JEE Physics Formulas Cheatsheet

Cheatsheet Content

1. Mechanics Kinematics 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: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ Equations of Motion (constant $a$): $v = u + at$ $s = ut + \frac{1}{2}at^2$ $v^2 = u^2 + 2as$ $s_n = u + \frac{a}{2}(2n-1)$ (displacement in $n$-th second) Relative Velocity: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ Newton's Laws of Motion Newton's First Law: Inertia Newton's Second Law: $\vec{F}_{net} = m\vec{a}$ Newton's Third Law: $\vec{F}_{AB} = -\vec{F}_{BA}$ Impulse: $\vec{J} = \Delta \vec{p} = \vec{F}_{avg}\Delta t = \int \vec{F} dt$ Conservation of Momentum: If $\vec{F}_{net} = 0$, then $\Delta \vec{p} = 0 \implies \vec{p}_{initial} = \vec{p}_{final}$ Friction: Static: $f_s \le \mu_s N$ Kinetic: $f_k = \mu_k N$ Work, Energy & Power Work Done: $W = \vec{F} \cdot \vec{d} = Fd \cos\theta$ (constant force) Work Done (variable force): $W = \int \vec{F} \cdot d\vec{r}$ Kinetic Energy: $K = \frac{1}{2}mv^2$ Potential Energy (Gravitational): $U_g = mgh$ Potential Energy (Spring): $U_s = \frac{1}{2}kx^2$ Work-Energy Theorem: $W_{net} = \Delta K$ Conservation of Mechanical Energy: $E = K + U = \text{constant}$ (if only conservative forces act) Power: $P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$ 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: $v = r\omega$, $a_t = r\alpha$, $a_c = \frac{v^2}{r} = r\omega^2$ Moment of Inertia: $I = \sum m_i r_i^2 = \int r^2 dm$ 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: If $\vec{\tau}_{net} = 0$, then $\vec{L}_{initial} = \vec{L}_{final}$ Rotational Kinetic Energy: $K_R = \frac{1}{2}I\omega^2$ Rolling without slipping: $v_{CM} = R\omega$, $a_{CM} = R\alpha$ Parallel Axis Theorem: $I = I_{CM} + Md^2$ Perpendicular Axis Theorem (for planar objects): $I_z = I_x + I_y$ Gravitation Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$ Gravitational Acceleration: $g = G \frac{M}{R^2}$ Variation of $g$: With height: $g_h = g(1 - \frac{2h}{R})$ for $h \ll R$ With depth: $g_d = g(1 - \frac{d}{R})$ Gravitational Potential Energy: $U = -G \frac{M m}{r}$ Gravitational Potential: $V = -G \frac{M}{r}$ 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 with Sun at one focus. 2nd Law: Equal areas in equal times ($\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$). 3rd Law: $T^2 \propto R^3$. Properties of Matter Young's Modulus: $Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}$ Bulk Modulus: $B = \frac{\Delta P}{-\Delta V/V}$ Shear Modulus: $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{x/h}$ Poisson's Ratio: $\sigma = -\frac{\text{lateral strain}}{\text{longitudinal strain}}$ Pressure: $P = \frac{F}{A}$ Pressure in fluid: $P = P_0 + \rho gh$ 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: $A_1 v_1 = A_2 v_2$ Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ Surface Tension: $T = \frac{F}{L}$ Excess Pressure in bubble: $\Delta P = \frac{2T}{R}$ (liquid drop), $\Delta P = \frac{4T}{R}$ (soap bubble) Capillary Rise: $h = \frac{2T \cos\theta}{\rho g r}$ Viscosity (Stokes' Law): $F = 6\pi\eta rv$ Reynolds Number: $Re = \frac{\rho vd}{\eta}$ 2. Heat & Thermodynamics Thermal Expansion Linear: $\Delta L = L_0 \alpha \Delta T$ Area: $\Delta A = A_0 \beta \Delta T$, where $\beta \approx 2\alpha$ Volume: $\Delta V = V_0 \gamma \Delta T$, where $\gamma \approx 3\alpha$ Heat Transfer Conduction: $\frac{dQ}{dt} = -KA \frac{dT}{dx}$ Convection: $\frac{dQ}{dt} = hA(T_s - T_f)$ Radiation (Stefan-Boltzmann Law): $P = e\sigma A T^4$ Wein's Displacement Law: $\lambda_m T = b$ Newton's Law of Cooling: $\frac{dT}{dt} = -k(T - T_s)$ Thermodynamics First Law of Thermodynamics: $\Delta U = Q - W$ Work Done: $W = \int P dV$ Specific Heat: $Q = mc\Delta T$ Molar Heat Capacity: $C_P - C_V = R$ (Mayer's relation) Ratio of Specific Heats: $\gamma = C_P/C_V$ Isothermal Process: $PV = \text{constant}$ ($T=\text{constant}$), $W = nRT \ln(\frac{V_f}{V_i})$ Adiabatic Process: $PV^\gamma = \text{constant}$, $T V^{\gamma-1} = \text{constant}$, $P^{1-\gamma} T^\gamma = \text{constant}$ Isobaric Process: $W = P\Delta V$ ($P=\text{constant}$) Isochoric Process: $W = 0$ ($V=\text{constant}$) Efficiency of Heat Engine: $\eta = 1 - \frac{Q_C}{Q_H} = 1 - \frac{T_C}{T_H}$ (Carnot) Coefficient of Performance (Refrigerator): $K = \frac{Q_C}{W} = \frac{T_C}{T_H - T_C}$ Kinetic Theory of Gases Ideal Gas Equation: $PV = nRT = Nk_B T$ Average Kinetic Energy per molecule: $K_{avg} = \frac{3}{2}k_B T$ RMS Speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3k_B T}{m}}$ Pressure of an ideal gas: $P = \frac{1}{3}\frac{N}{V}m v_{rms}^2$ Degrees of Freedom: $f$ Internal Energy: $U = \frac{f}{2}nRT$ Molar Heat Capacity: $C_V = \frac{f}{2}R$, $C_P = (\frac{f}{2}+1)R$ 3. Oscillations & Waves Simple Harmonic Motion (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: $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$ 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: $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ Waves 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: $v = \sqrt{\frac{B}{\rho}}$ Speed of sound in solid rod: $v = \sqrt{\frac{Y}{\rho}}$ Intensity: $I = \frac{P}{A} \propto A^2 \omega^2$ Doppler Effect: $f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$ Standing Waves: String fixed at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$ 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: $f_{beat} = |f_1 - f_2|$ 4. Electrostatics Coulomb's Law: $F = k \frac{|q_1 q_2|}{r^2}$ where $k = \frac{1}{4\pi\epsilon_0}$ Electric Field: $\vec{E} = \frac{\vec{F}}{q_0} = k \frac{q}{r^2} \hat{r}$ Electric Field (continuous charge dist.): $\vec{E} = \int d\vec{E}$ Electric Potential: $V = \frac{U}{q_0} = k \frac{q}{r}$ Relation between E and V: $\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$ Electric Dipole Moment: $\vec{p} = q\vec{d}$ Torque on a dipole: $\vec{\tau} = \vec{p} \times \vec{E}$ Potential Energy of a dipole: $U = -\vec{p} \cdot \vec{E}$ Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ Capacitance Capacitance: $C = \frac{Q}{V}$ Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ With Dielectric: $C = \frac{K\epsilon_0 A}{d}$ In Series: $\frac{1}{C_{eq}} = \sum \frac{1}{C_i}$ In Parallel: $C_{eq} = \sum C_i$ Energy Stored: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ Energy Density: $u = \frac{1}{2}\epsilon_0 E^2$ 5. Current Electricity Current: $I = \frac{dQ}{dt} = nAve_d$ Ohm's Law: $V = IR$ Resistance: $R = \rho \frac{L}{A}$ Resistivity (temperature dependence): $\rho_T = \rho_0 [1 + \alpha(T - T_0)]$ 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: $\sum I_{in} = \sum I_{out}$ Loop Rule: $\sum \Delta V = 0$ Electrical Power: $P = VI = I^2R = \frac{V^2}{R}$ Cells in Series (same polarity): $E_{eq} = \sum E_i$, $r_{eq} = \sum r_i$ Cells in Parallel (same E): $E_{eq} = E$, $\frac{1}{r_{eq}} = \sum \frac{1}{r_i}$ Wheatstone Bridge (balanced): $\frac{R_1}{R_2} = \frac{R_3}{R_4}$ Meter Bridge: $\frac{R}{S} = \frac{l}{(100-l)}$ Potentiometer: Comparing EMFs: $\frac{E_1}{E_2} = \frac{l_1}{l_2}$ Internal Resistance: $r = R(\frac{l_1}{l_2} - 1)$ 6. 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 (long straight wire): $B = \frac{\mu_0 I}{2\pi r}$ Magnetic Field (circular loop at center): $B = \frac{\mu_0 I}{2R}$ Magnetic Field (solenoid): $B = \mu_0 n I$ Magnetic Field (toroid): $B = \frac{\mu_0 N I}{2\pi r}$ Ampere's 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 current carrying wire: $\vec{F} = I(\vec{l} \times \vec{B})$ Force between two parallel current wires: $F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$ Torque on a current loop: $\vec{\tau} = \vec{M} \times \vec{B}$, where $\vec{M} = NI\vec{A}$ (magnetic dipole moment) Moving Coil Galvanometer: $NIAB = k\theta$ Cyclotron Frequency: $f = \frac{qB}{2\pi m}$ Magnetism & Matter Magnetic Field Lines (properties) Earth's Magnetism: Angle of Declination, Angle of Dip, Horizontal Component $B_H = B_E \cos I$ Magnetic Permeability: $\mu = \mu_0 \mu_r = \mu_0 (1 + \chi)$ Magnetic Susceptibility: $\chi$ Magnetic Intensity: $H = \frac{B}{\mu_0} - M$ Relation: $B = \mu_0(H+M)$ Types of Materials: Diamagnetic, Paramagnetic, Ferromagnetic 7. Electromagnetic Induction & AC EMI Magnetic Flux: $\Phi_B = \int \vec{B} \cdot d\vec{A} = BA\cos\theta$ Faraday's Law of EMI: $\mathcal{E} = -\frac{d\Phi_B}{dt}$ Motional EMF: $\mathcal{E} = (Blv)$ (for conductor moving perpendicular to B) Self Inductance: $\Phi_B = LI$, $\mathcal{E} = -L \frac{dI}{dt}$ Mutual Inductance: $\Phi_{21} = MI_1$, $\mathcal{E}_2 = -M \frac{dI_1}{dt}$ Energy Stored in Inductor: $U = \frac{1}{2}LI^2$ Energy Density: $u = \frac{B^2}{2\mu_0}$ $RL$ Circuit: $I = I_0(1 - e^{-t/\tau})$ (charging), $I = I_0 e^{-t/\tau}$ (discharging), where $\tau = L/R$ Alternating Current (AC) AC Voltage: $V = V_m \sin(\omega t)$ AC Current: $I = I_m \sin(\omega t + \phi)$ RMS Values: $V_{rms} = \frac{V_m}{\sqrt{2}}$, $I_{rms} = \frac{I_m}{\sqrt{2}}$ Inductive Reactance: $X_L = \omega L$ Capacitive Reactance: $X_C = \frac{1}{\omega C}$ Impedance (RLC Series): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ Power in AC Circuit: $P_{avg} = V_{rms} I_{rms} \cos\phi$ ($\cos\phi$ is power factor) Resonance Frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$ 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) 8. Electromagnetic Waves Speed 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: $c = E/B$ Energy Density: $u_{avg} = \frac{1}{2}\epsilon_0 E_{rms}^2 + \frac{1}{2\mu_0} B_{rms}^2 = \epsilon_0 E_{rms}^2 = \frac{B_{rms}^2}{\mu_0}$ Poynting Vector: $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (direction of propagation and energy flow) Intensity: $I = c u_{avg}$ Momentum carried: $p = \frac{U}{c}$ Pressure: $P = I/c$ (perfect absorption), $P = 2I/c$ (perfect reflection) EM Spectrum: Radio, Microwave, IR, Visible, UV, X-ray, Gamma ray (increasing freq, decreasing wavelength) 9. Optics Ray Optics Reflection: Angle of incidence = Angle of 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): $n_1 \sin i = n_2 \sin r$ Refractive Index: $n = \frac{c}{v}$ Critical Angle: $\sin C = \frac{n_2}{n_1}$ (for TIR, $n_1 > n_2$) Lens Formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ (converging $f>0$, diverging $f Lens Maker's Formula: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ Power of Lens: $P = \frac{1}{f}$ (in diopters, $f$ in meters) Total Power (lenses in contact): $P = P_1 + P_2$ Magnification (lenses in contact): $m = m_1 m_2$ Prism: $\delta = (n-1)A$ (for small angle prism), $\delta_{min} = 2i - A$, $n = \frac{\sin((A+\delta_m)/2)}{\sin(A/2)}$ Wave Optics Huygens' Principle Interference: Path difference: $\Delta x = d \sin\theta$ Constructive: $\Delta x = n\lambda$ Destructive: $\Delta x = (n + \frac{1}{2})\lambda$ Fringe Width: $\beta = \frac{\lambda D}{d}$ Diffraction (Single Slit): Minima: $a \sin\theta = n\lambda$ Maxima (approx): $a \sin\theta = (n+\frac{1}{2})\lambda$ Angular Width of Central Maxima: $2\theta = \frac{2\lambda}{a}$ Polarization (Brewster's Law): $\tan i_p = n$ Malus' Law: $I = I_0 \cos^2\theta$ 10. Modern Physics Dual Nature of Radiation and Matter Photon Energy: $E = hf = \frac{hc}{\lambda}$ Photon Momentum: $p = \frac{E}{c} = \frac{h}{\lambda}$ Photoelectric Effect (Einstein): $K_{max} = hf - \phi_0$, where $\phi_0 = hf_0$ (work function) Stopping Potential: $eV_s = K_{max}$ De Broglie Wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$ De Broglie Wavelength of electron accelerated by V: $\lambda = \frac{12.27}{\sqrt{V}} \text{ Å}$ Atomic Structure Bohr's Postulates Radius of $n^{th}$ orbit: $r_n = 0.529 \frac{n^2}{Z} \text{ Å}$ Energy of $n^{th}$ orbit: $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$ Wavelength of emitted photon: $\frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ (Rydberg formula) Hydrogen Spectrum Series: Lyman ($n_1=1$), Balmer ($n_1=2$), Paschen ($n_1=3$), etc. Nuclear Physics Mass Defect: $\Delta m = (Z m_p + N m_n) - M_{nucleus}$ 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: $A = -\frac{dN}{dt} = \lambda N$ Units: 1 amu = $931.5 \text{ MeV/c}^2$, 1 Curie = $3.7 \times 10^{10} \text{ Bq}$ 11. Semiconductors & Electronic Devices Diode: Forward bias, Reverse bias Rectifiers: Half-wave, Full-wave Transistor (BJT): Current relations: $I_E = I_B + I_C$, $I_C = \alpha I_E + I_{CBO}$, $I_C = \beta I_B + I_{CEO}$ Current gains: $\alpha = \frac{I_C}{I_E}$, $\beta = \frac{I_C}{I_B}$ Relation: $\beta = \frac{\alpha}{1-\alpha}$ Logic Gates: AND, OR, NOT, NAND, NOR, XOR, XNOR (Truth Tables) 12. Communication Systems Modulation: Amplitude Modulation (AM), Frequency Modulation (FM) Bandwidth requirements Range of TV transmission: $d = \sqrt{2Rh_T}$ Ground waves, Sky waves, Space waves Constants Speed of light, $c = 3 \times 10^8 \text{ m/s}$ Planck's constant, $h = 6.626 \times 10^{-34} \text{ J s}$ Gravitational constant, $G = 6.67 \times 10^{-11} \text{ N m}^2 / \text{kg}^2$ Elementary charge, $e = 1.602 \times 10^{-19} \text{ C}$ Permittivity of free space, $\epsilon_0 = 8.854 \times 10^{-12} \text{ C}^2 / \text{N m}^2$ Permeability of free space, $\mu_0 = 4\pi \times 10^{-7} \text{ T m} / \text{A}$ Boltzmann constant, $k_B = 1.38 \times 10^{-23} \text{ J/K}$ Avogadro's number, $N_A = 6.022 \times 10^{23} \text{ mol}^{-1}$ Universal Gas Constant, $R = 8.314 \text{ J/(mol K)}$ Mass of electron, $m_e = 9.1 \times 10^{-31} \text{ kg}$ Mass of proton, $m_p = 1.672 \times 10^{-27} \text{ kg}$