JEE Physics Formulas Cheatshee

Cheatsheet Content

### Units and Dimensions - **Fundamental Quantities:** Mass (kg), Length (m), Time (s), Electric Current (A), Temperature (K), Luminous Intensity (cd), Amount of Substance (mol). - **Derived Quantities:** All other quantities (e.g., Velocity, Force, Energy). - **Dimensional Analysis:** - Principle of Homogeneity: Dimensions of terms on both sides of an equation must be the same. - Uses: Check correctness of formula, derive relations between physical quantities. - Limitations: Cannot determine dimensionless constants, relations involving trigonometric/logarithmic functions. - **Common Dimensions:** - Velocity: $[L T^{-1}]$ - Acceleration: $[L T^{-2}]$ - Force: $[M L T^{-2}]$ - Energy/Work/Torque: $[M L^2 T^{-2}]$ - Power: $[M L^2 T^{-3}]$ - Pressure/Stress: $[M L^{-1} T^{-2}]$ - Frequency: $[T^{-1}]$ - Moment of Inertia: $[M L^2]$ - Gravitational Constant (G): $[M^{-1} L^3 T^{-2}]$ - Planck's Constant (h): $[M L^2 T^{-1}]$ - Electric Charge (Q): $[A T]$ - Electric Potential (V): $[M L^2 T^{-3} A^{-1}]$ - Resistance (R): $[M L^2 T^{-3} A^{-2}]$ - Magnetic Field (B): $[M T^{-2} A^{-1}]$ ### Kinematics #### 1. Motion in One Dimension - **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 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 $n^{th}$ second) - **Relative Velocity:** $v_{AB} = v_A - v_B$ #### 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 (Launch from origin):** - Initial velocity: $u$ at angle $\theta$ with horizontal - Horizontal component: $u_x = u\cos\theta$ (constant velocity) - Vertical component: $u_y = u\sin\theta$ (constant acceleration $-g$) - Time of Flight: $T = \frac{2u\sin\theta}{g}$ - Maximum Height: $H = \frac{u^2\sin^2\theta}{2g}$ - Horizontal Range: $R = \frac{u^2\sin(2\theta)}{g}$ (Max for $\theta=45^\circ$) - Equation of Trajectory: $y = x\tan\theta - \frac{g x^2}{2u^2\cos^2\theta}$ - **Relative Velocity in 2D:** $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ - **River-Boat Problems:** - To cross in minimum time: Boat velocity perpendicular to river flow. $t_{min} = \frac{d}{v_b}$, Drift $x = v_r t_{min} = d \frac{v_r}{v_b}$. - To cross without drift: Boat velocity at angle $\alpha$ upstream. $\sin\alpha = \frac{v_r}{v_b}$. $t = \frac{d}{\sqrt{v_b^2 - v_r^2}}$. ### Newton's Laws of Motion & Friction - **Newton's First Law:** Inertia. 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:** For every action, there is an equal and opposite reaction. $\vec{F}_{AB} = -\vec{F}_{BA}$ - **Impulse:** $\vec{J} = \int \vec{F} dt = \Delta\vec{p} = m(\vec{v}_f - \vec{v}_i)$ - **Conservation of Linear Momentum:** If $\vec{F}_{net} = 0$, then $\vec{p}_{total} = \text{constant}$. - **Friction:** - Static Friction: $f_s \le \mu_s N$ (where $N$ is normal force, $\mu_s$ is coefficient of static friction). - Kinetic Friction: $f_k = \mu_k N$ (where $\mu_k$ is coefficient of kinetic friction). - Rolling Friction: $f_r ### Work, Energy, and Power - **Work Done by a Constant Force:** $W = \vec{F} \cdot \vec{d} = Fd\cos\theta$ - **Work Done by a Variable Force:** $W = \int \vec{F} \cdot d\vec{r}$ - **Kinetic Energy:** $K = \frac{1}{2}mv^2$ - **Work-Energy Theorem:** $W_{net} = \Delta K = K_f - K_i$ - **Potential Energy:** - Gravitational: $U_g = mgh$ - Spring: $U_s = \frac{1}{2}kx^2$ - **Conservation of Mechanical Energy:** If only conservative forces do work, $E = K + U = \text{constant}$. - **Power:** - Average Power: $P_{avg} = \frac{W}{\Delta t}$ - Instantaneous Power: $P = \vec{F} \cdot \vec{v} = Fv\cos\theta$ - **Collisions:** - **Elastic Collision:** Both momentum and kinetic energy are conserved. - In 1D: $v_1 - v_2 = -(u_1 - u_2)$ (relative velocity of approach = - relative velocity of separation) - Velocities after collision (1D): - $v_1 = \frac{(m_1-m_2)u_1 + 2m_2u_2}{m_1+m_2}$ - $v_2 = \frac{(m_2-m_1)u_2 + 2m_1u_1}{m_1+m_2}$ - **Inelastic Collision:** Momentum is conserved, but kinetic energy is NOT conserved ($K_{final} ### Rotational Motion - **Angular Displacement:** $\Delta\theta$ (radians) - **Angular Velocity:** $\omega = \frac{d\theta}{dt}$ - **Angular Acceleration:** $\alpha = \frac{d\omega}{dt}$ - **Relations between Linear and Angular Variables:** - $v = r\omega$ - $a_t = r\alpha$ (tangential acceleration) - $a_c = \frac{v^2}{r} = r\omega^2$ (centripetal acceleration) - Total acceleration: $\vec{a} = \vec{a}_t + \vec{a}_c$ - **Equations of Rotational Motion (Constant Angular Acceleration):** - $\omega = \omega_0 + \alpha t$ - $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ - $\omega^2 = \omega_0^2 + 2\alpha\theta$ - **Moment of Inertia (I):** - For a point mass: $I = mr^2$ - For a system of particles: $I = \sum m_i r_i^2$ - For a rigid body: $I = \int r^2 dm$ - Parallel Axis Theorem: $I = I_{CM} + Md^2$ - Perpendicular Axis Theorem (for planar bodies): $I_z = I_x + I_y$ - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ - $\tau = I\alpha$ (Newton's second law for rotation) - **Angular Momentum:** $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$ - **Conservation of Angular Momentum:** If $\vec{\tau}_{net} = 0$, then $\vec{L}_{total} = \text{constant}$. - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ - **Total Kinetic Energy (Rolling without slipping):** $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{CM}^2 + \frac{1}{2}I_{CM}\omega^2$ - **Work Done by Torque:** $W = \int \tau d\theta$ - **Power in Rotational Motion:** $P = \tau\omega$ ### Gravitation - **Newton's Law of Universal Gravitation:** $F = G\frac{m_1 m_2}{r^2}$ (G is gravitational constant, $G = 6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2$) - **Acceleration due to Gravity (g):** - On Earth's surface: $g = G\frac{M_e}{R_e^2}$ - Variation with Altitude: $g_h = g(1 - \frac{2h}{R_e})$ for $h \ll R_e$ - Variation with Depth: $g_d = g(1 - \frac{d}{R_e})$ - Variation with Latitude: $g_\phi = g - R_e\omega^2\cos^2\phi$ - **Gravitational Potential Energy:** $U = -\frac{GMm}{r}$ (Relative to infinity) - **Gravitational Potential:** $V = -\frac{GM}{r}$ - **Escape Velocity:** $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$ - **Orbital Velocity:** $v_o = \sqrt{\frac{GM}{r}}$ (for circular orbit of radius r) - **Time Period of Satellite:** $T = \frac{2\pi r}{v_o} = 2\pi\sqrt{\frac{r^3}{GM}}$ - **Kepler's Laws:** 1. Law of Orbits: All planets move in elliptical orbits with the Sun at one of the foci. 2. Law of Areas: The line joining a planet to the Sun sweeps equal areas in equal intervals of time (conservation of angular momentum). 3. Law of Periods: The square of the time period of revolution of any planet around the Sun is proportional to the cube of the semi-major axis of its elliptical orbit ($T^2 \propto r^3$). ### Properties of Matter #### 1. Elasticity - **Stress:** $\sigma = \frac{F}{A}$ - **Strain:** $\epsilon = \frac{\Delta L}{L}$ (longitudinal), $\epsilon_V = \frac{\Delta V}{V}$ (volume), $\phi = \frac{\Delta x}{L}$ (shear) - **Hooke's Law:** Stress $\propto$ Strain (within elastic limit) - **Young's Modulus (Y):** $Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{F/A}{\Delta L/L}$ - **Bulk Modulus (B):** $B = \frac{\text{Volume Stress}}{\text{Volume Strain}} = \frac{-P}{\Delta V/V}$ - **Shear Modulus (G) / Modulus of Rigidity:** $G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{F/A}{\phi}$ - **Poisson's Ratio ($\nu$):** $\nu = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = -\frac{\Delta D/D}{\Delta L/L}$ - **Energy Stored in a Stretched Wire:** $U = \frac{1}{2} \text{Stress} \times \text{Strain} \times \text{Volume} = \frac{1}{2} Y (\text{Strain})^2 \text{Volume}$ #### 2. Fluid Mechanics - **Density:** $\rho = \frac{m}{V}$ - **Pressure:** $P = \frac{F}{A}$ - **Pressure at Depth h:** $P = P_0 + \rho gh$ - **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. - **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 flow) - **Bernoulli's Principle:** $P + \rho gh + \frac{1}{2}\rho v^2 = \text{constant}$ - **Torricelli's Law:** Velocity of efflux $v = \sqrt{2gh}$ - **Viscosity:** - Stoke's Law: Viscous drag force $F = 6\pi\eta rv$ - Terminal Velocity: $v_t = \frac{2r^2(\rho - \rho_0)g}{9\eta}$ - **Surface Tension (T):** - $T = \frac{F}{L}$ - Surface Energy: $E = T \times \text{Area}$ - Excess Pressure: - Liquid drop: $\Delta P = \frac{2T}{R}$ - Soap bubble: $\Delta P = \frac{4T}{R}$ - Air bubble in liquid: $\Delta P = \frac{2T}{R}$ - Capillary Rise: $h = \frac{2T\cos\theta}{\rho rg}$ ### Thermal Physics #### 1. Heat and Thermodynamics - **Temperature Scales:** - Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$ - Celsius to Kelvin: $K = C + 273.15$ - **Thermal Expansion:** - Linear: $\Delta L = L_0 \alpha \Delta T$ - Area: $\Delta A = A_0 \beta \Delta T$, where $\beta = 2\alpha$ - Volume: $\Delta V = V_0 \gamma \Delta T$, where $\gamma = 3\alpha$ - **Specific Heat Capacity:** $Q = mc\Delta T$ - **Latent Heat:** $Q = mL$ (L is latent heat of fusion or vaporization) - **Heat Transfer:** - Conduction: $\frac{dQ}{dt} = -KA\frac{dT}{dx}$ (Fourier's Law) - Convection: Complex, depends on fluid properties. - Radiation: Stefan-Boltzmann Law, $\frac{dQ}{dt} = e\sigma A T^4$ (e is emissivity, $\sigma$ is Stefan-Boltzmann constant). - Wien's Displacement Law: $\lambda_{max} T = b$ (b is Wien's constant). - Newton's Law of Cooling: $\frac{dT}{dt} \propto (T - T_0)$ - **Thermodynamics:** - **Zeroth Law:** If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other. - **First Law:** $\Delta U = Q - W$ (Q is heat added to system, W is work done BY system). - Work done by gas: $W = \int P dV$ - **Second Law:** Heat cannot spontaneously flow from a colder body to a hotter body. Enthalpy of universe always increases. - Efficiency of Heat Engine: $\eta = 1 - \frac{Q_C}{Q_H} = 1 - \frac{T_C}{T_H}$ (Carnot engine) - Coefficient of Performance (Refrigerator): $COP = \frac{Q_C}{W} = \frac{T_C}{T_H - T_C}$ - **Third Law:** As temperature approaches absolute zero, the entropy of a system approaches a minimum constant. - **Thermodynamic Processes:** - Isothermal: $\Delta T = 0$, $PV = \text{constant}$, $W = nRT \ln(\frac{V_f}{V_i})$ - Adiabatic: $Q = 0$, $PV^\gamma = \text{constant}$, $T V^{\gamma-1} = \text{constant}$, $P^{1-\gamma} T^\gamma = \text{constant}$ - Isobaric: $\Delta P = 0$, $W = P\Delta V$ - Isochoric: $\Delta V = 0$, $W = 0$, $\Delta U = Q$ - **Specific Heats of Gases:** - $C_P - C_V = R$ (Mayer's relation) - $\gamma = C_P/C_V$ - For monoatomic gas: $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$, $\gamma = 5/3$ - For diatomic gas: $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$, $\gamma = 7/5$ (at moderate temps) - Internal Energy of Ideal Gas: $U = \frac{f}{2}nRT$ (f is degrees of freedom) #### 2. Kinetic Theory of Gases - **Ideal Gas Equation:** $PV = nRT = Nk_B T$ (n is moles, N is number of molecules, $k_B$ is Boltzmann constant) - **Average Kinetic Energy of a Gas Molecule:** $K_{avg} = \frac{3}{2}k_B T$ - **RMS Speed:** $v_{rms} = \sqrt{\frac{3RT}{M_{mol}}} = \sqrt{\frac{3k_B T}{m}}$ - **Mean Free Path:** $\lambda = \frac{1}{\sqrt{2}\pi d^2 n}$ (d is molecular diameter, n is number density) ### Oscillations and Waves #### 1. Simple Harmonic Motion (SHM) - **Displacement:** $x(t) = A\sin(\omega t + \phi)$ (A is amplitude, $\omega$ is angular frequency, $\phi$ is phase constant) - **Velocity:** $v(t) = A\omega\cos(\omega t + \phi) = \pm\omega\sqrt{A^2 - x^2}$ - **Acceleration:** $a(t) = -A\omega^2\sin(\omega t + \phi) = -\omega^2 x$ - **Angular Frequency:** $\omega = \sqrt{\frac{k}{m}}$ (for spring-mass system) - **Time Period:** $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$ - **Time Period of Simple Pendulum:** $T = 2\pi\sqrt{\frac{L}{g}}$ (for small angles) - **Time Period of Physical Pendulum:** $T = 2\pi\sqrt{\frac{I}{mgL}}$ (L is distance from pivot to CM) - **Energy in SHM:** $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2 = K + U = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ #### 2. Waves - **Wave Equation:** $y(x,t) = A\sin(kx - \omega t + \phi)$ (for a travelling wave) - Wave number: $k = \frac{2\pi}{\lambda}$ - Angular frequency: $\omega = 2\pi f = \frac{2\pi}{T}$ - Wave speed: $v = f\lambda = \frac{\omega}{k}$ - **Speed of Transverse Wave on a String:** $v = \sqrt{\frac{T}{\mu}}$ (T is tension, $\mu$ is linear mass density) - **Speed of Longitudinal Wave (Sound):** - In fluid: $v = \sqrt{\frac{B}{\rho}}$ (B is bulk modulus) - In solid rod: $v = \sqrt{\frac{Y}{\rho}}$ (Y is Young's modulus) - In gas (Newton-Laplace): $v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M_{mol}}}$ - **Intensity of Wave:** $I = 2\pi^2 f^2 A^2 \rho v$ - **Principle of Superposition:** When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements. - **Standing Waves:** Formed by superposition of two identical waves travelling in opposite directions. - Fixed ends (string): $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L} = n f_1$ ($n=1,2,3,...$) - Open organ pipe: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L} = n f_1$ ($n=1,2,3,...$) - Closed organ pipe: $\lambda_n = \frac{4L}{2n-1}$, $f_n = \frac{(2n-1)v}{4L} = (2n-1)f_1$ ($n=1,2,3,...$ i.e., only odd harmonics) - **Beats:** Produced when two waves of slightly different frequencies ($f_1, f_2$) interfere. - Beat frequency: $f_{beat} = |f_1 - f_2|$ - **Doppler Effect (Sound):** - Apparent frequency: $f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$ - Use '+' for observer moving towards source, '-' for observer moving away. - Use '-' for source moving towards observer, '+' for source moving away. - For light: $f' = f \sqrt{\frac{c \pm v}{c \mp v}}$ (v is relative speed, c is speed of light) ### Electrostatics - **Coulomb's Law:** $F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$ (force between two point charges) - $\epsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$ (permittivity of free space) - $\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2$ - **Electric Field:** $\vec{E} = \frac{\vec{F}}{q_0}$ (force per unit test charge) - For a point charge q: $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$ - **Electric Potential:** $V = \frac{U}{q_0}$ (potential energy per unit test charge) - For a point charge q: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ - 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:** - Dipole Moment: $\vec{p} = q(2\vec{a})$ (from -q to +q) - Electric Field on Axial Line: $E = \frac{1}{4\pi\epsilon_0} \frac{2pr}{(r^2 - a^2)^2} \approx \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3}$ (for $r \gg a$) - Electric Field on Equatorial Line: $E = \frac{1}{4\pi\epsilon_0} \frac{p}{(r^2 + a^2)^{3/2}} \approx \frac{1}{4\pi\epsilon_0} \frac{p}{r^3}$ (for $r \gg a$) - Torque on Dipole in Uniform E-field: $\vec{\tau} = \vec{p} \times \vec{E}$ - Potential Energy of Dipole: $U = -\vec{p} \cdot \vec{E}$ - **Gauss's Law:** $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ - Electric field for infinite line charge: $E = \frac{\lambda}{2\pi\epsilon_0 r}$ - Electric field for infinite plane sheet: $E = \frac{\sigma}{2\epsilon_0}$ - Electric field for charged conducting sphere: - $r > R$: $E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$ - $r \le R$: $E = 0$ (inside conductor) - **Capacitance:** $C = \frac{Q}{V}$ - Parallel Plate Capacitor: $C = \frac{\epsilon_0 A}{d}$ - With dielectric: $C_k = k C_0 = \frac{k\epsilon_0 A}{d}$ - Spherical Capacitor: $C = 4\pi\epsilon_0 \frac{ab}{b-a}$ (concentric shells) - Isolated Spherical Conductor: $C = 4\pi\epsilon_0 R$ - Cylindrical Capacitor: $C = \frac{2\pi\epsilon_0 L}{\ln(b/a)}$ - **Capacitors in Series:** $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ - **Capacitors in Parallel:** $C_{eq} = C_1 + C_2 + ...$ - **Energy Stored in Capacitor:** $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ - **Energy Density in Electric Field:** $u_E = \frac{1}{2}\epsilon_0 E^2$ ### Current Electricity - **Electric Current:** $I = \frac{dQ}{dt} = nA v_d e$ (n is charge carrier density, $v_d$ is drift velocity) - **Ohm's Law:** $V = IR$ - **Resistance:** $R = \rho \frac{L}{A}$ ($\rho$ is resistivity) - Temperature dependence: $R_T = R_0(1 + \alpha \Delta T)$ - **Conductance:** $G = \frac{1}{R}$ - **Conductivity:** $\sigma = \frac{1}{\rho}$ - **Resistors in Series:** $R_{eq} = R_1 + R_2 + ...$ - **Resistors in Parallel:** $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$ - **Kirchhoff's Laws:** - **Junction Rule (KCL):** Sum of currents entering a junction equals sum of currents leaving ($ \sum I = 0 $). (Conservation of Charge) - **Loop Rule (KVL):** Sum of potential changes around any closed loop is zero ($ \sum \Delta V = 0 $). (Conservation of Energy) - **Electromotive Force (EMF) and Internal Resistance:** $V = \mathcal{E} - Ir$ (terminal voltage) - **Cells in Series:** $\mathcal{E}_{eq} = \mathcal{E}_1 + \mathcal{E}_2 + ...$, $r_{eq} = r_1 + r_2 + ...$ - **Cells in Parallel:** $\mathcal{E}_{eq} = \frac{\mathcal{E}_1/r_1 + \mathcal{E}_2/r_2 + ...}{1/r_1 + 1/r_2 + ...}$, $\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} + ...$ - **Electrical Power:** $P = VI = I^2 R = \frac{V^2}{R}$ - **Electrical Energy:** $E = Pt = VIt$ - **Wheatstone Bridge:** Balanced condition $\frac{P}{Q} = \frac{R}{S}$ - **Meter Bridge:** Balanced condition $\frac{R}{S} = \frac{l}{(100-l)}$ - **Potentiometer:** - To compare EMFs: $\frac{\mathcal{E}_1}{\mathcal{E}_2} = \frac{L_1}{L_2}$ - To find internal resistance: $r = R \left(\frac{L_1}{L_2} - 1\right)$ ### Magnetism #### 1. Magnetic Effects of Current - **Biot-Savart Law:** $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$ - Magnetic field at center of circular loop: $B = \frac{\mu_0 I}{2R}$ - Magnetic field on axis of circular loop: $B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$ - Magnetic field due to infinitely long straight wire: $B = \frac{\mu_0 I}{2\pi r}$ - Magnetic field due to finite straight wire: $B = \frac{\mu_0 I}{4\pi r}(\sin\theta_1 + \sin\theta_2)$ - **Ampere's Circuital Law:** $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ - Magnetic field inside a long solenoid: $B = \mu_0 n I$ (n is turns per unit length) - Magnetic field inside a toroid: $B = \frac{\mu_0 N I}{2\pi r}$ - **Force on a Moving Charge (Lorentz Force):** $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ - Magnetic force: $\vec{F}_m = q(\vec{v} \times \vec{B})$ - **Force on a Current-Carrying Conductor:** $\vec{F} = I(\vec{L} \times \vec{B})$ - **Force between Two Parallel Current-Carrying Wires:** $F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$ (attractive if currents are in same direction, repulsive if opposite) - **Torque on a Current Loop in Magnetic Field:** $\vec{\tau} = \vec{M} \times \vec{B}$ - Magnetic dipole moment: $\vec{M} = NI\vec{A}$ - **Moving Coil Galvanometer:** Current $I \propto \theta$ (deflection) #### 2. Earth's Magnetism & Magnetic Properties of Materials - **Magnetic Elements of Earth:** - Angle of Declination ($\theta$): Angle between geographic and magnetic meridians. - Angle of Dip ($\delta$): Angle between Earth's total magnetic field and the horizontal. - Horizontal Component ($B_H$): $B_H = B_E \cos\delta$ - Vertical Component ($B_V$): $B_V = B_E \sin\delta$ - $\tan\delta = \frac{B_V}{B_H}$ - **Magnetic Intensity (H):** $H = \frac{B}{\mu}$ ($\mu$ is permeability) - **Magnetization (M):** Magnetic moment per unit volume. - **Magnetic Susceptibility ($\chi_m$):** $\chi_m = \frac{M}{H}$ - **Relative Permeability ($\mu_r$):** $\mu_r = 1 + \chi_m = \frac{\mu}{\mu_0}$ - **Types of Magnetic Materials:** - **Diamagnetic:** $\chi_m$ is small and negative ($\mu_r 1$). Weakly attracted. Curie's Law: $\chi_m \propto 1/T$. - **Ferromagnetic:** $\chi_m$ is large and positive ($\mu_r \gg 1$). Strongly attracted. Exhibit hysteresis. ### Electromagnetic Induction & Alternating Current #### 1. Electromagnetic Induction (EMI) - **Magnetic Flux:** $\Phi_B = \int \vec{B} \cdot d\vec{A} = BA\cos\theta$ - **Faraday's Law of Induction:** $\mathcal{E} = -\frac{d\Phi_B}{dt}$ (Induced EMF) - **Lenz's Law:** The direction of induced current is such that it opposes the cause producing it. - **Motional EMF:** $\mathcal{E} = Blv$ (for a conductor of length L moving with velocity v perpendicular to B) - **Self-Inductance (L):** $\Phi_B = LI$ - Self-induced EMF: $\mathcal{E}_L = -L\frac{dI}{dt}$ - Energy stored in an inductor: $U_L = \frac{1}{2}LI^2$ - Energy density in magnetic field: $u_B = \frac{B^2}{2\mu_0}$ - **Mutual Inductance (M):** $\Phi_{21} = M_{21} I_1$ - Mutually induced EMF: $\mathcal{E}_2 = -M\frac{dI_1}{dt}$ - Coefficient of Coupling: $k = \frac{M}{\sqrt{L_1 L_2}}$ #### 2. Alternating Current (AC) - **AC Voltage/Current:** $V = V_m\sin(\omega t)$, $I = I_m\sin(\omega t + \phi)$ - **RMS Values:** $V_{rms} = \frac{V_m}{\sqrt{2}}$, $I_{rms} = \frac{I_m}{\sqrt{2}}$ - **Reactance:** - Inductive Reactance: $X_L = \omega L = 2\pi f L$ - Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$ - **Impedance (Z):** - R-L series circuit: $Z = \sqrt{R^2 + X_L^2}$ - R-C series circuit: $Z = \sqrt{R^2 + X_C^2}$ - L-C-R series circuit: $Z = \sqrt{R^2 + (X_L - X_C)^2}$ - **Phase Angle ($\phi$):** $\tan\phi = \frac{X_L - X_C}{R}$ - **Power in AC Circuit:** - Average Power: $P_{avg} = V_{rms} I_{rms} \cos\phi$ - Power Factor: $\cos\phi = \frac{R}{Z}$ - **Resonance in L-C-R Series Circuit:** - Occurs when $X_L = X_C \implies \omega_0 L = \frac{1}{\omega_0 C}$ - Resonant frequency: $f_0 = \frac{1}{2\pi\sqrt{LC}}$ - At resonance: $Z = R$, $\phi = 0$, $\cos\phi = 1$ (max power) - **Quality Factor (Q-factor):** $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}}$ - **Transformer:** - $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$ (for ideal transformer) - Efficiency: $\eta = \frac{\text{Output Power}}{\text{Input Power}} = \frac{V_s I_s}{V_p I_p}$ ### Electromagnetic Waves - **Properties:** - Are transverse waves. - Do not require a medium to propagate. - Travel at the speed of light in vacuum: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$. - Wavelength ($\lambda$), Frequency ($f$), Speed ($c$): $c = f\lambda$. - Electric and magnetic fields are perpendicular to each other and to the direction of propagation. - $E_0 = c B_0$ (Peak values of E and B fields). - **Energy Density:** - Electric: $u_E = \frac{1}{2}\epsilon_0 E^2$ - Magnetic: $u_B = \frac{1}{2\mu_0} B^2$ - Total: $u = u_E + u_B = \epsilon_0 E^2 = \frac{B^2}{\mu_0}$ (in EM wave $u_E = u_B$) - **Intensity (Average Power per unit Area):** $I = \frac{P_{avg}}{A} = \frac{1}{2}\epsilon_0 E_0^2 c = \frac{B_0^2}{2\mu_0} c = \frac{E_0 B_0}{2\mu_0}$ - **Momentum Carried by EM Waves:** $p = \frac{U}{c}$ (U is total energy) - **Radiation Pressure:** $P_{rad} = \frac{I}{c}$ (for perfectly absorbing surface) - $P_{rad} = \frac{2I}{c}$ (for perfectly reflecting surface) - **Electromagnetic Spectrum:** Radio waves, Microwaves, Infrared, Visible light (ROYGBIV), Ultraviolet, X-rays, Gamma rays (increasing frequency, decreasing wavelength). ### Optics #### 1. Ray Optics - **Reflection:** - Angle of incidence = Angle of reflection ($\angle i = \angle r$). - Plane Mirror: Image is virtual, erect, laterally inverted, same size, same distance behind mirror as object in front. - Spherical Mirrors (Concave/Convex): - Mirror Formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ - Magnification: $m = -\frac{v}{u} = \frac{h_i}{h_o}$ - Focal length: $f = R/2$ - Sign Convention: Cartesian system, light from left. - **Refraction:** - Snell's Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$ - Refractive Index: $n = \frac{c}{v}$ - Apparent Depth: $D_{app} = \frac{D_{real}}{n_{relative}}$ - Total Internal Reflection (TIR): Occurs when light travels from denser to rarer medium and angle of incidence $>$ critical angle ($\theta_c = \sin^{-1}(\frac{n_2}{n_1})$). - **Lenses (Thin Lenses):** - Lens Maker's Formula: $\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ - Lens Formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$ - Magnification: $m = \frac{v}{u} = \frac{h_i}{h_o}$ - Power of Lens: $P = \frac{1}{f}$ (in dioptres, f in meters) - Lenses in Contact: $P_{eq} = P_1 + P_2$, $\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$ - **Prism:** - Angle of Deviation: $\delta = (i_1 + i_2) - A$ - For minimum deviation: $i_1 = i_2$, $r_1 = r_2 = A/2$. - $\mu = \frac{\sin((A+\delta_m)/2)}{\sin(A/2)}$ - **Dispersion:** Splitting of white light into its constituent colors. - Angular Dispersion: $\theta = \delta_v - \delta_r = (n_v - n_r)A$ - Dispersive Power: $\omega = \frac{\delta_v - \delta_r}{\delta_y} = \frac{n_v - n_r}{n_y - 1}$ - **Optical Instruments:** - **Compound Microscope:** $M = M_o M_e = \left(\frac{L}{f_o}\right)\left(1 + \frac{D}{f_e}\right)$ (for final image at D) - $M = M_o M_e = \left(\frac{L}{f_o}\right)\left(\frac{D}{f_e}\right)$ (for final image at infinity) - **Astronomical Telescope:** - Normal Adjustment (final image at infinity): $M = -\frac{f_o}{f_e}$, $L = f_o + f_e$ - Final image at D: $M = -\frac{f_o}{f_e}\left(1 + \frac{f_e}{D}\right)$, $L = f_o + u_e$ #### 2. Wave Optics - **Interference (Young's Double Slit Experiment - YDSE):** - Path difference: $\Delta x = d\sin\theta = \frac{yd}{D}$ - Condition for Constructive Interference (Bright Fringes): $\Delta x = n\lambda$ - Condition for Destructive Interference (Dark Fringes): $\Delta x = (n + \frac{1}{2})\lambda$ - Fringe Width: $\beta = \frac{\lambda D}{d}$ - **Diffraction (Single Slit):** - Condition for Minima: $a\sin\theta = n\lambda$ (n=1,2,3,...) - Condition for Maxima: $a\sin\theta = (n + \frac{1}{2})\lambda$ (approximate) - Angular width of central maximum: $2\theta = \frac{2\lambda}{a}$ - **Polarization:** - Brewster's Law: $\tan i_p = n$ (angle of polarization) - Malus's Law: $I = I_0 \cos^2\theta$ - Intensity of unpolarized light passing through polarizer is $I_0/2$. ### Modern Physics #### 1. Dual Nature of Radiation and Matter - **Planck's Quantum Theory:** $E = hf = \frac{hc}{\lambda}$ (h is Planck's constant, $h = 6.626 \times 10^{-34} \text{ Js}$) - **Photoelectric Effect:** - Einstein's Photoelectric Equation: $h f = \phi_0 + K_{max}$ - $\phi_0 = h f_0$ (work function, $f_0$ is threshold frequency) - $K_{max} = eV_s$ (maximum kinetic energy, $V_s$ is stopping potential) - Number of photoelectrons $\propto$ Intensity of light. - $K_{max}$ depends on frequency, not intensity. - **De Broglie Wavelength:** $\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2mK}} = \frac{h}{\sqrt{2me V}}$ (for accelerated charge q) - **Heisenberg's Uncertainty Principle:** - $\Delta x \Delta p_x \ge \frac{\hbar}{2}$ - $\Delta E \Delta t \ge \frac{\hbar}{2}$ ($\hbar = h/2\pi$) #### 2. Atoms and Nuclei - **Bohr's Model of Hydrogen Atom:** - Radius of $n^{th}$ orbit: $r_n = 0.529 n^2 \text{ Å}$ - Velocity of electron in $n^{th}$ orbit: $v_n = \frac{2.18 \times 10^6}{n} \text{ m/s}$ - Energy of electron in $n^{th}$ orbit: $E_n = -\frac{13.6}{n^2} \text{ eV}$ - Wavelength of emitted/absorbed photon: $\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ (R is Rydberg constant). - Lyman Series ($n_1=1$, transitions to ground state, UV region) - Balmer Series ($n_1=2$, transitions to 2nd state, Visible region) - Paschen Series ($n_1=3$, IR region) - Brackett Series ($n_1=4$, IR region) - Pfund Series ($n_1=5$, IR region) - **X-rays:** - Minimum wavelength (Cut-off wavelength): $\lambda_{min} = \frac{hc}{eV} = \frac{12400}{V \text{(Volts)}} \text{ Å}$ - Mosley's Law: $\sqrt{f} = a(Z-b)$ (for characteristic X-rays) - **Nuclear Physics:** - Atomic Mass Unit (amu): $1 \text{ amu} = 931.5 \text{ MeV/c}^2 = 1.66 \times 10^{-27} \text{ kg}$ - Mass Defect: $\Delta m = (Z m_p + N m_n) - m_{nucleus}$ - Binding Energy: $BE = \Delta m c^2$ - **Radioactivity:** - Decay Law: $N(t) = N_0 e^{-\lambda t}$ ($\lambda$ is decay constant) - Half-life: $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$ - Mean Life: $\tau = \frac{1}{\lambda}$ - Activity: $A = -\frac{dN}{dt} = \lambda N$ - Alpha Decay: $(Z, A) \to (Z-2, A-4) + \alpha$ - Beta Decay ($\beta^-$): $(Z, A) \to (Z+1, A) + e^- + \bar{\nu}$ - Beta Decay ($\beta^+$): $(Z, A) \to (Z-1, A) + e^+ + \nu$ - Gamma Decay: Emission of photon from excited nucleus. - **Nuclear Fission & Fusion:** - Fission: Heavy nucleus splits into lighter nuclei. - Fusion: Light nuclei combine to form heavier nucleus. #### 3. Semiconductors - **Energy Bands:** Conduction band, Valence band, Band gap ($E_g$). - **Conductors:** $E_g \approx 0$. - **Insulators:** $E_g > 3 \text{ eV}$. - **Semiconductors:** $E_g \approx 1 \text{ eV}$. - **Types of Semiconductors:** - Intrinsic: Pure semiconductor (e.g., Si, Ge). $n_e = n_h = n_i$. - Extrinsic: Doped semiconductor. - N-type: Doped with pentavalent impurities (e.g., P, As). Majority carriers are electrons. - P-type: Doped with trivalent impurities (e.g., B, Al). Majority carriers are holes. - **P-N Junction Diode:** - Forward Bias: Current flows. - Reverse Bias: No current (ideally). - Rectifier: Converts AC to DC. - **Transistor (BJT - NPN/PNP):** - Common Emitter Configuration: - Current Gain ($\beta$): $\beta = \frac{I_C}{I_B}$ - Voltage Gain: $A_V = -\beta \frac{R_L}{R_{in}}$ - Power Gain: $A_P = \beta A_V$ - **Logic Gates:** AND, OR, NOT, NAND, NOR, XOR, XNOR.