Class 11 physics
Units & measurement
- Sum or differenceΔ(A ± B) = ΔA + ΔB
- Product or quotientΔZ/Z = ΔA/A + ΔB/B
- PowerΔZ/Z = n(ΔA/A)
- Vernier least count1 MSD − 1 VSD
- Screw gauge least countpitch / no. of divisions
- Force, energy[MLT⁻²], [ML²T⁻²]
Kinematics
- Velocityv = u + at
- Displacements = ut + ½at²
- No timev² = u² + 2as
- Nth secondsn = u + a(2n − 1)/2
- Projectile: timeT = 2u sinθ / g
- Projectile: heightH = u² sin²θ / 2g
- Projectile: rangeR = u² sin2θ / g
- Centripetal accelerationa = v²/r = ω²r
Laws of motion
- Newton’s second lawF = ma = dp/dt
- ImpulseJ = FΔt = Δp
- Static frictionfs ≤ μsN
- Kinetic frictionfk = μkN
- Angle of reposetanθ = μ
- Banked road (no friction)v² = rg tanθ
- Max speed, flat roadv = √(μrg)
Work, energy & power
- WorkW = Fs cosθ
- Kinetic energyK = ½mv²
- Work–energy theoremW = ΔK
- Gravitational PEU = mgh
- Spring PEU = ½kx²
- PowerP = W/t = Fv
- Restitutione = (v₂ − v₁)/(u₁ − u₂)
- Elastic collision (u₂ = 0)v₁ = (m₁ − m₂)u₁/(m₁ + m₂)
Rotational motion
- Torqueτ = Iα = r × F
- Angular momentumL = Iω
- Rotational KEK = ½Iω²
- Ring / discMR², ½MR²
- Solid sphere⅖MR²
- Parallel axisI = Icm + Md²
- Perpendicular axisIz = Ix + Iy
- Rolling KE½mv²(1 + k²/R²)
Gravitation
- Newton’s lawF = Gm₁m₂/r²
- Surface gravityg = GM/R²
- At height hg′ = g(1 − 2h/R)
- At depth dg′ = g(1 − d/R)
- Escape velocityve = √(2gR)
- Orbital velocityv = √(GM/r)
- Potential energyU = −GMm/r
- Kepler’s third lawT² ∝ r³
Properties of solids
- StressF/A
- Young’s modulusY = FL / AΔL
- Bulk modulusB = −VΔP/ΔV
- Shear modulusG = shear stress / shear strain
- Energy per volume½ × stress × strain
- Linear expansionΔL = αLΔT
Fluid mechanics
- Pressure at depthP = P₀ + ρgh
- ContinuityA₁v₁ = A₂v₂
- BernoulliP + ½ρv² + ρgh = const
- Torricelliv = √(2gh)
- Viscous forceF = ηA(dv/dx)
- Stokes’ lawF = 6πηrv
- Terminal velocity2r²(ρ − σ)g / 9η
- Surface tensionS = F/l
- Excess pressure2S/r (drop), 4S/r (bubble)
- Capillary riseh = 2S cosθ / rρg
Thermodynamics & heat
- HeatQ = mcΔT, Q = mL
- Ideal gasPV = nRT
- First lawΔQ = ΔU + W
- Isothermal workW = nRT ln(V₂/V₁)
- AdiabaticPVᵞ = const
- Mayer’s relationCp − Cv = R
- Carnot efficiencyη = 1 − T₂/T₁
- Stefan’s lawP = eσAT⁴
- Wien’s lawλmT = b
Kinetic theory
- PressureP = ⅓ρv²rms
- RMS speed√(3RT/M)
- Average KE per molecule(3/2)kBT
- Internal energyU = (f/2)nRT
- Specific heatCv = (f/2)R
- Mean free path1 / (√2 πd²n)
Oscillations
- SHMx = A sin(ωt + φ)
- Speedv = ω√(A² − x²)
- SpringT = 2π√(m/k)
- PendulumT = 2π√(l/g)
- Total energyE = ½kA² = ½mω²A²
- Springs in series1/k = 1/k₁ + 1/k₂
Waves & sound
- Wave speedv = fλ
- Stringv = √(T/μ)
- Gasv = √(γP/ρ)
- Beatsf = |f₁ − f₂|
- Doppler effectf′ = f(v ± vo)/(v ∓ vs)
- Open pipef = nv/2L
- Closed pipef = (2n − 1)v/4L
- Fixed stringf = nv/2L
Class 12 physics
Electrostatics
- Coulomb’s lawF = kq₁q₂/r²
- Field of point chargeE = kq/r²
- PotentialV = kq/r
- Gauss’s lawΦ = q/ε₀
- Dipole torqueτ = pE sinθ
- Dipole field (axial)2kp/r³
- Dipole field (equatorial)kp/r³
- Infinite sheetE = σ/2ε₀
Capacitors
- CapacitanceC = Q/V
- Parallel plateC = ε₀A/d
- With dielectricC = Kε₀A/d
- Series1/C = 1/C₁ + 1/C₂
- ParallelC = C₁ + C₂
- EnergyU = ½CV² = Q²/2C
- Energy density½ε₀E²
Current electricity
- Ohm’s lawV = IR
- ResistanceR = ρl/A
- Drift velocityI = nAevd
- Terminal voltageV = E − Ir
- PowerP = VI = I²R
- TemperatureR = R₀(1 + αΔT)
- Meter bridgeR/S = l/(100 − l)
- PotentiometerE₁/E₂ = l₁/l₂
- Internal resistancer = R(l₁ − l₂)/l₂
Moving charges & magnetism
- Lorentz forceF = q(E + v × B)
- Radius in fieldr = mv/qB
- Force on wireF = BIl sinθ
- Long wireB = μ₀I/2πr
- Circular loop centreB = μ₀I/2R
- SolenoidB = μ₀nI
- Parallel wiresF/l = μ₀I₁I₂/2πd
- Coil torqueτ = NIAB sinθ
- Ammeter shuntS = IgG/(I − Ig)
Magnetism & matter
- Magnetic momentm = NIA
- MagnetisationM = m/V
- Field in matterB = μ₀(H + M)
- Susceptibilityχ = M/H
- Permeabilityμr = 1 + χ
- Curie’s lawχ = C/T
Electromagnetic induction
- FluxΦ = BA cosθ
- Faraday–Lenz lawε = −dΦ/dt
- Motional emfε = Blv
- Self-inductionε = −L dI/dt
- Solenoid inductanceL = μ₀n²Al
- Energy in inductorU = ½LI²
Alternating current
- RMS valueVrms = V₀/√2
- ReactancesXL = ωL, XC = 1/ωC
- ImpedanceZ = √(R² + (XL − XC)²)
- Phasetanφ = (XL − XC)/R
- Resonanceω₀ = 1/√(LC)
- Average powerP = VrmsIrms cosφ
- TransformerVs/Vp = Ns/Np
Electromagnetic waves
- Speed of lightc = 1/√(μ₀ε₀)
- Field ratioE₀/B₀ = c
- Displacement currentId = ε₀ dΦE/dt
- IntensityI = ½ε₀cE₀²
- Radiation pressureI/c (absorbed), 2I/c (reflected)
Ray optics
- Mirror formula1/v + 1/u = 1/f
- Lens formula1/v − 1/u = 1/f
- Snell’s lawμ = sin i / sin r
- Lens maker’s1/f = (μ − 1)(1/R₁ − 1/R₂)
- PowerP = 1/f (metre)
- Critical anglesinC = 1/μ
- Prismμ = sin[(A + δm)/2] / sin(A/2)
- Telescopem = f₀/fe
- Microscopem ≈ (L/f₀)(D/fe)
Wave optics
- Fringe widthβ = λD/d
- Path differenceΔ = yd/D
- ConstructiveΔ = nλ
- DestructiveΔ = (n + ½)λ
- Single slit minimaa sinθ = nλ
- Central maximum width2λD/a
- Brewster’s lawtan iB = μ
- Malus’ lawI = I₀cos²θ
Dual nature of radiation & matter
- Photon energyE = hν = 1240/λ(nm) eV
- Photon momentump = h/λ
- Einstein’s equationhν = φ₀ + Kmax
- Stopping potentialeV₀ = Kmax
- de Broglieλ = h/p = h/√(2mK)
- Electron through Vλ = 12.27/√V Å
Atoms
- Bohr radiusrn = 0.529 n²/Z Å
- Energy levelsEn = −13.6 Z²/n² eV
- Orbital speedvn = 2.19×10⁶ Z/n m/s
- Angular momentumL = nh/2π
- Spectral lines1/λ = R(1/n₁² − 1/n₂²)
- Rydberg constantR = 1.097×10⁷ m⁻¹
Nuclei
- Nuclear radiusR = R₀A^⅓, R₀ = 1.2 fm
- Mass defectΔm = Zmp + (A − Z)mn − M
- Binding energyEb = Δm c²
- Energy equivalent1 u = 931.5 MeV/c²
- Decay lawN = N₀e−λt
- Half-lifeT½ = 0.693/λ
- ActivityA = λN
Semiconductor electronics
- Transistor currentsIE = IB + IC
- Current gainα = IC/IE, β = IC/IB
- Relationβ = α/(1 − α)
- Mass action lawnenh = ni²
- Total currentI = Ie + Ih
- Voltage gain (CE)Av = βRL/Rin
Constants and conversions
Fundamental constants
- Speed of lightc = 3×10⁸ m/s
- Gravitational constantG = 6.67×10⁻¹¹ N m²/kg²
- Planck’s constanth = 6.63×10⁻³⁴ J s
- Electron chargee = 1.6×10⁻¹⁹ C
- Electron mass9.11×10⁻³¹ kg
- Proton mass1.67×10⁻²⁷ kg
More constants
- Boltzmann constantkB = 1.38×10⁻²³ J/K
- Avogadro numberNA = 6.02×10²³ /mol
- Gas constantR = 8.31 J/mol K
- Permittivityε₀ = 8.85×10⁻¹² F/m
- Permeabilityμ₀ = 4π×10⁻⁷ T m/A
- Stefan–Boltzmannσ = 5.67×10⁻⁸ W/m² K⁴
Conversions
- Electron-volt1 eV = 1.6×10⁻¹⁹ J
- Atomic mass unit1 u = 1.66×10⁻²⁷ kg
- Coulomb constant1/4πε₀ = 9×10⁹ N m²/C²
- Gravityg = 9.8 m/s²
- hc1240 eV nm
This sheet lists the standard NCERT results without derivations. Always check the units and sign conventions before you substitute values, and use the full chapter notes when you need the reasoning behind a formula.
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