JEE Main (Paper 1: B.E./B.Tech)

Rotational Motion

• Torque and angular momentumτ = r × F = Iα, L = Iω, τ = dL/dt
• Parallel and perpendicular axesI = I_cm + Md², I_z = I_x + I_y (planar body)
• Standard MOI (about centre)Ring MR², Disc ½MR², Solid sphere (2/5)MR², Hollow sphere (2/3)MR², Rod (1/12)ML²
• Rolling kinetic energyKE = ½Mv²(1 + k²/R²), with v = ωR
• Rolling down an inclinea = g sinθ / (1 + k²/R²), v = √(2gh / (1 + k²/R²))
• Centre of massx_cm = Σmᵢxᵢ / Σmᵢ

💡 For rolling, just plug k²/R²: ring 1, disc and cylinder ½, solid sphere 2/5, hollow sphere 2/3.

Thermodynamics and Kinetic Theory

• First lawΔQ = ΔU + W, ΔU = nC_vΔT
• Isothermal workW = nRT ln(V₂/V₁)
• Adiabatic processPV^γ = const, TV^(γ−1) = const, W = nR(T₁ − T₂)/(γ − 1)
• Mayer's relationC_p − C_v = R, γ = C_p/C_v = 1 + 2/f
• Carnot efficiencyη = 1 − T₂/T₁ (temperatures in kelvin)
• Gas speedsv_rms = √(3RT/M), KE per molecule = (f/2)kT

💡 Sign convention: W is work done BY the gas. Always convert °C to K before using Carnot or gas laws.

Electrostatics and Capacitors

• Coulomb field and potentialF = kq₁q₂/r², E = kq/r², V = kq/r, k = 1/(4πε₀) = 9 × 10⁹ N m²/C²
• Gauss's law∮E·dA = q_enclosed/ε₀
• Potential energy of a pairU = kq₁q₂/r, W = qΔV
• Dipolep = q(2a), E_axis = 2kp/r³, E_equator = kp/r³, τ = p × E
• Parallel plate capacitorC = Kε₀A/d; with slab of thickness t: C = ε₀A/(d − t + t/K)
• Energy and combinationsU = ½CV² = Q²/2C; series 1/C = Σ1/Cᵢ, parallel C = ΣCᵢ

💡 Battery connected means V stays fixed; battery removed means Q stays fixed. Decide this first in every dielectric question.

Current Electricity

• Ohm's law and resistanceV = IR, R = ρL/A
• Drift speedI = neAv_d, J = σE
• Stretched wire (volume constant)R ∝ L²; length n times gives R' = n²R
• Cell with internal resistanceI = E/(R + r), terminal voltage V = E − Ir
• Cells in parallelE_eq = (E₁/r₁ + E₂/r₂)/(1/r₁ + 1/r₂), 1/r_eq = 1/r₁ + 1/r₂
• Wheatstone and metre bridgeP/Q = R/S at balance; metre bridge X/R = l/(100 − l)
• PowerP = VI = I²R = V²/R

💡 Small stretch: %ΔR ≈ 2 × %ΔL. In cell problems, check whether the cells oppose each other before adding EMFs.

Magnetic Effects of Current

• Long straight wireB = μ₀I/(2πr)
• Circular loopCentre: B = μ₀NI/(2R); axis: B = μ₀NIR²/(2(R² + x²)^(3/2))
• Solenoid and toroidB = μ₀nI (n = turns per metre), toroid B = μ₀NI/(2πr)
• Lorentz forceF = q(v × B), radius r = mv/(qB), T = 2πm/(qB)
• Force on wire and between wiresF = I(L × B), F/L = μ₀I₁I₂/(2πd)
• Magnetic moment and torquem = NIA, τ = m × B

💡 Time period in a magnetic field does not depend on speed; a faster particle just moves in a bigger circle.

EMI and Alternating Current

• Faraday and motional EMFε = −N dΦ/dt, ε = Bvl
• Charge through a coilQ = NΔΦ/R (independent of time taken)
• Inductor energyU = ½LI², ε = −L dI/dt
• Reactances and impedanceX_L = ωL, X_C = 1/(ωC), Z = √(R² + (X_L − X_C)²)
• Resonance and powerω₀ = 1/√(LC), P = V_rms I_rms cosφ, cosφ = R/Z

💡 At resonance X_L = X_C and Z = R. Doubling the frequency doubles X_L and halves X_C.

Ray Optics

• Mirror and lens formulaMirror 1/v + 1/u = 1/f; lens 1/v − 1/u = 1/f
• Lens maker's formula1/f = (μ_rel − 1)(1/R₁ − 1/R₂), μ_rel = μ_lens/μ_medium
• Refraction and TIRμ₁ sin i = μ₂ sin r, sin C = 1/μ, apparent depth = real depth/μ
• PrismA = r₁ + r₂, δ = i + e − A, μ = sin((A + δ_m)/2) / sin(A/2)
• Lens combination and power1/F = 1/f₁ + 1/f₂, P = 1/f (f in metres, P in dioptre)

💡 A lens in a liquid uses μ_rel = μ_glass/μ_liquid; if μ_rel is close to 1, the focal length becomes very large.

Wave Optics

• YDSE fringe widthβ = λD/d, angular width = λ/d
• Path and phase differenceΔx = yd/D, φ = (2π/λ)Δx
• Resultant intensityI = I₁ + I₂ + 2√(I₁I₂) cosφ; equal slits: I = I₀cos²(φ/2)
• Single slitMinima at a sinθ = nλ, central maximum width = 2λD/a
• PolarisationMalus: I = I₀cos²θ, Brewster: tan θ_B = μ

💡 In a medium of index μ, wavelength becomes λ/μ, so the fringe width also shrinks by μ.

Modern Physics: Dual Nature

• Photon energyE = hν = hc/λ; E(eV) ≈ 1240/λ(nm)
• Einstein's equationKE_max = hν − φ = eV₀
• Thresholdφ = hν₀ = hc/λ₀
• de Broglie wavelengthλ = h/p = h/√(2mK)
• Electron through potential Vλ = h/√(2meV) ≈ 1.227/√V nm

💡 Intensity changes the number of photoelectrons (current), never the stopping potential.

Modern Physics: Atoms and Nuclei

• Bohr model (hydrogen-like)rₙ = 0.529 n²/Z Å, vₙ ∝ Z/n, Eₙ = −13.6 Z²/n² eV
• Rydberg formula1/λ = RZ²(1/n₁² − 1/n₂²)
• Mass defect and binding energyBE = Δm c², 1 u = 931.5 MeV/c²
• Nuclear radiusR = R₀A^(1/3), so nuclear density is the same for all nuclei
• Radioactive decayN = N₀e^(−λt) = N₀(½)^(t/T½), T½ = 0.693/λ, mean life τ = 1/λ

💡 Count half-lives first: after n half-lives, fraction left = 1/2ⁿ and fraction decayed = 1 − 1/2ⁿ.