LI^2$ Magnetic Energy Density: $u_B = \frac{B^2}{2\mu_0}$ 30. Electromagnetic Oscillations & AC LC Oscillations: $q(t) = Q \cos(\omega t + \phi)$, $I(t) = - \omega Q \sin(\omega t + \phi)$ Angular Frequency of LC: $\omega = \frac{1}{\sqrt{LC}}$ LRC Circuits (Damped): $q(t) = Qe^{-Rt/2L} \cos(\omega' t + \phi)$ Forced Oscillations/AC Circuits: Capacitive Reactance: $X_C = \frac{1}{\omega C}$ Inductive Reactance: $X_L = \omega L$ Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ RMS Values: $V_{rms} = V_m/\sqrt{2}$, $I_{rms} = I_m/\sqrt{2}$ Power (AC): $P_{avg} = I_{rms}V_{rms}\cos\phi$ (Power Factor $\cos\phi$) Resonance: $X_L = X_C \implies \omega_0 = \frac{1}{\sqrt{LC}}$ Transformers: $V_S/V_P = N_S/N_P$ (ideal) 31. Maxwell's Equations & EM Waves Maxwell's Equations (Integral Form): $\oint \vec{E} \cdot d\vec{A} = q_{enc}/\epsilon_0$ (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 EM Waves: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Relation E and B in EM Wave: $E = cB$ 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: $P_{rad} = I/c$ (absorbed), $2I/c$ (reflected) 32. Light & Reflection Law of Reflection: $\theta_i = \theta_r$ Plane Mirrors: Virtual, upright, same size, same distance behind mirror. Spherical Mirrors: Mirror Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Focal Length: $f = R/2$ (concave +R, convex -R) Magnification: $m = -\frac{i}{p}$ Sign Conventions: $p$: + real object, - virtual object $i$: + real image, - virtual image $f$: + concave, - convex $m$: + upright, - inverted 33. 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 $\theta_1 > \theta_c$, where $\sin\theta_c = n_2/n_1$ (for $n_1 > n_2$) Thin Lenses: Lensmaker's Equation: $\frac{1}{f} = (n-1)(\frac{1}{r_1} - \frac{1}{r_2})$ Thin Lens Equation: $\frac{1}{p} + \frac{1}{i} = \frac{1}{f}$ Magnification: $m = -\frac{i}{p}$ Optical Instruments: Magnifying glass, compound microscope, refracting telescope. 34. Interference Young's Double-Slit Experiment: Bright Fringes (Max): $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2,...$) Dark Fringes (Min): $d\sin\theta = (m + \frac{1}{2})\lambda$ ($m=0, \pm 1, \pm 2,...$) Fringe Spacing (approx): $\Delta y = \frac{\lambda L}{d}$ Thin-Film Interference: Path difference, phase changes upon reflection. Conditions for Maxima/Minima: Depends on film thickness, indices of refraction, reflections. 35. Diffraction Single-Slit Diffraction: Dark Fringes (Min): $a\sin\theta = m\lambda$ ($m=\pm 1, \pm 2,...$) Central maximum is twice as wide as other maxima. Diffraction Grating: Bright Fringes (Max): $d\sin\theta = m\lambda$ ($m=0, \pm 1, \pm 2,...$) Dispersion: $D = \frac{d\theta}{d\lambda}$ Resolving Power: $R = \frac{\lambda_{avg}}{\Delta\lambda} = Nm$ Rayleigh's Criterion (Resolution): $\theta_R = 1.22 \frac{\lambda}{D}$ 36. Relativity Postulates: The laws of physics are the same for all inertial reference frames. The speed of light in vacuum ($c$) has the same value in all inertial reference frames. Time Dilation: $\Delta t = \gamma \Delta t_0$ ($\Delta t_0$ proper time) Length Contraction: $L = L_0/\gamma$ ($L_0$ proper length) Lorentz Factor: $\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$ 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) Energy-Momentum Relation: $E^2 = (pc)^2 + (mc^2)^2$ 37. Photons & Matter Waves Planck's Quantum Hypothesis: $E = hf$ Photoelectric Effect: $K_{max} = hf - \Phi$ ($\Phi$ work function) Photon Momentum: $p = E/c = h/\lambda$ Compton Effect: $\Delta\lambda = \frac{h}{mc}(1 - \cos\phi)$ de Broglie Wavelength: $\lambda = h/p$ Heisenberg's Uncertainty Principle: $\Delta x \Delta p_x \ge \hbar/2$ $\Delta E \Delta t \ge \hbar/2$ 38. Quantum Mechanics Schrödinger Equation: Time-dependent: $i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + U(x)\Psi$ Time-independent: $-\frac{\hbar^2}{2m}\frac{d^2 \psi}{dx^2} + U(x)\psi = E\psi$ Probability Density: $P(x) = |\psi(x)|^2$ Normalization Condition: $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$ Quantization of Energy: For bound systems (e.g., infinite potential well, harmonic oscillator) Quantum Numbers (Hydrogen Atom): $n, l, m_l, m_s$ 39. Atomic Physics Bohr Model (Hydrogen): Energy Levels: $E_n = -\frac{13.6 \text{ eV}}{n^2}$ Radii: $r_n = n^2 a_0$ ($a_0$ Bohr radius) Photon Energy: $\Delta E = E_f - E_i = hf$ Pauli Exclusion Principle: No two electrons can occupy the same quantum state. X-rays: Continuous (Bremsstrahlung) and characteristic radiation. Lasers: Population inversion, stimulated emission, optical cavity. 40. Nuclear Physics Atomic Nucleus: Protons ($Z$), Neutrons ($N$), Mass Number ($A = Z+N$) Mass Defect: $\Delta m = (Z m_p + N m_n) - M_{nucleus}$ Binding Energy: $E_b = \Delta m c^2$ Radioactive Decay: $N(t) = N_0 e^{-\lambda t}$ 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 to proton + electron + antineutrino Beta ($\beta^+$): Proton to neutron + positron + neutrino Gamma ($\gamma$): Emission of high-energy photon Fission: Heavy nucleus splits into lighter nuclei. Fusion: Light nuclei combine to form heavier nucleus. 41. Particle Physics & Cosmology Fundamental Forces: Strong, Electromagnetic, Weak, Gravitational Standard Model: Quarks, Leptons, Force Carriers (gluons, photons, W/Z bosons) Quarks: up, down, charm, strange, top, bottom Leptons: electron, muon, tau, and their neutrinos Hadrons: Baryons (3 quarks, e.g., proton, neutron), Mesons (quark-antiquark pair) Big Bang Theory: Universe originated from an extremely hot, dense state. Cosmic Microwave Background (CMB): Evidence for Big Bang. Dark Matter & Dark Energy: Components of the universe.