JEE Physics Formulas

Cheatsheet Content

### Units & Dimensions - **Fundamental Quantities & Units:** - Length: meter (m) - Mass: kilogram (kg) - Time: second (s) - Electric Current: ampere (A) - Temperature: Kelvin (K) - Luminous Intensity: candela (cd) - Amount of Substance: mole (mol) - **Dimensional Formulae:** - Velocity: $[L^1 T^{-1}]$ - Acceleration: $[L^1 T^{-2}]$ - Force: $[M^1 L^1 T^{-2}]$ - Work/Energy: $[M^1 L^2 T^{-2}]$ - Power: $[M^1 L^2 T^{-3}]$ - Pressure: $[M^1 L^{-1} T^{-2}]$ - Frequency: $[T^{-1}]$ - Angular Velocity: $[T^{-1}]$ - Torque: $[M^1 L^2 T^{-2}]$ - Momentum: $[M^1 L^1 T^{-1}]$ - **Principle of Homogeneity:** Dimensions of each term in an equation must be the same. ### Kinematics - **Equations of Motion (Constant Acceleration):** - $v = u + at$ - $s = ut + \frac{1}{2}at^2$ - $v^2 = u^2 + 2as$ - $s_n = u + \frac{a}{2}(2n - 1)$ (Displacement in nth second) - **Relative Velocity:** - $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ - **Projectile Motion:** - Time of Flight: $T = \frac{2u \sin\theta}{g}$ - Max Height: $H = \frac{u^2 \sin^2\theta}{2g}$ - Range: $R = \frac{u^2 \sin(2\theta)}{g}$ - Equation of Trajectory: $y = x \tan\theta - \frac{gx^2}{2u^2 \cos^2\theta}$ ### Newton's Laws & Friction - **Newton's First Law:** Inertia. - **Newton's Second Law:** $\vec{F} = m\vec{a}$ - **Newton's Third Law:** Action-reaction pairs. - **Impulse:** $\vec{J} = \Delta\vec{p} = \vec{F}_{avg}\Delta t$ - **Static Friction:** $f_s \le \mu_s N$ - **Kinetic Friction:** $f_k = \mu_k N$ ($\mu_k ### Work, Energy & Power - **Work Done:** $W = \vec{F} \cdot \vec{s} = Fs \cos\theta$ - **Kinetic Energy:** $K = \frac{1}{2}mv^2$ - **Potential Energy:** - Gravitational: $U_g = mgh$ - 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:** $E = K + U = \text{constant}$ (for conservative forces) ### Rotational Motion - **Angular Displacement:** $\theta$ - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ - **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:** $I = \sum m_i r_i^2$ (discrete), $I = \int r^2 dm$ (continuous) - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ - **Newton's Second Law for Rotation:** $\tau = I\alpha$ - **Angular Momentum:** $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ - **Conservation of Angular Momentum:** If $\tau_{net} = 0$, then $\vec{L} = \text{constant}$. - **Rotational Kinetic Energy:** $K_R = \frac{1}{2}I\omega^2$ ### Gravitation - **Newton's Law of Gravitation:** $F = \frac{Gm_1 m_2}{r^2}$ - **Gravitational Acceleration:** $g = \frac{GM}{R^2}$ - **Variation of g:** - With height: $g' = g(1 - \frac{2h}{R})$ (for $h \ll R$) - With depth: $g' = g(1 - \frac{d}{R})$ - **Gravitational Potential Energy:** $U = -\frac{Gm_1 m_2}{r}$ - **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 Third Law: $T^2 \propto r^3$) ### Properties of Solids & Fluids - **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:** $G = \frac{\text{Shear Stress}}{\text{Shear Strain}}$ - **Pascal's Law:** Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. - **Pressure in Fluid Column:** $P = P_0 + \rho gh$ - **Archimedes' Principle:** Buoyant force $F_B = \rho_{fluid} V_{submerged} g$ - **Equation of Continuity:** $A_1 v_1 = A_2 v_2$ (for incompressible fluid) - **Bernoulli's Principle:** $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ - **Viscosity (Stokes' Law):** $F = 6\pi \eta r v$ - **Surface Tension:** $S = \frac{F}{L}$ - **Excess Pressure in Bubble/Drop:** - Liquid Drop: $\Delta P = \frac{2S}{R}$ - Soap Bubble: $\Delta P = \frac{4S}{R}$ - **Capillary Rise:** $h = \frac{2S \cos\theta}{\rho gr}$ ### Thermal Physics - **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 Transfer:** - Conduction: $\frac{dQ}{dt} = -KA \frac{dT}{dx}$ - Convection: Involves fluid movement. - Radiation (Stefan-Boltzmann Law): $P = e\sigma A T^4$ - **Specific Heat Capacity:** $Q = mc\Delta T$ - **Latent Heat:** $Q = mL$ - **First Law of Thermodynamics:** $\Delta U = Q - W$ - **Work Done by Gas:** $W = \int P dV$ - **Ideal Gas Equation:** $PV = nRT = NkT$ - **Kinetic Theory of Gases:** - Average Kinetic Energy per molecule: $\langle K \rangle = \frac{3}{2}kT$ - RMS speed: $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$ - **Mayer's Relation:** $C_P - C_V = R$ - **Adiabatic Process:** $PV^\gamma = \text{constant}$, $T V^{\gamma-1} = \text{constant}$, $T^\gamma P^{1-\gamma} = \text{constant}$ - $\gamma = C_P/C_V$ ### Simple Harmonic Motion & Waves - **SHM Equation:** $x(t) = A \sin(\omega t + \phi)$ - **Velocity:** $v(t) = A\omega \cos(\omega t + \phi)$ - **Acceleration:** $a(t) = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x$ - **Angular Frequency:** $\omega = \sqrt{k/m}$ (spring-mass), $\omega = \sqrt{g/L}$ (simple pendulum) - **Time Period:** $T = 2\pi/\omega$ - **Wave Equation:** $y(x,t) = A \sin(kx - \omega t + \phi)$ - Wave Speed: $v = f\lambda = \omega/k$ - **Speed of Sound:** $v = \sqrt{B/\rho}$ (solids/liquids), $v = \sqrt{\gamma P/\rho}$ (gases) - **Intensity:** $I = \frac{1}{2}\rho \omega^2 A^2 v$ - **Doppler Effect:** $f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$ (+$v_o$ for observer approaching, -$v_s$ for source approaching) - **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: $\lambda_n = \frac{4L}{(2n-1)}$, $f_n = \frac{(2n-1)v}{4L}$ ### Electrostatics - **Coulomb's Law:** $F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ - **Electric Field due to Point Charge:** $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ - **Electric Potential:** $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ - **Relation between E and V:** $\vec{E} = -\nabla V$ - **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:** $U = -\vec{p} \cdot \vec{E}$ - **Gauss's Law:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - **Capacitance:** $C = Q/V$ - **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:** $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ ### Current Electricity - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ - **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: $\sum I = 0$ (at a junction) - Loop Rule: $\sum \Delta V = 0$ (around a closed loop) - **Power Dissipated:** $P = VI = I^2R = \frac{V^2}{R}$ - **Wheatstone Bridge:** Balanced condition $\frac{R_1}{R_2} = \frac{R_3}{R_4}$ - **Meter Bridge:** $\frac{R}{S} = \frac{l}{(100-l)}$ - **Potentiometer:** - Comparison of EMFs: $\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{l_1}{l_2}$ - Internal Resistance: $r = R \left(\frac{l_1}{l_2} - 1\right)$ ### Magnetism - **Lorentz Force:** $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ - **Magnetic Force on Current-Carrying Conductor:** $\vec{F} = I(\vec{L} \times \vec{B})$ - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$ - **Magnetic Field:** - Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ - Circular Loop (center): $B = \frac{\mu_0 I}{2R}$ - Solenoid: $B = \mu_0 n I$ - **Ampere's Circuital Law:** $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ - **Torque on Current Loop:** $\vec{\tau} = \vec{M} \times \vec{B}$ where $\vec{M} = NI\vec{A}$ (magnetic dipole moment) - **Moving Coil Galvanometer:** $NIAB = k\phi$ - **Earth's Magnetic Field:** - Horizontal component: $B_H = B_E \cos I$ - Vertical component: $B_V = B_E \sin I$ - Dip angle: $\tan I = B_V/B_H$ ### Electromagnetic Induction & AC - **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:** $\mathcal{E} = Blv$ - **Lenz's Law:** Induced current opposes the change in magnetic flux. - **Self-Inductance:** $\Phi_B = LI$, $\mathcal{E} = -L \frac{dI}{dt}$ - **Mutual Inductance:** $\Phi_{21} = M_{21} I_1$, $\mathcal{E}_2 = -M_{21} \frac{dI_1}{dt}$ - **Energy Stored in Inductor:** $U = \frac{1}{2}LI^2$ - **AC Circuits:** - RMS Value: $I_{rms} = I_0/\sqrt{2}$, $V_{rms} = V_0/\sqrt{2}$ - Inductive Reactance: $X_L = \omega L$ - Capacitive Reactance: $X_C = \frac{1}{\omega C}$ - Impedance (LCR Series): $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - Phase Angle: $\tan\phi = \frac{X_L - X_C}{R}$ - Power Factor: $\cos\phi = R/Z$ - Resonant Frequency: $f_r = \frac{1}{2\pi\sqrt{LC}}$ ### Electromagnetic Waves - **Speed of EM waves in vacuum:** $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ - **Speed in medium:** $v = \frac{1}{\sqrt{\mu \epsilon}}$ - **Relationship between E and B field:** $E = cB$ - **Poynting Vector:** $\vec{S} = \frac{1}{\mu_0}(\vec{E} \times \vec{B})$ (Intensity of EM wave) - **EM Spectrum:** Radio, Microwave, Infrared, Visible, Ultraviolet, X-ray, Gamma ray (increasing freq, decreasing wavelength) ### Ray Optics - **Mirror Formula:** $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ - **Magnification:** $m = -\frac{v}{u} = \frac{h_i}{h_o}$ - **Refractive Index:** $n = \frac{c}{v}$ - **Snell's Law:** $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - **Critical Angle:** $\sin C = n_2/n_1$ (for $n_1 > n_2$) - **Lens Maker's Formula:** $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ - **Lens Formula:** $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ - **Power of Lens:** $P = \frac{1}{f}$ (in diopters, f in meters) - **Prism:** $\delta = (n-1)A$ (for small angle prism) - **Dispersion:** Angular dispersion $\theta = (n_v - n_r)A$ ### Wave Optics - **Huygens' Principle:** Every point on a wavefront is a source of secondary wavelets. - **Interference (Young's Double Slit Experiment):** - Path Difference: $\Delta x = d \sin\theta$ - Bright Fringes (constructive): $\Delta x = n\lambda$ - Dark Fringes (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: $a \sin\theta = (n + \frac{1}{2})\lambda$ (approx) - Angular width of central maxima: $2\theta = \frac{2\lambda}{a}$ - **Polarization:** Malus' Law: $I = I_0 \cos^2\theta$ ### Modern Physics - **Photoelectric Effect:** - Energy of Photon: $E = hf = \frac{hc}{\lambda}$ - Einstein's Photoelectric Equation: $K_{max} = hf - \phi_0$ (where $\phi_0$ is work function) - **De Broglie Wavelength:** $\lambda = \frac{h}{p} = \frac{h}{mv}$ - **Bohr's Model of Hydrogen Atom:** - Radius of nth orbit: $r_n = \frac{n^2 a_0}{Z}$ ($a_0 = 0.529 \text{ Å}$) - Energy of nth orbit: $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$ - Wavenumber of spectral lines: $\frac{1}{\lambda} = RZ^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ - **Radioactivity:** - Decay Law: $N = N_0 e^{-\lambda t}$ - Half-life: $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$ - Mean Life: $\tau = \frac{1}{\lambda}$ - **Mass-Energy Equivalence:** $E = mc^2$ ### Semiconductors & Communication Systems - **Diode:** - Forward biasing: Low resistance - Reverse biasing: High resistance - **Transistor (BJT):** - Current Gain ($\alpha$): $I_C/I_E$ - Current Gain ($\beta$): $I_C/I_B$ ($\beta = \frac{\alpha}{1-\alpha}$) - **Logic Gates:** AND, OR, NOT, NAND, NOR, XOR, XNOR (Truth Tables) - **Communication Systems:** - Modulation: Amplitude Modulation (AM), Frequency Modulation (FM) - Bandwidth, Transmission Mediums