JEE Physics Formula Cheatsheet

Cheatsheet Content

1. Units and Dimensions Fundamental Quantities: Mass (M), Length (L), Time (T), Electric Current (A), Temperature (K), Luminous Intensity (cd), Amount of Substance (mol). Dimensional Analysis: Check homogeneity, derive relations, convert units. Significant Figures: Rules for addition/subtraction and multiplication/division. Error Analysis: Absolute Error: $\Delta A = |A_{mean} - A_i|$ Relative Error: $\frac{\Delta A_{mean}}{A_{mean}}$ Percentage Error: $\frac{\Delta A_{mean}}{A_{mean}} \times 100\%$ Propagation of Errors: If $Z = A \pm B$, then $\Delta Z = \Delta A + \Delta B$ If $Z = A^m B^n / C^p$, then $\frac{\Delta Z}{Z} = m \frac{\Delta A}{A} + n \frac{\Delta B}{B} + p \frac{\Delta C}{C}$ 2. Kinematics 2.1. Motion in One Dimension 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: $v = \frac{dx}{dt}$ Instantaneous Acceleration: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ Equations of Motion (constant acceleration): $v = u + at$ $s = ut + \frac{1}{2}at^2$ $v^2 = u^2 + 2as$ $s_n = u + \frac{a}{2}(2n - 1)$ (distance in $n^{th}$ second) 2.2. Motion in Two and Three Dimensions 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: Time of Flight: $T = \frac{2u \sin\theta}{g}$ Maximum Height: $H = \frac{u^2 \sin^2\theta}{2g}$ Horizontal Range: $R = \frac{u^2 \sin(2\theta)}{g}$ Equation of Trajectory: $y = x \tan\theta - \frac{gx^2}{2u^2 \cos^2\theta}$ Relative Velocity: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ Uniform Circular Motion: Angular Velocity: $\omega = \frac{d\theta}{dt} = \frac{v}{r}$ Centripetal Acceleration: $a_c = \frac{v^2}{r} = r\omega^2$ Centripetal Force: $F_c = \frac{mv^2}{r} = mr\omega^2$ Time Period: $T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$ 3. Laws of Motion and Friction Newton's First Law: Law of Inertia Newton's Second Law: $\vec{F} = m\vec{a}$ Newton's Third Law: Action-Reaction pairs Impulse: $\vec{J} = \vec{F}_{avg} \Delta t = \Delta \vec{p}$ Conservation of Linear Momentum: $\vec{p}_{initial} = \vec{p}_{final}$ (if $\vec{F}_{ext} = 0$) Friction: Static Friction: $f_s \le \mu_s N$ Kinetic Friction: $f_k = \mu_k N$ ($\mu_k Angle of Repose: $\tan\theta = \mu_s$ Banking of Roads: $\tan\theta = \frac{v^2}{rg}$ (ideal) 4. Work, Energy, and Power Work Done: $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ Work Done by Variable Force: $W = \int \vec{F} \cdot d\vec{r}$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Potential Energy: Gravitational: $U_g = mgh$ Elastic (spring): $U_s = \frac{1}{2}kx^2$ Work-Energy Theorem: $W_{net} = \Delta K$ Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ Conservation of Mechanical Energy: $K_i + U_i = K_f + U_f$ (for conservative forces) Elastic Collision (1D): $v_1' = \frac{(m_1-m_2)v_1 + 2m_2v_2}{m_1+m_2}$ $v_2' = \frac{(m_2-m_1)v_2 + 2m_1v_1}{m_1+m_2}$ Coefficient of Restitution: $e = \frac{v_{2f}-v_{1f}}{v_{1i}-v_{2i}}$ ($e=1$ for elastic, $e=0$ for perfectly inelastic) 5. Rotational Motion Angular Displacement: $\Delta\theta$ Angular Velocity: $\omega = \frac{d\theta}{dt}$ Angular Acceleration: $\alpha = \frac{d\omega}{dt}$ Relations between linear and angular: $v = r\omega$, $a_t = r\alpha$, $a_c = r\omega^2$ Equations of Rotational Motion (constant $\alpha$): $\omega = \omega_0 + \alpha t$ $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ $\omega^2 = \omega_0^2 + 2\alpha\theta$ Moment of Inertia: $I = \sum m_i r_i^2 = \int r^2 dm$ Parallel Axis Theorem: $I = I_{CM} + Md^2$ Perpendicular Axis Theorem (for planar bodies): $I_z = I_x + I_y$ Torque: $\vec{\tau} = \vec{r} \times \vec{F}$ ; $\tau = I\alpha$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ Conservation of Angular Momentum: $\vec{L}_{initial} = \vec{L}_{final}$ (if $\vec{\tau}_{ext} = 0$) Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$ Rolling Motion (without slipping): $v_{CM} = R\omega$, $a_{CM} = R\alpha$ Total Kinetic Energy (rolling): $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$ 6. Gravitation Newton's Law of Gravitation: $F = \frac{Gm_1m_2}{r^2}$ Acceleration due to gravity: $g = \frac{GM}{R^2}$ Variation of $g$: With Altitude: $g' = g(1 - \frac{2h}{R})$ (for $h \ll R$) ; $g' = \frac{GM}{(R+h)^2}$ (general) With Depth: $g' = g(1 - \frac{d}{R})$ With Latitude: $g' = g - R\omega^2 \cos^2\lambda$ Gravitational Potential Energy: $U = -\frac{GMm}{r}$ Gravitational Potential: $V = -\frac{GM}{r}$ Escape Velocity: $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$ Orbital Velocity: $v_o = \sqrt{\frac{GM}{r}}$ Time Period of Satellite: $T = 2\pi \sqrt{\frac{r^3}{GM}}$ Kepler's Laws: 1st Law: Orbits are ellipses. 2nd Law: Equal areas in equal times ($\frac{dA}{dt} = \frac{L}{2m}$ constant). 3rd Law: $T^2 \propto r^3$. 7. Properties of Matter 7.1. Elasticity Stress: $\sigma = \frac{F}{A}$ Strain: $\epsilon = \frac{\Delta L}{L}$ (longitudinal), $\frac{\Delta V}{V}$ (volume), $\theta$ (shear) Young's Modulus: $Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L}$ Bulk Modulus: $B = \frac{\text{Volume Stress}}{\text{Volume Strain}} = \frac{-\Delta P}{\Delta V/V}$ Shear Modulus (Rigidity): $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\theta}$ Poisson's Ratio: $\nu = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}$ Energy Stored per unit volume: $U = \frac{1}{2} \text{Stress} \times \text{Strain} = \frac{1}{2} Y (\text{Strain})^2$ 7.2. Fluid Mechanics Density: $\rho = \frac{m}{V}$ Pressure: $P = \frac{F}{A}$ Pressure in a fluid column: $P = P_0 + \rho gh$ Pascal's Law: Pressure applied to enclosed fluid is transmitted undiminished. Archimedes' Principle: Buoyant Force $F_B = \rho_{fluid} V_{submerged} g$ Equation of Continuity: $A_1v_1 = A_2v_2$ Bernoulli's Principle: $P + \rho gh + \frac{1}{2}\rho v^2 = \text{constant}$ Torricelli's Law (Efflux velocity): $v = \sqrt{2gh}$ Viscosity: $F = -\eta A \frac{dv}{dy}$ (Newton's Law of Viscosity) Stokes' Law: $F_v = 6\pi\eta rv$ (for spherical body) Terminal Velocity: $v_t = \frac{2r^2(\rho_p - \rho_f)g}{9\eta}$ Reynolds Number: $R_e = \frac{\rho v D}{\eta}$ 7.3. Surface Tension Surface Tension: $T = \frac{F}{L}$ (force per unit length) Surface Energy: $E = T \Delta A$ Excess Pressure: Liquid drop: $\Delta P = \frac{2T}{R}$ Soap bubble: $\Delta P = \frac{4T}{R}$ Capillary Rise: $h = \frac{2T \cos\theta}{\rho rg}$ 8. Heat and Thermodynamics 8.1. Thermal Properties of Matter Thermal Expansion: Linear: $\Delta L = L_0 \alpha \Delta T$ Area: $\Delta A = A_0 \beta \Delta T$ ($\beta = 2\alpha$) Volume: $\Delta V = V_0 \gamma \Delta T$ ($\gamma = 3\alpha$) Heat Capacity: $C = \frac{dQ}{dT}$ Specific Heat Capacity: $c = \frac{C}{m} = \frac{1}{m}\frac{dQ}{dT}$ Latent Heat: $Q = mL$ Heat Transfer: Conduction: $\frac{dQ}{dt} = -KA \frac{dT}{dx}$ (Fourier's Law) Convection: $\frac{dQ}{dt} = hA \Delta T$ (Newton's Law of Cooling) Radiation: $\frac{dQ}{dt} = e\sigma A T^4$ (Stefan-Boltzmann Law) Wien's Displacement Law: $\lambda_m T = b$ 8.2. Kinetic Theory of Gases Ideal Gas Equation: $PV = nRT = NkT$ Average Kinetic Energy per molecule: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ Pressure of an ideal gas: $P = \frac{1}{3}\frac{nm}{V}v_{rms}^2$ Degrees of Freedom ($f$): Monatomic (3), Diatomic (5), Polyatomic (6) Internal Energy: $U = \frac{f}{2}nRT$ Molar Specific Heat: $C_V = \frac{f}{2}R$ $C_P = (\frac{f}{2}+1)R = C_V + R$ (Mayer's Relation) Ratio of Specific Heats: $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$ 8.3. Thermodynamics First Law of Thermodynamics: $\Delta Q = \Delta U + \Delta W$ Work Done: $W = \int P dV$ Isobaric: $W = P\Delta V$ Isothermal: $W = nRT \ln(\frac{V_f}{V_i})$ Adiabatic: $W = \frac{P_iV_i - P_fV_f}{\gamma - 1} = \frac{nR(T_i - T_f)}{\gamma - 1}$ Adiabatic Process: $PV^\gamma = \text{constant}$, $TV^{\gamma-1} = \text{constant}$, $P^{1-\gamma}T^\gamma = \text{constant}$ Efficiency of Heat Engine: $\eta = 1 - \frac{Q_{cold}}{Q_{hot}} = 1 - \frac{T_{cold}}{T_{hot}}$ (Carnot) Coefficient of Performance (Refrigerator): $K = \frac{Q_{cold}}{W} = \frac{T_{cold}}{T_{hot} - T_{cold}}$ Second Law of Thermodynamics: Entropy of isolated system never decreases. 9. Oscillations and Waves 9.1. Simple Harmonic Motion (SHM) Displacement: $x = A \sin(\omega t + \phi)$ Velocity: $v = A\omega \cos(\omega t + \phi) = \pm \omega \sqrt{A^2 - x^2}$ Acceleration: $a = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$ (spring-mass), $\omega = \sqrt{\frac{g}{L}}$ (simple pendulum) Time Period: $T = \frac{2\pi}{\omega}$ Total Energy: $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ 9.2. Wave Motion Wave Equation: $y(x,t) = A \sin(kx - \omega t + \phi)$ Wave Speed: $v = f\lambda = \frac{\omega}{k}$ Speed of transverse wave on string: $v = \sqrt{\frac{T}{\mu}}$ Speed of sound in fluid: $v = \sqrt{\frac{B}{\rho}}$ ; in solid: $v = \sqrt{\frac{Y}{\rho}}$ Intensity: $I = \frac{1}{2}\rho v \omega^2 A^2$ Interference: Constructive: $\Delta x = n\lambda$ Destructive: $\Delta x = (n + \frac{1}{2})\lambda$ Standing Waves: Fixed ends (string): $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ Open organ pipe: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ Closed organ pipe: $\lambda_n = \frac{4L}{2n-1}$, $f_n = \frac{(2n-1)v}{4L}$ Beats: $f_{beat} = |f_1 - f_2|$ Doppler Effect: $f' = f \frac{v \pm v_o}{v \mp v_s}$ (upper signs for approaching, lower for receding) 10. Electrostatics Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{|q_1q_2|}{r^2}$ Electric Field: $\vec{E} = \frac{\vec{F}}{q_0} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}\hat{r}$ Electric Dipole Moment: $\vec{p} = q(2\vec{a})$ Torque on a dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of a dipole: $U = -\vec{p} \cdot \vec{E}$ Electric Potential: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ Relation between E and V: $\vec{E} = -\nabla V$ (for 1D: $E = -\frac{dV}{dx}$) Electric Flux: $\Phi_E = \int \vec{E} \cdot d\vec{A}$ Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ Capacitance: $C = \frac{Q}{V}$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ (with dielectric: $C = \frac{K\epsilon_0 A}{d}$) Capacitors in Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Capacitors in Parallel: $C_{eq} = \sum C_i$ Energy Stored in Capacitor: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ Energy Density in E-field: $u_E = \frac{1}{2}\epsilon_0 E^2$ 11. Current Electricity Current: $I = \frac{dQ}{dt} = nAve_d$ Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ Resistivity: $\rho = \rho_0(1 + \alpha(T - T_0))$ 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: $\sum I = 0$ Loop Rule: $\sum \Delta V = 0$ Electric Power: $P = VI = I^2R = \frac{V^2}{R}$ Cells in Series/Parallel: Series: $E_{eq} = \sum E_i$, $r_{eq} = \sum r_i$ Parallel (identical): $E_{eq} = E$, $r_{eq} = \frac{r}{n}$ Wheatstone Bridge: $\frac{P}{Q} = \frac{R}{S}$ (balanced condition) Meter Bridge: $\frac{R}{S} = \frac{l}{(100-l)}$ Potentiometer: Compare EMFs: $\frac{E_1}{E_2} = \frac{l_1}{l_2}$ Internal Resistance: $r = R(\frac{E}{V} - 1) = R(\frac{l_1}{l_2} - 1)$ 12. Magnetic Effects of Current and Magnetism 12.1. Magnetic Effects of Current Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$ Magnetic field: Straight wire: $B = \frac{\mu_0 I}{2\pi r}$ Circular loop (center): $B = \frac{\mu_0 I}{2R}$ Solenoid: $B = \mu_0 n I$ Toroid: $B = \frac{\mu_0 N I}{2\pi r}$ Ampere's Circuital Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ Lorentz Force: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ Magnetic Force on a current-carrying wire: $\vec{F} = I(\vec{l} \times \vec{B})$ Magnetic moment of current loop: $\vec{M} = I\vec{A}$ Torque on current loop: $\vec{\tau} = \vec{M} \times \vec{B}$ Moving Coil Galvanometer: $I = k\theta$ 12.2. Magnetism Magnetic Susceptibility: $\chi_m = \frac{M}{H}$ Relative Permeability: $\mu_r = 1 + \chi_m$ Magnetic Permeability: $\mu = \mu_0 \mu_r$ Magnetic Field Lines: Properties. Diamagnetism, Paramagnetism, Ferromagnetism: Characteristics. 13. Electromagnetic Induction and Alternating Current 13.1. Electromagnetic Induction (EMI) Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Motional EMF: $\mathcal{E} = (Blv)$ (if $\vec{B}, \vec{l}, \vec{v}$ are mutually perpendicular) Self-Inductance: $\Phi_B = LI$, $\mathcal{E} = -L\frac{dI}{dt}$ Mutual Inductance: $\Phi_{B2} = M I_1$, $\mathcal{E}_2 = -M\frac{dI_1}{dt}$ Energy Stored in Inductor: $U = \frac{1}{2}LI^2$ Energy Density in B-field: $u_B = \frac{B^2}{2\mu_0}$ 13.2. Alternating Current (AC) Instantaneous Voltage/Current: $V = V_0 \sin(\omega t)$, $I = I_0 \sin(\omega t + \phi)$ RMS Values: $V_{rms} = \frac{V_0}{\sqrt{2}}$, $I_{rms} = \frac{I_0}{\sqrt{2}}$ Reactance: Inductive Reactance: $X_L = \omega L$ Capacitive Reactance: $X_C = \frac{1}{\omega C}$ Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ Power in AC Circuit: $P_{avg} = V_{rms}I_{rms}\cos\phi$ (Power Factor: $\cos\phi = \frac{R}{Z}$) Resonance (Series RLC): $X_L = X_C \Rightarrow \omega_0 = \frac{1}{\sqrt{LC}}$ Quality Factor: $Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$ Transformer: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$ (ideal) 14. Electromagnetic Waves Speed of EM waves in vacuum: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$ Speed in medium: $v = \frac{1}{\sqrt{\mu \epsilon}}$ Relation between E and B fields: $E = cB$ Energy Density: $u = u_E + u_B = \frac{1}{2}\epsilon_0 E^2 + \frac{B^2}{2\mu_0} = \epsilon_0 E^2 = \frac{B^2}{\mu_0}$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (Energy flow per unit area per unit time) Momentum carried by EM wave: $p = \frac{U}{c}$ EM Spectrum: Radio, Microwave, IR, Visible, UV, X-ray, Gamma ray (increasing freq/energy, decreasing wavelength) 15. Optics 15.1. Ray Optics Reflection: $\angle i = \angle r$ Mirror Formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ (concave $f 0$) Magnification: $m = -\frac{v}{u} = \frac{h_i}{h_o}$ Refraction: Snell's Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$ Critical Angle: $\sin\theta_c = \frac{n_2}{n_1}$ (for $n_1 > n_2$) Lens Maker's Formula: $\frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2})$ Lens Formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ (concave $f 0$) Power of Lens: $P = \frac{1}{f (\text{in meters})}$ Lenses in Contact: $P_{eq} = P_1 + P_2$, $\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$ Prism: $\delta = (n-1)A$ (for small angle prism) Deviation by prism: $\delta = i+e-A$ Minimum deviation: $n = \frac{\sin((A+\delta_m)/2)}{\sin(A/2)}$ 15.2. Wave Optics Huygens' Principle Young's Double Slit Experiment (YDSE): Fringe Width: $\beta = \frac{\lambda D}{d}$ Condition for Maxima: $d \sin\theta = n\lambda$ Condition for Minima: $d \sin\theta = (n+\frac{1}{2})\lambda$ Diffraction (Single Slit): Condition for Minima: $a \sin\theta = n\lambda$ Width of Central Maxima: $\frac{2\lambda D}{a}$ Malus's Law: $I = I_0 \cos^2\theta$ Brewster's Law: $\tan\theta_p = n$ 16. Modern Physics 16.1. Dual Nature of Radiation and Matter Photon Energy: $E = hf = \frac{hc}{\lambda}$ Momentum of Photon: $p = \frac{h}{\lambda} = \frac{E}{c}$ Photoelectric Effect: $K_{max} = hf - \phi_0$ (Einstein's Equation) Threshold Frequency: $f_0 = \frac{\phi_0}{h}$ Stopping Potential: $eV_s = K_{max}$ de Broglie Wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$ de Broglie for electron acc. through V: $\lambda = \frac{h}{\sqrt{2meV}}$ 16.2. Atoms Rutherford's Model: $\alpha$-scattering Bohr's Model: Quantization of Angular Momentum: $L = n\frac{h}{2\pi}$ Radius of $n^{th}$ orbit: $r_n = \frac{n^2 h^2 \epsilon_0}{\pi m Z e^2} = 0.529 \frac{n^2}{Z} \mathring{A}$ Energy of $n^{th}$ orbit: $E_n = -\frac{m Z^2 e^4}{8 \epsilon_0^2 h^2 n^2} = -13.6 \frac{Z^2}{n^2} \text{ eV}$ Wavelength of emitted photon: $\frac{1}{\lambda} = RZ^2(\frac{1}{n_1^2} - \frac{1}{n_2^2})$ (Rydberg Formula) X-ray Production: $eV = hf_{max} = \frac{hc}{\lambda_{min}}$ Moseley's Law: $\sqrt{f} = a(Z-b)$ 16.3. Nuclei Atomic Mass Unit: $1 \text{ amu} = 1.66 \times 10^{-27} \text{ kg} = 931.5 \text{ MeV}/c^2$ Nuclear Radius: $R = R_0 A^{1/3}$ ($R_0 \approx 1.2 \times 10^{-15} \text{ m}$) Mass Defect: $\Delta m = (Zm_p + (A-Z)m_n) - M_{nucleus}$ Binding Energy: $E_b = \Delta m c^2$ Radioactive Decay Law: $N = N_0 e^{-\lambda t}$ Half-life: $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$ Mean Life: $\tau = \frac{1}{\lambda}$ Activity: $A = \lambda N$ Nuclear Fission and Fusion 16.4. Semiconductors Conductors, Insulators, Semiconductors: Band Theory Intrinsic Semiconductors: $n_e = n_h = n_i$ Extrinsic Semiconductors: N-type: Majority electrons (donors) P-type: Majority holes (acceptors) Diode: Forward Bias: Low resistance Reverse Bias: High resistance (breakdown voltage) Rectifiers (Half-wave, Full-wave) Logic Gates: AND, OR, NOT, NAND, NOR, XOR, XNOR (Truth tables)