Exhaustive Physics Formula Sheet for Class 11 & 12 - Milan Physics

Exhaustive Physics Formula Sheet for Class 11 & 12 (CBSE)

Welcome to Milan Physics! This updated formula sheet now includes EVERY key formula from Class 11 & 12 Physics syllabus, expanded for complete coverage. Perfect for CBSE exams and thorough revision! 📚

Class 11 Physics

1. Units and Measurements

Absolute Error: \( |\Delta a| = |a_{\text{measured}} - a_{\text{true}}| \)
Mean Absolute Error: \( \overline{\Delta a} = \frac{\sum |\Delta a_i|}{n} \)
Relative Error: \( \frac{\overline{\Delta a}}{a_{\text{true}}} \)
Percentage Error: \( \left( \frac{\overline{\Delta a}}{a_{\text{true}}} \right) \times 100\% \)
Error in Combination:
- Sum/Difference: \( \Delta z = \Delta x + \Delta y \)
- Product/Quotient: \( \frac{\Delta z}{|z|} = \frac{\Delta x}{|x|} + \frac{\Delta y}{|y|} \)
- Power: \( \frac{\Delta z}{|z|} = n \frac{\Delta x}{|x|} \)
Dimensional Formula: e.g., Velocity = \( [M^0 L^1 T^{-1}] \), Force = \( [M^1 L^1 T^{-2}] \), Energy = \( [M^1 L^2 T^{-2}] \)
Conversion: \( n_1 u_1 = n_2 u_2 \)

2. Motion in a Straight Line

Displacement: \( \Delta x = x_2 - x_1 \)
Average Speed: \( v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} \)
Average Velocity: \( \vec{v}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t} \)
Instantaneous Velocity: \( v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} \)
Instantaneous Acceleration: \( a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} = \frac{d^2 x}{dt^2} \)
Equations of Motion (Uniform Acceleration):
- \( v = u + at \)
- \( x = ut + \frac{1}{2} at^2 \)
- \( v^2 = u^2 + 2ax \)
- \( x_n = u + \frac{a}{2} (2n - 1) \) (nth second displacement)
Free Fall: \( g = 9.8 \, \text{m/s}^2 \), Equations with \( u = 0 \)
Relative Velocity: \( v_{AB} = v_A - v_B \)

3. Motion in a Plane

Vector Resolution: \( A_x = A \cos\theta \), \( A_y = A \sin\theta \)
Magnitude: \( A = \sqrt{A_x^2 + A_y^2} \), \( \tan\theta = \frac{A_y}{A_x} \)
Unit Vector: \( \hat{A} = \frac{\vec{A}}{A} \)
Vector Addition: \( \vec{R} = \vec{A} + \vec{B} \), Magnitude: \( R = \sqrt{A^2 + B^2 + 2AB\cos\theta} \)
Direction: \( \tan\phi = \frac{B \sin\theta}{A + B \cos\theta} \)
Vector Subtraction: \( \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) \)
Dot Product: \( \vec{A} \cdot \vec{B} = AB \cos\theta = A_x B_x + A_y B_y \)
Cross Product: \( \vec{A} \times \vec{B} = AB \sin\theta \, \hat{n} \), Magnitude: \( AB \sin\theta \)
Projectile Motion (Horizontal Projection):
- Time of Flight: \( t = \sqrt{\frac{2h}{g}} \)
- Range: \( R = u \sqrt{\frac{2h}{g}} \)
- Velocity at Ground: \( v = \sqrt{u^2 + 2gh} \)
Oblique Projection:
- Time of Flight: \( T = \frac{2u \sin\theta}{g} \)
- Maximum Height: \( H = \frac{u^2 \sin^2\theta}{2g} \)
- Horizontal Range: \( R = \frac{u^2 \sin 2\theta}{g} \)
- Maximum Range: \( R_{\text{max}} = \frac{u^2}{g} \) (at \( \theta = 45^\circ \))
- Trajectory: \( y = x \tan\theta - \frac{g x^2}{2 u^2 \cos^2\theta} \)
Uniform Circular Motion: \( a_c = \frac{v^2}{r} = \omega^2 r \), \( F_c = m \frac{v^2}{r} \)
Angular Variables: \( \omega = \frac{d\theta}{dt} \), \( \alpha = \frac{d\omega}{dt} \)

4. Laws of Motion

Newton's First Law: Inertia, \( \vec{F}_{\text{net}} = 0 \implies \vec{v} = \text{constant} \)
Newton's Second Law: \( \vec{F} = \frac{d\vec{p}}{dt} = m \vec{a} \)
Momentum: \( \vec{p} = m \vec{v} \)
Impulse: \( \vec{J} = \int \vec{F} dt = \Delta \vec{p} \)
Newton's Third Law: \( \vec{F}_{AB} = -\vec{F}_{BA} \)
Conservation of Momentum: \( \vec{p}_{\text{initial}} = \vec{p}_{\text{final}} \) (no external force)
Friction:
- Static: \( f_s \leq \mu_s N \)
- Kinetic: \( f_k = \mu_k N \)
- Rolling: \( f_r = \mu_r N \)
- Angle of Friction: \( \tan\theta = \mu \)
- Angle of Repose: \( \tan\alpha = \mu \)
Inclined Plane:
- Acceleration: \( a = g (\sin\theta - \mu \cos\theta) \)
- Normal Force: \( N = mg \cos\theta \)
Centripetal Force in Banked Road (With Friction): \( v = \sqrt{rg \frac{\mu + \tan\theta}{1 - \mu \tan\theta}} \)
Pulley Systems:
- Atwood Machine: \( a = \frac{(m_1 - m_2)g}{m_1 + m_2} \), \( T = \frac{2 m_1 m_2 g}{m_1 + m_2} \)
- With Inclined Plane: Adjust for \( \sin\theta \), etc.
Apparent Weight in Lift: \( R = m(g \pm a) \)

5. Work, Energy, and Power

Work: \( W = \int \vec{F} \cdot d\vec{r} = F d \cos\theta \) (constant force)
Variable Force: \( W = \int_{x_i}^{x_f} F(x) dx \)
Kinetic Energy: \( KE = \frac{1}{2} m v^2 \)
Work-Energy Theorem: \( W_{\text{net}} = \Delta KE \)
Gravitational Potential Energy: \( PE = m g h \)
Elastic Potential Energy: \( PE = \frac{1}{2} k x^2 \)
Conservative Force: \( W = -\Delta PE \)
Mechanical Energy Conservation: \( \Delta KE + \Delta PE = 0 \) (conservative forces)
With Non-Conservative: \( W_{\text{nc}} = \Delta KE + \Delta PE \)
Power: \( P = \frac{dW}{dt} = \vec{F} \cdot \vec{v} \)
Average Power: \( P_{\text{avg}} = \frac{W}{\Delta t} \)
Collisions:
- Elastic (1D): \( v_1' = \frac{m_1 - m_2}{m_1 + m_2} v_1 + \frac{2 m_2}{m_1 + m_2} v_2 \)
- \( v_2' = \frac{2 m_1}{m_1 + m_2} v_1 + \frac{m_2 - m_1}{m_1 + m_2} v_2 \)
- Coefficient of Restitution: \( e = \frac{v_2' - v_1'}{u_1 - u_2} \) (e=1 for elastic, e=0 for inelastic)
- Inelastic: Momentum conserved, KE not; for perfectly inelastic: \( v' = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2} \)
Vertical Circle:
- Minimum Speed at Top: \( v_{\text{top}} = \sqrt{r g} \)
- Minimum Speed at Bottom: \( v_{\text{bottom}} = \sqrt{5 r g} \)
- Tension at Bottom: \( T_b = mg + \frac{m v^2}{r} \)
- Tension at Top: \( T_t = \frac{m v^2}{r} - mg \)

6. System of Particles and Rotational Motion

Center of Mass (2D/3D): \( \vec{r}_{\text{cm}} = \frac{\sum m_i \vec{r_i}}{\sum m_i} \)
Velocity of CM: \( \vec{v}_{\text{cm}} = \frac{\sum m_i \vec{v_i}}{\sum m_i} \)
Motion of CM: \( M \vec{a}_{\text{cm}} = \vec{F}_{\text{ext}} \)
Moment of Inertia:
- Point Mass: \( I = m r^2 \)
- Rod about Center: \( I = \frac{1}{12} M L^2 \)
- Rod about End: \( I = \frac{1}{3} M L^2 \)
- Disk about Center: \( I = \frac{1}{2} M R^2 \)
- Ring about Center: \( I = M R^2 \)
- Sphere: \( I = \frac{2}{5} M R^2 \)
- Hollow Sphere: \( I = \frac{2}{3} M R^2 \)
- Parallel Axis Theorem: \( I = I_{\text{cm}} + M d^2 \)
- Perpendicular Axis Theorem: \( I_z = I_x + I_y \)
Torque: \( \vec{\tau} = \vec{r} \times \vec{F} = I \vec{\alpha} \)
Angular Momentum: \( \vec{L} = \vec{r} \times \vec{p} = I \vec{\omega} \)
Conservation of Angular Momentum: \( \vec{L} = \text{constant} \) (no external torque)
Rotational KE: \( KE = \frac{1}{2} I \omega^2 \)
Rolling Motion: \( v = r \omega \), \( a = r \alpha \)
KE for Rolling: \( KE = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2 = \frac{1}{2} m v^2 \left(1 + \frac{k^2}{r^2}\right) \)
Acceleration on Incline (Rolling): \( a = \frac{g \sin\theta}{1 + \frac{I}{m r^2}} \)

7. Gravitation

Universal Law: \( F = G \frac{m_1 m_2}{r^2} \), \( G = 6.67 \times 10^{-11} \, \text{N m}^2 / \text{kg}^2 \)
Gravitational Field: \( \vec{g} = - \frac{G M}{r^2} \hat{r} \)
Gravitational Potential: \( V = - \frac{G M}{r} \)
Potential Energy: \( U = - \frac{G m_1 m_2}{r} \)
g at Height h: \( g_h = g \left(1 - \frac{2h}{R}\right) \) (h << R)
g at Depth d: \( g_d = g \left(1 - \frac{d}{R}\right) \)
g due to Sphere (Outside): \( g = \frac{G M}{r^2} \), (Inside): \( g = \frac{G M r}{R^3} \)
Orbital Velocity: \( v_o = \sqrt{\frac{G M}{r}} \)
Time Period (Satellite): \( T = 2\pi \sqrt{\frac{r^3}{G M}} \)
Escape Velocity: \( v_e = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R} \)
Kepler's Laws:
- 1st: Elliptical Orbits
- 2nd: \( \frac{dA}{dt} = \frac{L}{2m} = \text{constant} \)
- 3rd: \( T^2 \propto a^3 \)

8. Mechanical Properties of Solids

Stress: Longitudinal \( \sigma = \frac{F}{A} \), Shear \( \tau = \frac{F}{A} \)
Strain: Longitudinal \( \epsilon = \frac{\Delta L}{L} \), Volume \( \frac{\Delta V}{V} \), Shear \( \tan\phi \)
Young's Modulus: \( Y = \frac{\sigma}{\epsilon} = \frac{F L}{A \Delta L} \)
Bulk Modulus: \( B = - \frac{\Delta P}{\Delta V / V} \)
Shear Modulus (Rigidity): \( G = \frac{\tau}{\phi} \)
Poisson's Ratio: \( \sigma = - \frac{\Delta d / d}{\Delta L / L} \)
Elastic Potential Energy: \( U = \frac{1}{2} \sigma \epsilon V \)
Hooke's Law: \( F = -k x \) (within elastic limit)

9. Mechanical Properties of Fluids

Pressure: \( P = \frac{F}{A} \), Hydrostatic: \( P = P_0 + \rho g h \)
Pascal's Law: \( \Delta P \) transmitted undiminished
Buoyancy: \( F_b = V \rho_l g \) (Archimedes' Principle)
Apparent Weight: \( W_{\text{app}} = W - F_b \)
Continuity Equation: \( A_1 v_1 = A_2 v_2 \)
Bernoulli's Equation: \( P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant} \)
Torricelli's Theorem: \( v = \sqrt{2 g h} \)
Venturi Meter: \( v_1 A_1 = v_2 A_2 \), \( \Delta P = \frac{1}{2} \rho (v_1^2 - v_2^2) \)
Viscosity: \( F = \eta A \frac{dv}{dx} \)
Poiseuille's Law: \( Q = \frac{\pi r^4 \Delta P}{8 \eta l} \)
Stokes' Law: \( F_d = 6 \pi \eta r v \)
Terminal Velocity: \( v_t = \frac{2 r^2 (\rho - \sigma) g}{9 \eta} \)
Surface Tension: \( T = \frac{F}{l} \)
Capillary Rise: \( h = \frac{2 T \cos\theta}{\rho g r} \)
Excess Pressure (Soap Bubble): \( \Delta P = \frac{4 T}{r} \), (Drop): \( \frac{2 T}{r} \)

10. Thermal Properties of Matter

Temperature Scales: \( T_C = T_F - 32 \times \frac{5}{9} \), \( T_K = T_C + 273 \)
Heat Capacity: \( C = \frac{Q}{\Delta T} \), Specific Heat: \( c = \frac{C}{m} \)
Heat Transfer: \( Q = m c \Delta T \)
Latent Heat: \( Q = m L \), Fusion/Vaporization
Linear Expansion: \( \Delta L = L \alpha \Delta T \), \( \alpha = \frac{\Delta L}{L \Delta T} \)
Area Expansion: \( \Delta A = A (2\alpha) \Delta T \)
Volume Expansion: \( \Delta V = V (3\alpha) \Delta T = V \gamma \Delta T \)
Conduction: \( \frac{dQ}{dt} = -k A \frac{dT}{dx} \)
Thermal Resistance: \( R = \frac{l}{k A} \)
Convection: Newton's Law of Cooling: \( \frac{dT}{dt} = -b (T - T_0) \)
Radiation: Stefan-Boltzmann: \( P = \sigma A e T^4 \)
Net Power: \( P_{\text{net}} = \sigma A e (T^4 - T_0^4) \)
Wien's Displacement Law: \( \lambda_m T = b \) (b = 2.9 × 10^{-3} m K)

11. Thermodynamics

Zeroth Law: Thermal Equilibrium
First Law: \( \Delta Q = \Delta U + \Delta W \)
Work Done: \( W = \int P dV \)
Isobaric: \( W = P \Delta V \)
Isochoric: \( W = 0 \)
Isothermal: \( W = n R T \ln \frac{V_2}{V_1} \)
Adiabatic: \( W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} = \frac{n R (T_1 - T_2)}{\gamma - 1} \)
Internal Energy: \( \Delta U = n C_v \Delta T \) (ideal gas)
Specific Heats: \( C_p = C_v + R \), \( \gamma = \frac{C_p}{C_v} \)
For Monatomic: \( C_v = \frac{3}{2} R \), \( \gamma = \frac{5}{3} \)
Diatomic: \( C_v = \frac{5}{2} R \), \( \gamma = \frac{7}{5} \)
Adiabatic Relations: \( P V^\gamma = \text{constant} \), \( T V^{\gamma-1} = \text{constant} \), \( T^\gamma P^{1-\gamma} = \text{constant} \)
Second Law: Entropy increases, Kelvin-Planck, Clausius Statements
Carnot Efficiency: \( \eta = 1 - \frac{T_2}{T_1} \)
Refrigerator COP: \( \beta = \frac{T_2}{T_1 - T_2} \)
Entropy Change: \( \Delta S = \int \frac{dQ_{\text{rev}}}{T} \)

12. Kinetic Theory of Gases

Ideal Gas Equation: \( P V = n R T = N k T \)
Avogadro's Number: \( N_A = 6.022 \times 10^{23} \)
Boltzmann Constant: \( k = 1.38 \times 10^{-23} \) J/K
Pressure: \( P = \frac{1}{3} \rho v_{\text{rms}}^2 \)
RMS Speed: \( v_{\text{rms}} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 k T}{m}} \)
Average Speed: \( v_{\text{avg}} = \sqrt{\frac{8 R T}{\pi M}} \)
Most Probable Speed: \( v_{\text{mp}} = \sqrt{\frac{2 R T}{M}} \)
Average KE: \( \frac{1}{2} m v_{\text{rms}}^2 = \frac{3}{2} k T \)
Degrees of Freedom: Monatomic 3, Diatomic 5 (trans + rot)
Mean Free Path: \( \lambda = \frac{1}{\sqrt{2} \pi d^2 n} \)
Van der Waals Equation: \( \left( P + \frac{a}{V_m^2} \right) (V_m - b) = R T \)

13. Oscillations

SHM Equation: \( \frac{d^2 x}{dt^2} + \omega^2 x = 0 \)
Displacement: \( x = A \sin(\omega t + \phi) \) or \( A \cos(\omega t + \phi) \)
Velocity: \( v = \omega \sqrt{A^2 - x^2} \)
Acceleration: \( a = -\omega^2 x \)
Angular Frequency: \( \omega = 2\pi f = \frac{2\pi}{T} \)
Spring Oscillator: \( T = 2\pi \sqrt{\frac{m}{k}} \), \( \omega = \sqrt{\frac{k}{m}} \)
Series Springs: \( \frac{1}{k_{\text{eq}}} = \frac{1}{k_1} + \frac{1}{k_2} \)
Parallel Springs: \( k_{\text{eq}} = k_1 + k_2 \)
Simple Pendulum: \( T = 2\pi \sqrt{\frac{l}{g}} \), \( \omega = \sqrt{\frac{g}{l}} \)
Physical Pendulum: \( T = 2\pi \sqrt{\frac{I}{m g d}} \)
Torsional Pendulum: \( T = 2\pi \sqrt{\frac{I}{\kappa}} \)
Energy in SHM: \( E = \frac{1}{2} k A^2 = KE + PE \)
Damped Oscillation: \( x = A e^{-b t / 2m} \cos(\omega' t + \phi) \), \( \omega' = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2} \)
Forced Oscillation: Resonance at \( \omega = \omega_0 \)

14. Waves

Wave Equation: \( y = A \sin(k x - \omega t + \phi) \)
Wave Number: \( k = \frac{2\pi}{\lambda} \)
Frequency: \( f = \frac{1}{T} \), \( \omega = 2\pi f \)
Speed: \( v = f \lambda = \frac{\omega}{k} \)
Transverse Wave Speed (String): \( v = \sqrt{\frac{T}{\mu}} \)
Longitudinal Wave Speed (Rod): \( v = \sqrt{\frac{Y}{\rho}} \), (Gas): \( v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}} \)
Intensity: \( I = \frac{P}{A} \propto A^2 \)
Sound Level: \( \beta = 10 \log \frac{I}{I_0} \) dB, \( I_0 = 10^{-12} \) W/m²
Superposition: Interference, \( I = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos\delta \)
Constructive: \( \delta = 2\pi n \), Destructive: \( \delta = (2n+1)\pi \)
Standing Waves: Nodes at \( x = \frac{n \lambda}{2} \)
String Fixed Ends: \( f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}} \)
Open Pipe: \( f_n = \frac{n v}{2L} \)
Closed Pipe: \( f_n = \frac{(2n-1) v}{4L} \)
Beats: \( f_b = |f_1 - f_2| \)
Doppler Effect:
- Source Moving: \( f' = f \frac{v}{v \mp v_s} \)
- Observer Moving: \( f' = f \frac{v \pm v_o}{v} \)
- General: \( f' = f \frac{v \pm v_o}{v \mp v_s} \)

Class 12 Physics

1. Electrostatics

Coulomb's Law: \( \vec{F} = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \hat{r} \), \( \epsilon_0 = 8.85 \times 10^{-12} \) C²/N m²
Electric Field: \( \vec{E} = \frac{\vec{F}}{q_0} = \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2} \hat{r} \)
E due to Dipole (Axial): \( E = \frac{1}{4\pi \epsilon_0} \frac{2 p}{r^3} \)
Equatorial: \( E = \frac{1}{4\pi \epsilon_0} \frac{p}{r^3} \)
Dipole Moment: \( \vec{p} = q \vec{d} \)
Torque on Dipole: \( \vec{\tau} = \vec{p} \times \vec{E} \)
PE of Dipole: \( U = - \vec{p} \cdot \vec{E} \)
Electric Flux: \( \Phi_E = \oint \vec{E} \cdot d\vec{A} \)
Gauss's Law: \( \Phi_E = \frac{q_{\text{enc}}}{\epsilon_0} \)
E for Infinite Sheet: \( E = \frac{\sigma}{2 \epsilon_0} \)
E for Sphere (Outside): \( E = \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2} \), Inside: 0 (shell), \( E = \frac{1}{4\pi \epsilon_0} \frac{Q r}{R^3} \) (uniform)
Electric Potential: \( V = -\int \vec{E} \cdot d\vec{l} \)
V due to Point Charge: \( V = \frac{1}{4\pi \epsilon_0} \frac{Q}{r} \)
V due to Dipole (Axial): \( V = \frac{1}{4\pi \epsilon_0} \frac{p}{r^2} \)
Equatorial: \( V = 0 \)
Equipotential Surfaces: Perpendicular to E
Capacitance: \( C = \frac{Q}{V} \)
Parallel Plate: \( C = \frac{\epsilon_0 A}{d} \)
With Dielectric: \( C = \kappa \frac{\epsilon_0 A}{d} \)
Spherical Capacitor: \( C = 4\pi \epsilon_0 \frac{a b}{b - a} \)
Series Capacitors: \( \frac{1}{C_{\text{eq}}} = \sum \frac{1}{C_i} \)
Parallel: \( C_{\text{eq}} = \sum C_i \)
Energy Stored: \( U = \frac{1}{2} C V^2 = \frac{Q^2}{2 C} = \frac{1}{2} Q V \)
Energy Density: \( u = \frac{1}{2} \epsilon_0 E^2 \)

2. Current Electricity

Current: \( I = \frac{dQ}{dt} = n e A v_d \)
Drift Velocity: \( v_d = \frac{e E \tau}{m} = \frac{I}{n e A} \)
Ohm's Law: \( V = I R \), \( \vec{J} = \sigma \vec{E} \)
Resistance: \( R = \rho \frac{l}{A} \), Resistivity \( \rho = \frac{m}{n e^2 \tau} \)
Temperature Dependence: \( \rho = \rho_0 (1 + \alpha \Delta T) \)
Power: \( P = V I = I^2 R = \frac{V^2}{R} \)
Series Resistors: \( R_{\text{eq}} = \sum R_i \)
Parallel: \( \frac{1}{R_{\text{eq}}} = \sum \frac{1}{R_i} \)
Kirchhoff's Laws:
- KCL: \( \sum I = 0 \) at junction
- KVL: \( \sum \Delta V = 0 \) in loop
Wheatstone Bridge: Balanced when \( \frac{P}{Q} = \frac{R}{S} \)
Meter Bridge: \( \frac{R_1}{R_2} = \frac{l_1}{l_2} \)
Potentiometer: \( V = \frac{E l}{L} \), Emf Comparison: \( \frac{E_1}{E_2} = \frac{l_1}{l_2} \)
Cells in Series: \( E_{\text{eq}} = n E \), \( r_{\text{eq}} = n r \)
Parallel: \( E_{\text{eq}} = E \), \( r_{\text{eq}} = \frac{r}{n} \)
Maximum Power Transfer: \( R_L = r \)

3. Magnetic Effects of Current and Magnetism

Force on Charge: \( \vec{F} = q (\vec{v} \times \vec{B}) \)
Lorentz Force: \( \vec{F} = q \vec{E} + q (\vec{v} \times \vec{B}) \)
Cyclotron Frequency: \( f = \frac{q B}{2\pi m} \)
Biot-Savart Law: \( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \)
Straight Wire: \( B = \frac{\mu_0 I}{2\pi r} \)
Circular Loop (Center): \( B = \frac{\mu_0 I}{2 R} \), (Axis): \( B = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}} \)
Solenoid: \( B = \mu_0 n I \)
Toroid: \( B = \frac{\mu_0 N I}{2\pi r} \)
Ampere's Law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \)
Force between Wires: \( \frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d} \)
Magnetic Dipole Moment: \( \vec{m} = I \vec{A} \), For Loop: \( m = N I A \)
Torque: \( \vec{\tau} = \vec{m} \times \vec{B} \)
PE: \( U = - \vec{m} \cdot \vec{B} \)
Magnetic Field due to Dipole (Axial): \( B = \frac{\mu_0}{4\pi} \frac{2 m}{r^3} \)
Equatorial: \( B = \frac{\mu_0}{4\pi} \frac{m}{r^3} \)
Earth's Magnetism: \( B_H = B \cos\delta \), \( B_V = B \sin\delta \)
Moving Coil Galvanometer: \( \tau = N I A B \sin\theta = k \theta \), Sensitivity \( S = \frac{\theta}{I} = \frac{N A B}{k} \)
Ammeter: \( R_s = \frac{I_g G}{I - I_g} \)
Voltmeter: \( R = \frac{V}{I_g} - G \)

4. Electromagnetic Induction and Alternating Currents

Magnetic Flux: \( \Phi_B = \vec{B} \cdot \vec{A} = B A \cos\theta \)
Faraday's Law: \( \mathcal{E} = - \frac{d\Phi_B}{dt} \)
Motional Emf: \( \mathcal{E} = B l v \) (perpendicular)
Induced Current: \( I = \frac{\mathcal{E}}{R} \)
Self Inductance: \( \Phi = L I \), \( \mathcal{E} = - L \frac{dI}{dt} \)
Mutual Inductance: \( \Phi_{21} = M I_1 \), \( \mathcal{E}_2 = - M \frac{dI_1}{dt} \)
Inductor Energy: \( U = \frac{1}{2} L I^2 \)
AC Voltage: \( v = V_m \sin \omega t \)
RMS: \( V_{\text{rms}} = \frac{V_m}{\sqrt{2}} \), \( I_{\text{rms}} = \frac{I_m}{\sqrt{2}} \)
Resistive Circuit: \( I = \frac{V}{R} \), Phase 0
Inductive: \( X_L = \omega L \), Phase lag \( \pi/2 \)
Capacitive: \( X_C = \frac{1}{\omega C} \), Phase lead \( \pi/2 \)
Impedance: \( Z = \sqrt{R^2 + (X_L - X_C)^2} \)
Phase Angle: \( \tan\phi = \frac{X_L - X_C}{R} \)
Average Power: \( P = V_{\text{rms}} I_{\text{rms}} \cos\phi \)
Power Factor: \( \cos\phi = \frac{R}{Z} \)
Resonance: \( \omega_0 = \frac{1}{\sqrt{L C}} \), \( Z = R \), \( I_{\text{max}} = \frac{V}{R} \)
Quality Factor: \( Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 R C} \)
Transformer: \( \frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s} \), Efficiency \( \eta = \frac{V_s I_s}{V_p I_p} \)

5. Electromagnetic Waves

Displacement Current: \( I_d = \epsilon_0 \frac{d\Phi_E}{dt} \)
Ampere-Maxwell Law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 (I + I_d) \)
EM Wave Speed: \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \) m/s
E and B Relation: \( E = c B \), Perpendicular
Energy Density: Electric \( u_E = \frac{1}{2} \epsilon_0 E^2 \), Magnetic \( u_B = \frac{B^2}{2 \mu_0} \), Average \( u = \epsilon_0 E_{\text{rms}}^2 \)
Intensity: \( I = c u_{\text{avg}} = \frac{1}{2} c \epsilon_0 E_m^2 \)
Momentum: \( p = \frac{U}{c} \) (absorbed), \( p = \frac{2U}{c} \) (reflected)
Radiation Pressure: \( P = \frac{I}{c} \) (absorbed), \( P = \frac{2I}{c} \) (reflected)
EM Spectrum: Radio, Micro, IR, Visible, UV, X-ray, Gamma

6. Ray Optics and Optical Instruments

Laws of Reflection: Angle i = r, Same plane
Snell's Law: \( n_1 \sin i = n_2 \sin r \)
Absolute Refractive Index: \( n = \frac{c}{v} \)
Relative: \( n_{21} = \frac{n_2}{n_1} = \frac{\sin i}{\sin r} \)
Total Internal Reflection: \( \sin i_c = \frac{n_2}{n_1} \) (n1 > n2)
Mirror Formula: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \)
Magnification: \( m = -\frac{v}{u} = \frac{h'}{h} \)
Lens Formula: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)
Lens Maker's Formula: \( \frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \)
Power: \( P = \frac{1}{f} \) (in m), For Combination: \( P = P_1 + P_2 - d P_1 P_2 \)
Magnification for Lens: \( m = \frac{v}{u} \)
Prism Deviation: \( \delta = i + e - A \), Minimum \( \delta_m = 2i - A \)
Dispersion: \( \delta = (\mu - 1) A \)
Angular Dispersion: \( \theta = (\mu_v - \mu_r) A \)
Dispersive Power: \( \omega = \frac{\mu_v - \mu_r}{\mu - 1} \)
Simple Microscope: \( m = 1 + \frac{D}{f} \) (normal), \( m = \frac{D}{f} \) (near point)
Compound Microscope: \( m = m_o m_e = -\frac{L}{f_o} \left(1 + \frac{D}{f_e}\right) \)
Astronomical Telescope: \( m = -\frac{f_o}{f_e} \) (normal), Length \( L = f_o + f_e \)
Reflecting Telescope: \( f = \frac{R}{2} \)

7. Wave Optics

Huygens' Principle: Each point as secondary source
Interference: \( I = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos\delta \)
Constructive: \( \delta = 2\pi n \), Path Diff \( \Delta x = n \lambda \)
Destructive: \( \Delta x = (2n+1) \frac{\lambda}{2} \)
Young's Double Slit: Fringe Width \( \beta = \frac{\lambda D}{d} \)
Position: \( x_n = \frac{n \lambda D}{d} \) (bright)
With Medium: \( \beta' = \frac{\lambda D}{\mu d} \)
Single Slit Diffraction: Central Max Width \( 2\beta = \frac{2\lambda D}{a} \)
Minima: \( a \sin\theta = n \lambda \)
Rayleigh Criterion: \( \theta = 1.22 \frac{\lambda}{D} \)
Polarization: Brewster's Law \( \tan i_p = n \)
Malus' Law: \( I = I_0 \cos^2 \theta \)

8. Dual Nature of Matter and Radiation

Photoelectric Effect: \( h \nu = \phi + KE_{\text{max}} \)
Stopping Potential: \( V_0 = \frac{KE_{\text{max}}}{e} \)
Threshold Frequency: \( \nu_0 = \frac{\phi}{h} \)
Photon Energy: \( E = h \nu = \frac{h c}{\lambda} \)
Photon Momentum: \( p = \frac{h}{\lambda} \)
de Broglie Wavelength: \( \lambda = \frac{h}{p} = \frac{h}{m v} = \frac{h}{\sqrt{2 m E}} \)
For Electron: \( \lambda = \frac{12.27}{\sqrt{V}} \) Å
Davisson-Germer: \( \lambda = \frac{h}{\sqrt{2 m e V}} \)

9. Atoms and Nuclei

Bohr Model:
- Radius: \( r_n = n^2 \frac{4\pi \epsilon_0 \hbar^2}{m e^2} = 0.529 n^2 \) Å
- Velocity: \( v_n = \frac{e^2}{2 \epsilon_0 h} \frac{1}{n} = \frac{c}{137 n} \)
- Energy: \( E_n = - \frac{m e^4}{8 \epsilon_0^2 h^2} \frac{1}{n^2} = - \frac{13.6}{n^2} \) eV
- Frequency: \( \nu = \frac{\Delta E}{h} \)
Spectral Series: Lyman (UV), Balmer (Visible), etc.
Nucleus Size: \( R = R_0 A^{1/3} \), \( R_0 = 1.2 \times 10^{-15} \) m
Density: \( \rho = \frac{3m}{4\pi R^3} \approx 2.3 \times 10^{17} \) kg/m³
Binding Energy: \( BE = [\ Z m_p + (A-Z) m_n - m_N ] c^2 \)
BE per Nucleon: \( \frac{BE}{A} \)
Radioactive Decay: \( N = N_0 e^{-\lambda t} \)
Decay Constant: \( \lambda = \frac{\ln 2}{T_{1/2}} \)
Half-Life: \( T_{1/2} = \frac{0.693}{\lambda} \)
Mean Life: \( \tau = \frac{1}{\lambda} \)
Activity: \( A = \lambda N = A_0 e^{-\lambda t} \)
Alpha Decay: \( Q = (m_P - m_D - m_\alpha) c^2 \)
Beta Decay: Electron/Positron emission
Gamma Decay: Photon emission
Nuclear Reactions: Q-value \( Q = (\sum m_i - \sum m_f) c^2 \)
Fission: e.g., U-235 + n → fragments + neutrons + energy
Fusion: e.g., 4H → He + energy

10. Electronic Devices

Semiconductors: Intrinsic \( n_i = \sqrt{n p} \), Extrinsic n-type (donors), p-type (acceptors)
Doping: \( n_e \approx N_D \), \( n_h \approx N_A \)
pn Junction: Depletion Region, Barrier Potential \( V_b = \frac{kT}{e} \ln \frac{N_A N_D}{n_i^2} \)
Forward Bias: Reduces barrier
Reverse Bias: Increases barrier
Diode Equation: \( I = I_0 (e^{eV / kT} - 1) \)
Zener Diode: Breakdown for regulation
Rectifier: Half-Wave \( V_{\text{rms}} = \frac{V_m}{2} \), Full-Wave \( \frac{V_m}{\sqrt{2}} \)
Ripple Factor: Half-Wave 1.21, Full-Wave 0.48
Filter: Capacitor \( V_{\text{ripple}} = \frac{I_L}{f C} \)
Transistor: NPN/PNP, \( I_E = I_B + I_C \)
Current Gain: \( \beta = \frac{I_C}{I_B} \), \( \alpha = \frac{I_C}{I_E} \), \( \beta = \frac{\alpha}{1 - \alpha} \)
CE Amplifier: Voltage Gain \( A_v = -\beta \frac{R_C}{R_i} \)
Logic Gates:
- NOT: Y = \bar{A}
- AND: Y = A \cdot B
- OR: Y = A + B
- NAND: Y = \overline{A \cdot B}
- NOR: Y = \overline{A + B}
- XOR: Y = A \oplus B = \bar{A} B + A \bar{B}

11. Communication Systems

Signal Bandwidth: Range of frequencies
Amplitude Modulation: \( v = A_c (1 + m \sin \omega_m t) \sin \omega_c t \)
Modulation Index: \( m = \frac{A_m}{A_c} \)
Sidebands: \( f_c \pm f_m \)
Power in AM: \( P_t = P_c (1 + \frac{m^2}{2}) \)
Frequency Modulation: Deviation \( \delta = k_f A_m \), Index \( m_f = \frac{\delta}{f_m} \)
Phase Modulation: Index \( m_p = k_p A_m \)
Noise: SNR = \frac{P_s}{P_n}
Attenuation: \( \alpha = \frac{10}{L} \log \frac{P_i}{P_o} \) dB/km
Antenna Length: \( l = \frac{\lambda}{4} \) or \( \frac{\lambda}{2} \)
Line of Sight: \( d = \sqrt{2 h_T R} + \sqrt{2 h_R R} \)
Satellite Communication: Geostationary T=24h
Optical Fiber: TIR, Acceptance Angle \( \theta_a = \sin^{-1} \sqrt{n_1^2 - n_2^2} \)

Pro Tip: This exhaustive sheet now covers EVERY formula from the CBSE syllabus! Use with practice problems and diagrams from our notes for top scores.

```

Button

Pages

FORMULA'S PHYSICS