Classical & Modern Physics Fundamentals

Physics spans CM, EM, thermo, QM, and SR/GR. Understanding requires both mathematical rigor and physical intuition for real-world application.

## Classical Mechanics

Newton's Laws form the foundation: N1 (inertia) — object at rest stays at rest unless acted on by net external force. N2: F=ma (vector equation, apply component-wise). N3: action-reaction pairs act on DIFFERENT bodies (common misconception: they don't cancel). Weight W=mg where g=9.81 m/s² (varies w/ altitude and latitude).

Kinematics (constant accel): v = v₀ + at, x = x₀ + v₀t + ½at², v² = v₀² + 2a(x-x₀). Projectile motion: decompose into independent x (ax=0) and y (ay=-g) components. Range R = v₀²sin(2θ)/g, max at θ=45°. Air resistance makes real trajectories asymmetric.

Energy methods often simpler than force analysis. KE = ½mv², gravitational PE = mgh (near surface), elastic PE = ½kx². Work-energy theorem: Wnet = ΔKE. Conservation of energy: if only conservative forces act, Ei = Ef. Power P = dW/dt = F·v.

Momentum: p = mv (vector). Impulse J = FΔt = Δp. Conservation of momentum in isolated systems. Collisions: elastic (KE conserved) vs inelastic (KE not conserved). Perfectly inelastic → objects stick together, max KE loss. COM velocity unchanged in all collision types.

Rotational dynamics: τ = Iα (analog of F=ma). Moment of inertia I depends on mass distribution — solid cylinder I=½MR², solid sphere I=⅖MR², thin rod about center I=ML²/12. Parallel axis theorem: I = Icm + Md². Angular momentum L = Iω, conserved when no external torque (ice skater spinning). Rolling without slipping: v=ωR, both translational and rotational KE.

## Electromagnetism

Coulomb's Law: F = kq₁q₂/r² where k = 8.99×10⁹ N·m²/C². Superposition principle for multiple charges. Electric field E = F/q₀ (test charge). Field lines: originate on +, terminate on -, density proportional to field strength.

Gauss's Law: ∮E·dA = Qenc/ε₀. Choose Gaussian surface matching charge symmetry — spherical for point/sphere, cylindrical for infinite line/cylinder, planar for infinite plane. E from infinite plane of charge: E = σ/(2ε₀), uniform and independent of distance.

Electric potential: V = kQ/r (point charge), V = -∫E·dl. Potential is scalar → easier to compute than E for complex charge distributions. E = -∇V (gradient relationship). Equipotential surfaces perpendicular to field lines. Capacitance C = Q/V; parallel plate C = ε₀A/d. Energy stored U = ½CV² = ½QV.

Magnetic force: F = qv×B (Lorentz force, perpendicular to both v and B). Charged particle in uniform B → circular motion, r = mv/(qB), cyclotron freq ω = qB/m. Force on current-carrying wire: F = IL×B. Biot-Savart Law: dB = (μ₀/4π)(Idl×r̂/r²). Ampere's Law: ∮B·dl = μ₀Ienc.

Maxwell's Equations (integral form): 1) Gauss E: ∮E·dA = Q/ε₀, 2) Gauss B: ∮B·dA = 0 (no magnetic monopoles), 3) Faraday: ∮E·dl = -dΦB/dt (changing B creates E), 4) Ampere-Maxwell: ∮B·dl = μ₀(I + ε₀dΦE/dt). EM waves: E and B oscillate perpendicular to each other and propagation direction, c = 1/√(μ₀ε₀) = 3×10⁸ m/s.

## Thermodynamics & Statistical Mechanics

Zeroth Law: thermal equilibrium is transitive (defines temperature). First Law: ΔU = Q - W. For ideal gas: U depends only on T, U = nCvT. PV = nRT (ideal gas). Specific heats: Cp - Cv = R (per mole). γ = Cp/Cv: monatomic 5/3, diatomic 7/5.

Entropy: S = kB ln(Ω) (Boltzmann). ΔS = ∫dQ/T (reversible). Second Law: ΔSuniverse ≥ 0. Free energy: Helmholtz F = U - TS (const T,V), Gibbs G = H - TS (const T,P). Spontaneous process: ΔG < 0 at const T,P.

Maxwell-Boltzmann distribution: f(v) = 4π(m/2πkBT)^(3/2) v² exp(-mv²/2kBT). Most probable speed vp = √(2kBT/m), mean speed ⟨v⟩ = √(8kBT/πm), rms speed vrms = √(3kBT/m). Equipartition: each quadratic DOF contributes ½kBT to average energy.

## Quantum Mechanics

Wave-particle duality: photon energy E = hf = hc/λ, momentum p = h/λ (de Broglie). Photoelectric effect: KEmax = hf - φ (work function). Photons below threshold freq cannot eject electrons regardless of intensity — classical wave theory fails here.

Heisenberg uncertainty: ΔxΔp ≥ ℏ/2, ΔEΔt ≥ ℏ/2. Not measurement limitation but fundamental property of nature. Implies zero-point energy for confined particles.

Schrödinger equation (TISE): Ĥψ = Eψ where Ĥ = -ℏ²/(2m)∇² + V. Solutions give allowed energies and wavefunctions. |ψ|² = probability density. Particle in box: En = n²π²ℏ²/(2mL²), ψn = √(2/L)sin(nπx/L). Hydrogen atom: En = -13.6/n² eV, degeneracy = n².

QM postulates: 1) State described by wavefunction ψ, 2) Observables represented by Hermitian operators, 3) Measurement yields eigenvalue, collapses state to eigenstate, 4) Time evolution by Schrödinger equation. Superposition principle: any linear combination of solutions is also a solution.

Spin: intrinsic angular momentum, s=½ for electrons. Pauli exclusion: no two identical fermions in same quantum state. Explains periodic table structure, electron shell filling, and stability of matter. Bosons (integer spin) can occupy same state → BEC, lasers.

## Special Relativity

Postulates: 1) Laws of physics same in all inertial frames, 2) Speed of light c same for all observers. Consequences: time dilation Δt = γΔt₀ where γ = 1/√(1-v²/c²), length contraction L = L₀/γ, relativity of simultaneity.

Lorentz transformations: x' = γ(x-vt), t' = γ(t-vx/c²). Spacetime interval ds² = c²dt² - dx² - dy² - dz² is invariant. Four-momentum: E² = (pc)² + (mc²)². For massless particles: E = pc. Rest energy E₀ = mc². KE = (γ-1)mc².

Velocity addition: u' = (u-v)/(1-uv/c²). No material object can reach c — would require infinite energy. GPS satellites must account for both SR time dilation (clocks run slower due to velocity) and GR time dilation (clocks run faster due to weaker gravity) — corrections essential for meter-level accuracy.

## Waves and Optics

Wave equation: ∂²y/∂t² = v²∂²y/∂x². General solution y(x,t) = f(x-vt) + g(x+vt). Superposition → interference and standing waves. Standing wave: nodes fixed, antinodes at max amplitude. Resonant frequencies: string fixed both ends fn = nv/(2L).

Snell's Law: n₁sinθ₁ = n₂sinθ₂. Total internal reflection when θ > θc = arcsin(n₂/n₁), n₁>n₂. Thin lens: 1/f = 1/do + 1/di. Magnification m = -di/do. Diffraction: single slit minima at asinθ = mλ. Double slit maxima at dsinθ = mλ. Resolving power (Rayleigh): θmin = 1.22λ/D.