CHSE Odisha Elective Physics cheat sheet

Coulomb's law in vacuum: F = (1/4πε₀)·q₁q₂/r², with 1/4πε₀ = 9×10⁹ N·m²/C²; in a medium divide by dielectric constant K.

Electric field of a point charge E = (1/4πε₀)·q/r², directed radially outward for positive q; field inside a conductor in electrostatic equilibrium is zero.

Gauss's law: ∮E·dS = q_enclosed/ε₀; use it for spherical, cylindrical and infinite-plane symmetry.

Potential from a point charge V = (1/4πε₀)·q/r; E = −dV/dr, so field points from high to low potential.

Dipole moment p = q·2a; torque in uniform field τ = pE sinθ, potential energy U = −pE cosθ; axial field = (1/4πε₀)·2p/r³.

Capacitance C = Q/V; parallel plate C = ε₀KA/d; series 1/C = Σ1/Cᵢ, parallel C = ΣCᵢ.

Energy stored U = ½CV² = ½QV = Q²/2C; energy density in field u = ½ε₀E².

Dielectric slab of constant K filling the gap multiplies C by K; with battery disconnected Q is fixed, with battery connected V is fixed.

Spherical conductor of radius R: C = 4πε₀R; isolated sphere of 1 m radius has C ≈ 111 pF.

Ohm's law V = IR holds for metallic conductors at constant temperature; drift velocity v_d = I/(nAe), typically ~10⁻⁴ m/s.

Current density J = I/A = nev_d = σE; conductivity σ = 1/ρ, and ρ = m/(ne²τ) where τ is relaxation time.

Resistance R = ρL/A; temperature dependence R = R₀(1 + αΔT), α negative for semiconductors.

Series: R = ΣRᵢ, same current; parallel: 1/R = Σ1/Rᵢ, same voltage.

Kirchhoff's rules: junction rule ΣI = 0 (charge conservation), loop rule Σε = ΣIR (energy conservation).

Wheatstone bridge balanced when P/Q = R/S; metre bridge balance l₁/l₂ = R/S with unknown in the gap.

EMF and terminal voltage: V = ε − Ir; maximum power transfer to load when R = r, giving P = ε²/4r.

Potentiometer compares EMFs as ε₁/ε₂ = l₁/l₂ and measures internal resistance r = R(l₁ − l₂)/l₂.

Cells in series: ε_eq = Σε, r_eq = Σr; in parallel with identical cells ε_eq = ε, r_eq = r/n.

Biot–Savart law: dB = (μ₀/4π)·I dl sinθ/r²; μ₀/4π = 10⁻⁷ T·m/A.

Straight wire B = μ₀I/2πr; circular loop at centre B = μ₀I/2R; solenoid B = μ₀nI; toroid B = μ₀NI/2πr.

Force on moving charge F = qvB sinθ, circular path radius r = mv/qB, period T = 2πm/qB independent of speed.

Force on conductor F = BIL sinθ; between parallel wires F/L = μ₀I₁I₂/2πd, attractive for same-direction currents.

Ampere's circuital law ∮B·dl = μ₀I_enclosed; use for solenoids, toroids and coaxial cables.

Torque on current loop τ = NIAB sinθ = mB sinθ, where magnetic moment m = NIA; direction given by right-hand rule.

Moving-coil galvanometer: I = (k/NAB)θ; converting to ammeter uses a low shunt S = I_gG/(I − I_g), to voltmeter a high series R = V/I_g − G.

Bar magnet: axial field B = (μ₀/4π)·2M/r³, equatorial B = (μ₀/4π)·M/r³; torque τ = MB sinθ, U = −MB cosθ.

Earth's magnetic elements: declination, dip angle and horizontal component H = B cosδ; tangent law for deflection magnetometer B = H tanθ.

Faraday's law ε = −N dΦ/dt; Lenz's law fixes direction so induced current opposes the flux change.

Motional EMF in a rod moving with speed v in field B: ε = Bvl, current I = Bvl/R, retarding force F = B²l²v/R.

Self-inductance ε = −L dI/dt; solenoid L = μ₀n²Al; energy stored U = ½LI².

Mutual inductance of two coaxial coils M = μ₀n₁n₂Al; ε₂ = −M dI₁/dt.

AC through R: I and V in phase; through L: V leads I by 90°, X_L = ωL; through C: I leads V by 90°, X_C = 1/ωC.

Series LCR impedance Z = √(R² + (X_L − X_C)²); at resonance X_L = X_C, f₀ = 1/(2π√(LC)), Z = R and current is maximum.

Power in AC: P = V_rms I_rms cosφ, power factor cosφ = R/Z; wattless current when φ = 90°.

Transformer: V_s/V_p = N_s/N_p = I_p/I_s; step-up raises voltage, ideal transformer conserves power, no frequency change.

RMS and peak values: I_rms = I₀/√2, V_rms = V₀/√2; average of AC over a full cycle is zero.

Mirror formula 1/v + 1/u = 1/f with f = R/2; sign convention: distances measured from pole, against incident light negative.

Refraction: n₁ sin i = n₂ sin r; n = c/v = real depth/apparent depth; critical angle sin C = 1/n for n₂ < n₁.

Lens maker's formula 1/f = (n − 1)(1/R₁ − 1/R₂); thin lens 1/v − 1/u = 1/f; power P = 1/f (metres), unit dioptre.

Lens combination: 1/F = 1/f₁ + 1/f₂; two thin lenses separated by d give 1/F = 1/f₁ + 1/f₂ − d/f₁f₂.

Prism: A + D = i + e; at minimum deviation n = sin[(A + D_m)/2]/sin(A/2); for small A, D = (n − 1)A.

Dispersion: angular dispersion = (n_v − n_b)A; dispersive power ω = (n_v − n_b)/(n_y − 1).

Simple microscope M = 1 + D/f (image at near point D = 25 cm); compound microscope M ≈ (v₀/u₀)(D/f_e).

Astronomical telescope M = f₀/f_e (normal adjustment), length f₀ + f_e; for image at near point M = f₀/f_e(1 + f_e/D).

Total internal reflection occurs only when light travels from denser to rarer medium and i exceeds critical angle; basis of optical fibres and mirage.

Young's double slit: fringe width β = λD/d; bright fringes at path difference nλ, dark at (2n − 1)λ/2.

Intensity in interference I = I₁ + I₂ + 2√(I₁I₂)cosφ; for equal sources I_max/I_min = (a₁ + a₂)²/(a₁ − a₂)².

Immersing a double-slit setup in liquid of refractive index n reduces fringe width by factor n.

Single-slit diffraction: minima at a sinθ = nλ; central maximum width on screen = 2λD/a, twice a secondary maximum.

Resolving power of telescope = D/1.22λ; of microscope = 2n sinθ/λ; limit of resolution set by diffraction.

Polarisation by reflection: Brewster law tan i_p = n; reflected and refracted rays are perpendicular at polarising angle.

Malus law: intensity through analyser I = I₀ cos²θ; unpolarised light through a polariser loses half its intensity.

Huygens principle: every point on a wavefront is a secondary source; explains reflection, refraction and diffraction.

Coherent sources have constant phase difference; sustained interference requires them, and white light gives a central white fringe with coloured fringes.

Photon energy E = hν = hc/λ; h = 6.63×10⁻³⁴ J·s, and E(eV) = 1240/λ(nm).

Photoelectric equation hν = φ₀ + K_max; stopping potential eV₀ = K_max; threshold ν₀ = φ₀/h.

Photoelectric current saturates with intensity; K_max depends only on frequency, not intensity; emission is instantaneous.

de Broglie wavelength λ = h/p = h/mv = h/√(2mK); for an electron λ ≈ 12.27/√V Å with V in volts.

Momentum of photon p = E/c = h/λ; radiation pressure follows from momentum transfer.

Davisson–Germer experiment confirmed electron diffraction, verifying de Broglie's hypothesis.

Work function values are a few eV for metals; alkali metals have the lowest work functions and visible-light thresholds.

Photon flux for light of power P is n = P/hν per second; intensity = n hν per unit area per second.

Bohr model: mvr = nh/2π; radius r_n = 0.53 n²/Z Å; energy E_n = −13.6 Z²/n² eV.

Spectral lines: 1/λ = RZ²(1/n₁² − 1/n₂²), R = 1.097×10⁷ m⁻¹; Lyman series in UV, Balmer in visible.

Ionisation energy of hydrogen = 13.6 eV; energy needed for n = 1 to n = 2 is 10.2 eV.

Nuclear radius R = R₀A^(1/3), R₀ ≈ 1.2 fm; density is nearly constant for all nuclei (~2.3×10¹⁷ kg/m³).

Mass defect Δm and binding energy BE = Δm c²; 1 u = 931.5 MeV; BE per nucleon peaks near A = 56 (iron).

Radioactive decay N = N₀e^(−λt); half-life T = 0.693/λ; activity A = λN, unit becquerel.

Alpha decay reduces A by 4 and Z by 2; beta-minus raises Z by 1 with A unchanged; gamma emission changes neither.

Nuclear fission of U-235 releases about 200 MeV per event; fusion of light nuclei powers stars and releases energy per nucleon far larger.

Carbon dating uses half-life of ¹⁴C ≈ 5730 years; age t = (1/λ)ln(N₀/N).

Intrinsic semiconductor: n_e = n_h = n_i; mass action law n_e n_h = n_i²; conductivity rises with temperature.

Doped n-type has donor levels and electrons as majority carriers; p-type has acceptors and holes as majority carriers.

PN junction: depletion region, barrier potential ~0.7 V for Si and ~0.3 V for Ge; forward bias narrows the barrier, reverse bias widens it.

Diode I–V: negligible current below knee voltage, large forward current, small reverse saturation current; breakdown at reverse voltage.

Half-wave rectifier output frequency equals input; full-wave rectifier gives double the input frequency; a filter capacitor smooths ripple.

Zener diode operates in reverse breakdown as a voltage regulator; special diodes include LED, photodiode and solar cell.

Transistor: I_E = I_B + I_C; α = I_C/I_E, β = I_C/I_B, β = α/(1 − α); active, saturation and cut-off regions.

Common-emitter amplifier has 180° phase shift between input and output; voltage gain A_v = β × (R_C/r_in).

Logic gates: OR (Y = A + B), AND (Y = A·B), NOT (Y = Ā); NAND and NOR are universal gates.

Communication: modulation superimposes signal on carrier; bandwidth of AM = 2f_m; skywave uses ionospheric reflection, space wave is line-of-sight.

Torque τ = rF sinθ = Iα; angular momentum L = Iω; τ = dL/dt.

Moment of inertia: ring MR², disc MR²/2, solid sphere 2MR²/5, hollow sphere 2MR²/3, rod about centre ML²/12.

Parallel-axis theorem I = I_cm + Md²; perpendicular-axis theorem I_z = I_x + I_y for a plane lamina.

Rolling without slipping: v = ωR; total KE = ½Mv² + ½Iω² = ½Mv²(1 + k²/R²).

Rolling body on incline: a = g sinθ/(1 + I/MR²); sphere beats disc beats ring in a race down the incline.

Angular momentum conservation: I₁ω₁ = I₂ω₂ when external torque is zero, as in a spinning skater pulling arms in.

Equilibrium of rigid body requires both net force zero and net torque zero; centre of mass moves as if all mass and external forces acted there.

Radius of gyration k with I = Mk²; flywheel stores energy ½Iω² and smooths speed fluctuations.

Rotational work W = τθ and power P = τω; angular impulse = change in angular momentum.

Newton's law F = Gm₁m₂/r², G = 6.67×10⁻¹¹ N·m²/kg²; g = GM/R² ≈ 9.8 m/s² at Earth's surface.

Variation of g: with height g_h = g(1 − 2h/R) for h ≪ R; with depth g_d = g(1 − d/R); zero at Earth's centre.

Orbital speed v = √(GM/r); near-surface v ≈ 7.9 km/s; escape speed v_e = √(2GM/R) ≈ 11.2 km/s.

Kepler's laws: elliptical orbits, equal areas in equal times, T² ∝ a³; T² = 4π²a³/GM.

Gravitational potential V = −GM/r; potential energy U = −GMm/r; total energy of an orbit E = −GMm/2r.

Geostationary satellite: period 24 h, height ≈ 36,000 km, orbit in equatorial plane; weightlessness is free fall, not zero gravity.

Stress = force/area, strain = fractional change; Young's modulus Y = (F/A)/(ΔL/L), only for solids in longitudinal strain.

Hooke's law valid within proportional limit; elastic potential energy per volume = ½ × stress × strain.

Poisson's ratio σ = lateral strain/longitudinal strain, theoretical range −1 to 0.5; bulk modulus K = −P/(ΔV/V), compressibility = 1/K.

SHM: acceleration a = −ω²x, restoring force F = −kx; ω = 2π/T = √(k/m).

Displacement x = A sin(ωt + φ); velocity v = ω√(A² − x²), maximum at mean position; acceleration maximum at extremes.

Energy in SHM: KE = ½mω²(A² − x²), PE = ½mω²x², total = ½mω²A² constant.

Simple pendulum T = 2π√(L/g), independent of mass and small amplitude; seconds pendulum has T = 2 s, L ≈ 1 m.

Spring mass T = 2π√(m/k); springs in parallel add k, in series add 1/k.

Damped oscillation amplitude A = A₀e^(−bt/2m); forced oscillation resonates when driving frequency equals natural frequency.

Time period of a floating cylinder of cross-section A in liquid of density ρ: T = 2π√(m/Aρg).

Phase difference between displacement and velocity in SHM is π/2; between displacement and acceleration it is π.

Average velocity over a quarter cycle from mean to extreme is 2Aω/π, while instantaneous speed at mean is Aω.

Wave equation v = fλ; for a string v = √(T/μ) where μ is linear mass density.

Progressive wave y = A sin(ωt − kx) travels in +x; particle velocity = −v × slope of the waveform.

Speed of sound in air v = √(γP/ρ) = √(γRT/M); in gases independent of pressure at constant temperature, rises with temperature.

Standing waves on a string fixed at both ends: f_n = nv/2L; in a pipe open at both ends f_n = nv/2L; closed at one end only odd harmonics f_n = (2n − 1)v/4L.

Beats: beat frequency = |f₁ − f₂|; audible range roughly 20 Hz to 20 kHz.

Doppler effect: apparent frequency f' = f(v ± v_o)/(v ∓ v_s), signs chosen so approach raises pitch.

Intensity level in decibels = 10 log₁₀(I/I₀) with I₀ = 10⁻¹² W/m²; intensity ∝ amplitude² and ∝ 1/r² for a point source.

Superposition gives interference; path difference nλ gives constructive and (2n − 1)λ/2 destructive interference.

Newton's formula for sound speed in air ignored temperature; Laplace corrected it by the factor √γ, giving about 331 m/s at 0 °C.

Ideal gas law PV = nRT = NkT; k = 1.38×10⁻²³ J/K, R = 8.314 J/(mol·K).

Kinetic theory: PV = (1/3)mNc²_rms; average KE per molecule = (3/2)kT; v_rms = √(3RT/M).

Degrees of freedom: monatomic 3, diatomic 5, polyatomic 6; internal energy U = (f/2)nRT.

Specific heats: C_p − C_v = R; γ = C_p/C_v = 1 + 2/f, so 1.67 for monatomic and 1.4 for diatomic gases.

First law ΔQ = ΔU + ΔW; work done by gas W = ∫P dV, and W = PΔV at constant pressure.

Isothermal process: PV = constant, ΔU = 0; adiabatic: PV^γ = constant, TV^(γ−1) = constant, ΔQ = 0.

Carnot efficiency η = 1 − T₂/T₁; second law forbids a 100% efficient heat engine and spontaneous heat flow from cold to hot.

Coefficient of performance of refrigerator = T₂/(T₁ − T₂); efficiency of real engine always below Carnot value.

Conduction rate dQ/dt = kA(T₁ − T₂)/L; Stefan's law P = σAeT⁴ with σ = 5.67×10⁻⁸ W/(m²·K⁴).

Least count of vernier calliper = 1 main scale division − 1 vernier division; of screw gauge = pitch/number of circular scale divisions.

Zero error is subtracted algebraically from observed reading; negative zero error adds to the reading.

Metre bridge: unknown resistance R = S(l₁/l₂) where l₁ is the balancing length from the unknown gap; interchange gaps to remove end corrections.

Ohm's law experiment: plot V versus I, slope gives resistance; keep current small to avoid heating the wire.

Potentiometer measures EMF with no current drawn; balance point must lie on the wire, otherwise adjust rheostat or cell.

Focal length of convex lens by u–v method: plot 1/v against 1/u, intercepts give 1/f; or use uv/(u − v).

Resonance tube: v = 2f(l₂ − l₁) for first and second resonance; end correction e = (l₂ − 3l₁)/2.

Sonometer: f = (1/2L)√(T/m), verify by varying tension or length at constant frequency.

Error rules: for sums add absolute errors, for products add relative errors, and quote result with correct significant figures.