NEET Physics Formulae (Class 11 & 12)

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1. Mechanics (Class 11) 1.1 Units & Dimensions Dimensional Analysis Check: $[L]^a[M]^b[T]^c$ Error Analysis: If $Z = A \pm B$, $\Delta Z = \Delta A + \Delta B$. If $Z = A^p B^q / C^r$, $\frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}$ 1.2 Kinematics Average Speed: $\frac{\text{Total Distance}}{\text{Total Time}}$ Average Velocity: $\frac{\text{Total Displacement}}{\text{Total Time}}$ 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) Projectile Motion: Max Height $H = \frac{u^2 \sin^2\theta}{2g}$ Range $R = \frac{u^2 \sin(2\theta)}{g}$ Time of Flight $T = \frac{2u \sin\theta}{g}$ 1.3 Laws of Motion & Friction Newton's 2nd Law: $F = ma$ Impulse: $J = F \Delta t = \Delta p$ Momentum: $p = mv$ Friction: Static: $f_s \le \mu_s N$ Kinetic: $f_k = \mu_k N$ Centripetal Force: $F_c = \frac{mv^2}{r} = m\omega^2 r$ 1.4 Work, Energy & Power Work Done: $W = \vec{F} \cdot \vec{s} = Fs \cos\theta$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Potential Energy (Gravity): $U = mgh$ Potential Energy (Spring): $U = \frac{1}{2}kx^2$ Work-Energy Theorem: $W_{net} = \Delta K$ Power: $P = \frac{W}{t} = \vec{F} \cdot \vec{v}$ Conservation of Mechanical Energy: $E = K + U = \text{constant}$ (for conservative forces) 1.5 System of Particles & Rotational Motion Center of Mass: $X_{CM} = \frac{\sum m_i x_i}{\sum m_i}$ Torque: $\vec{\tau} = \vec{r} \times \vec{F} = I\vec{\alpha}$ Moment of Inertia: $I = \sum m_i r_i^2$ Kinetic Energy (Rotation): $K_{rot} = \frac{1}{2}I\omega^2$ Angular Momentum: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ Conservation of Angular Momentum: $I_1\omega_1 = I_2\omega_2$ 1.6 Gravitation Newton's Law: $F = \frac{Gm_1 m_2}{r^2}$ Gravitational Potential Energy: $U = -\frac{Gm_1 m_2}{r}$ Gravitational Potential: $V = -\frac{GM}{r}$ Escape Velocity: $v_e = \sqrt{\frac{2GM}{R}}$ Orbital Velocity: $v_o = \sqrt{\frac{GM}{r}}$ Kepler's 3rd Law: $T^2 \propto R^3$ 1.7 Properties of Bulk Matter Young's Modulus: $Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}$ Bulk Modulus: $B = \frac{\text{Stress}}{\text{Volume Strain}} = \frac{-P}{\Delta V/V}$ Shear Modulus: $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\phi}$ Pressure: $P = \frac{F}{A}$ Pressure in Fluid: $P = P_0 + \rho gh$ Archimedes' Principle: $F_{buoyant} = V_{immersed} \rho_{fluid} g$ Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ Viscosity: $F = -\eta A \frac{dv}{dy}$ (Newton's Law of Viscosity) Stokes' Law: $F = 6\pi\eta r v$ Surface Tension: $T = \frac{F}{L}$ Excess Pressure in a Drop: $\Delta P = \frac{2T}{R}$ (Bubble: $\frac{4T}{R}$) Capillary Rise: $h = \frac{2T \cos\theta}{r\rho g}$ 1.8 Thermodynamics Zeroth Law: Defines Temperature First Law: $\Delta U = Q - W$ Work Done (Isobaric): $W = P\Delta V$ Work Done (Isothermal): $W = nRT \ln\left(\frac{V_f}{V_i}\right)$ Work Done (Adiabatic): $W = \frac{P_i V_i - P_f V_f}{\gamma - 1}$ Specific Heat: $Q = mc\Delta T$ Mayer's Relation: $C_P - C_V = R$ Efficiency of Heat Engine: $\eta = 1 - \frac{Q_2}{Q_1} = 1 - \frac{T_2}{T_1}$ Coefficient of Performance (Refrigerator): $K = \frac{Q_2}{W} = \frac{T_2}{T_1 - T_2}$ 1.9 Kinetic Theory of Gases Ideal Gas Equation: $PV = nRT = NkT$ Average KE per molecule: $K_{avg} = \frac{3}{2}kT$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ Degrees of Freedom: $f$ Internal Energy: $U = \frac{f}{2}nRT$ Specific Heat Ratio: $\gamma = 1 + \frac{2}{f}$ 1.10 Oscillations & Waves SHM Displacement: $x(t) = A \sin(\omega t + \phi)$ Velocity: $v(t) = A\omega \cos(\omega t + \phi) = \omega\sqrt{A^2 - x^2}$ Acceleration: $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$ Time Period (Spring-Mass): $T = 2\pi\sqrt{\frac{m}{k}}$ Time Period (Simple Pendulum): $T = 2\pi\sqrt{\frac{L}{g}}$ Wave Speed: $v = f\lambda$ Speed of Transverse Wave on String: $v = \sqrt{\frac{T}{\mu}}$ Speed of Sound in Gas: $v = \sqrt{\frac{\gamma P}{\rho}}$ Doppler Effect: $f' = f_0 \left(\frac{v \pm v_o}{v \mp v_s}\right)$ Beats Frequency: $f_{beat} = |f_1 - f_2|$ Standing Waves (String fixed at both ends): $L = \frac{n\lambda}{2}$, $f_n = \frac{nv}{2L}$ Standing Waves (Open Organ Pipe): $L = \frac{n\lambda}{2}$, $f_n = \frac{nv}{2L}$ Standing Waves (Closed Organ Pipe): $L = \frac{(2n-1)\lambda}{4}$, $f_n = \frac{(2n-1)v}{4L}$ 2. Electrodynamics (Class 12) 2.1 Electrostatics Coulomb's Law: $F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$ Electric Field: $E = \frac{F}{q_0} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ Electric Potential: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ Potential Difference: $\Delta V = -\int \vec{E} \cdot d\vec{l}$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on Dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ Capacitance: $C = \frac{Q}{V}$ Capacitance of Parallel Plate: $C = \frac{\epsilon_0 A}{d}$ Energy Stored in Capacitor: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ Series Capacitors: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ Parallel Capacitors: $C_{eq} = \sum C_i$ 2.2 Current Electricity Current: $I = \frac{dQ}{dt}$ Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ Resistivity: $\rho_T = \rho_0 (1 + \alpha(T - T_0))$ Series Resistors: $R_{eq} = \sum R_i$ Parallel Resistors: $\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$ Joule's Heating: $H = I^2Rt = VIt = \frac{V^2}{R}t$ Power: $P = VI = I^2R = \frac{V^2}{R}$ Kirchhoff's Laws: Junction Rule: $\sum I = 0$ Loop Rule: $\sum \Delta V = 0$ Wheatstone Bridge: $\frac{P}{Q} = \frac{R}{S}$ (balanced) Meter Bridge: $\frac{R}{S} = \frac{L}{(100-L)}$ Potentiometer: $\frac{E_1}{E_2} = \frac{L_1}{L_2}$ 2.3 Magnetic Effects of Current & Magnetism 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): $B = \frac{\mu_0 I}{2\pi r}$ Magnetic Field (Circular Loop Center): $B = \frac{\mu_0 I}{2R}$ Magnetic Field (Solenoid): $B = \mu_0 n I$ Ampere's Circuital Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ Lorentz Force: $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ Force on Current Carrying Wire: $\vec{F} = I(\vec{l} \times \vec{B})$ Torque on Current Loop: $\vec{\tau} = \vec{M} \times \vec{B}$, where $\vec{M} = NI\vec{A}$ Cyclotron Frequency: $f = \frac{qB}{2\pi m}$ Magnetic Potential Energy: $U = -\vec{M} \cdot \vec{B}$ 2.4 Electromagnetic Induction & AC Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ Faraday's Law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Motional EMF: $\mathcal{E} = B L v$ Self Inductance: $\Phi_B = LI$, $\mathcal{E} = -L\frac{dI}{dt}$ Mutual Inductance: $\Phi_{B2} = MI_1$, $\mathcal{E}_2 = -M\frac{dI_1}{dt}$ Energy Stored in Inductor: $U = \frac{1}{2}LI^2$ RMS Current/Voltage: $I_{rms} = \frac{I_0}{\sqrt{2}}$, $V_{rms} = \frac{V_0}{\sqrt{2}}$ Reactance (Inductive): $X_L = \omega L$ Reactance (Capacitive): $X_C = \frac{1}{\omega C}$ Impedance (RLC Series): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ Resonant Frequency: $f_0 = \frac{1}{2\pi\sqrt{LC}}$ Power in AC Circuit: $P = V_{rms}I_{rms} \cos\phi$ Transformer: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$ 2.5 Electromagnetic Waves Speed of EM Wave: $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$ $c = E/B$ Energy Density: $u = u_E + u_B = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ 3. Optics (Class 12) 3.1 Ray Optics 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): $n_1 \sin\theta_1 = n_2 \sin\theta_2$ Lens Maker's Formula: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ Lens Formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ (Concave $f 0$) Power of Lens: $P = \frac{1}{f}$ (in diopters, $f$ in meters) Combination of Lenses: $P_{eq} = P_1 + P_2$, $\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$ Refraction at Spherical Surface: $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$ 3.2 Wave Optics Huygens' Principle Constructive Interference: Path difference $\Delta x = n\lambda$ Destructive Interference: Path difference $\Delta x = (n + \frac{1}{2})\lambda$ Young's Double Slit Experiment: Fringe Width: $\beta = \frac{\lambda D}{d}$ Position of Bright Fringe: $y_n = \frac{n\lambda D}{d}$ Position of Dark Fringe: $y_n = (n + \frac{1}{2})\frac{\lambda D}{d}$ Diffraction (Single Slit): Minima: $a \sin\theta = n\lambda$ Angular width of central maxima: $2\theta = \frac{2\lambda}{a}$ Brewster's Law: $\tan\theta_p = n$ 4. Modern Physics (Class 12) 4.1 Dual Nature of Radiation and Matter Photon Energy: $E = hf = \frac{hc}{\lambda}$ Photon Momentum: $p = \frac{h}{\lambda} = \frac{E}{c}$ Photoelectric Effect (Einstein): $K_{max} = hf - \phi_0$ Threshold Frequency: $f_0 = \frac{\phi_0}{h}$ Stopping Voltage: $eV_s = K_{max}$ de Broglie Wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$ de Broglie (Electron): $\lambda = \frac{1.227}{\sqrt{V}} \text{ nm}$ 4.2 Atoms & Nuclei Bohr's Model: Radius: $r_n = \frac{n^2 a_0}{Z}$ ($a_0 = 0.529 \text{ Å}$) Energy: $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$ Frequency of emitted photon: $hf = E_i - E_f$ Rydberg Formula: $\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ 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: $\tau = \frac{1}{\lambda}$ 4.3 Electronic Devices Diode Current (Ideal): $I = I_0 (e^{eV/kT} - 1)$ Transistor Current Relation: $I_E = I_B + I_C$ Current Gain ($\alpha$): $\alpha = \frac{I_C}{I_E}$ Current Gain ($\beta$): $\beta = \frac{I_C}{I_B}$ Relation between $\alpha$ and $\beta$: $\beta = \frac{\alpha}{1-\alpha}$, $\alpha = \frac{\beta}{1+\beta}$ Logic Gates: AND, OR, NOT, NAND, NOR, XOR, XNOR 4.4 Communication Systems Modulation Index (AM): $m_a = \frac{A_m}{A_c}$ Bandwidth (AM): $2f_m$ Range of TV transmission: $d = \sqrt{2Rh_T}$