Karnataka II PUC Physics (KSEAB) cheat sheet

Coulomb's law: F = (1/4πε₀)·q₁q₂/r², with 1/4πε₀ = 9×10⁹ N·m²/C²; force is central, acts along the line joining charges, and obeys Newton's third law.

Electric field E = F/q₀ (vector); for a point charge E = kq/r² directed radially outward for positive q and inward for negative q.

Superposition: net field at a point is the vector sum of fields from each charge; continuous distributions use E = k∫dq/r² with charge densities λ (line), σ (surface), ρ (volume).

Electric dipole moment p = q·2a, directed from −q to +q; axial field E = 2kp/r³, equatorial field E = kp/r³, both ∝ 1/r³.

Dipole in uniform field: torque τ = pE sinθ (vector τ = p × E); potential energy U = −pE cosθ, minimum at θ = 0.

Gauss's law: ∮E·dS = q_enclosed/ε₀; flux through a closed surface depends only on enclosed charge, not on shape or outside charges.

Field of an infinite line charge: E = λ/(2πε₀r); field of an infinite plane sheet: E = σ/(2ε₀), independent of distance; near a conductor surface E = σ/ε₀.

Field inside a uniformly charged spherical shell is zero; outside it behaves as a point charge at the centre. Field inside a uniformly charged solid sphere: E = ρr/(3ε₀), linear in r.

Potential V = W/q₀ (work per unit positive charge, zero reference at infinity); V = kq/r for a point charge; E = −dV/dr, so field points from high to low potential.

Potential energy of a two-charge system U = kq₁q₂/r; for a dipole in a field U = −pE cosθ; equipotential surfaces are always perpendicular to field lines.

Capacitance C = Q/V; parallel-plate air capacitor C = ε₀A/d; with dielectric slab of constant K filling the gap, C = Kε₀A/d.

Series: 1/C = 1/C₁ + 1/C₂ (same charge); parallel: C = C₁ + C₂ (same potential difference).

Energy stored U = ½CV² = ½QV = Q²/2C; energy density in the field u = ½ε₀E² (½Kε₀E² in a dielectric).

Isolated charged capacitor with dielectric inserted: Q constant, C increases by K, V and E fall by factor K, energy decreases.

Capacitor connected to a battery with dielectric inserted: V constant, C and Q increase by K, energy stored increases (battery supplies work).

Conductor properties: field inside is zero, charge resides on the outer surface, surface is equipotential; corona discharge occurs at sharp points where σ is largest.

Van de Graaff generator builds high potential by spraying charge onto an insulated dome; maximum potential limited by breakdown of surrounding air (≈3×10⁶ V/m).

Current I = dQ/dt; conventional current is opposite to electron drift; drift velocity v_d = I/(nAe), typically ~10⁻⁴ m/s, while the field signal travels near light speed.

Ohm's law V = IR for ohmic conductors at constant temperature; resistance R = ρL/A, conductivity σ = 1/ρ; ρ increases with temperature for metals.

Resistivity of a conductor rises with temperature (ρ = ρ₀[1+αΔT]); for semiconductors and electrolytes it falls with rising temperature (α negative).

Series: R = R₁+R₂+… (same current); parallel: 1/R = 1/R₁+1/R₂+… (same voltage); a shunt in parallel extends an ammeter's range, a series multiplier extends a voltmeter's range.

EMF and internal resistance: I = ε/(R+r); terminal voltage V = ε − Ir on discharge, V = ε + Ir on charge; maximum power transfer to R occurs when R = r.

Kirchhoff's rules: junction rule ΣI = 0 (charge conservation); loop rule ΣΔV = 0 (energy conservation), with IR taken negative along the current direction.

Wheatstone bridge is balanced when P/Q = R/S and no current flows through the galvanometer; the metre bridge applies this with unknown R = S·l/(100−l).

Potentiometer: at balance ε₁/ε₂ = l₁/l₂; it measures emf without drawing current, so it gives the true emf, unlike a voltmeter.

Combination errors: for R = R₁+R₂, ΔR = ΔR₁+ΔR₂; for products/quotients, relative errors add.

Magnetic force on a moving charge F = q(v × B), magnitude qvB sinθ; it does no work because it is always perpendicular to v.

Radius of circular motion r = mv/(qB); period T = 2πm/(qB) is independent of speed; pitch of a helix = v cosθ · T.

Force on a current element dF = I(dl × B); straight wire F = BIL sinθ; force per unit length between parallel wires = μ₀I₁I₂/(2πd), attractive for same-direction currents.

Torque on a current loop τ = NIAB sinθ = MB sinθ with magnetic moment M = NIA; the direction is given by the right-hand rule for the current loop.

Biot–Savart law dB = (μ₀/4π)·I dl sinθ/r²; μ₀ = 4π×10⁻⁷ T·m/A. Field at the centre of a circular loop: B = μ₀NI/(2R).

Ampere's circuital law ∮B·dl = μ₀I_enc; field of a long straight wire B = μ₀I/(2πr); inside a solenoid B = μ₀nI; at the end of a solenoid it is half this value.

Moving-coil galvanometer: I = (k/NAB)θ; converting to an ammeter needs a low shunt in parallel, to a voltmeter a high resistance in series.

Magnetic dipole in a uniform field: torque τ = M × B, potential energy U = −M·B; bar magnet axial field B = μ₀2M/(4πr³), equatorial B = μ₀M/(4πr³).

Earth's magnetism: declination, dip, and horizontal component H = B cosδ; relative permeability μ_r = 1 + χ; Curie law χ ∝ 1/T above the Curie temperature.

Hysteresis loop area equals energy loss per cycle; soft iron (narrow loop) suits transformer cores, steel (wide loop) suits permanent magnets.

Faraday's law ε = −N dΦ/dt; Lenz's law fixes the direction: induced current opposes the change in flux (energy conservation).

Motional emf ε = Bvl for a rod moving perpendicular to B; induced charge q = ΔΦ/R is independent of how fast the flux changes.

Self-inductance emf ε = −L dI/dt; energy stored in an inductor U = ½LI²; mutual inductance emf ε₂ = −M dI₁/dt.

AC quantities: peak and rms values I_rms = I₀/√2; mean value over a half cycle = 2I₀/π; form factor = 1.11.

Reactance: X_L = ωL, X_C = 1/ωC; impedance Z = √(R² + (X_L − X_C)²); tanφ = (X_L − X_C)/R.

Series LCR resonance at ω₀ = 1/√(LC); at resonance Z = R (minimum), current is maximum, and the circuit is purely resistive.

Power in AC: P = V_rms I_rms cosφ; the wattless component is V_rms I_rms sinφ; a pure inductor or capacitor has cosφ = 0.

Transformer: V_s/V_p = N_s/N_p = I_p/I_s; ideal transformer conserves power; step-up raises voltage and lowers current; eddy currents are reduced by laminations.

AC generator emf ε = NBAω sin(ωt), maximum when the coil plane is parallel to B; the emf is zero when the coil plane is perpendicular to B.

Maxwell's displacement current I_d = ε₀ dΦ_E/dt completes Ampere's law and predicts electromagnetic waves.

EM waves are transverse: E and B oscillate in phase, mutually perpendicular, and both perpendicular to the propagation direction; E × B gives the direction of travel.

Speed in vacuum c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s; in a medium v = 1/√(με) = c/n; ratio E₀/B₀ = c.

Intensity (average) I = ½ε₀E₀²c; radiation pressure = I/c for full absorption, 2I/c for perfect reflection.

Spectrum order of increasing wavelength: gamma, X-ray, UV, visible (violet→red), infrared, microwave, radio. Visible ranges roughly 400–700 nm.

Uses: microwaves for radar and ovens (water molecule resonance), infrared for remote controls and greenhouse warming, UV for sterilisation, X-rays for imaging.

EM waves carry energy and momentum but no charge; they need no medium and travel at c in vacuum regardless of the observer's frame.

Reflection: angle of incidence = angle of reflection; for a plane mirror the image is virtual, erect, laterally inverted, and as far behind as the object is in front.

Spherical mirror formula 1/v + 1/u = 1/f with f = R/2; magnification m = −v/u; sign convention: distances measured from the pole, along the incident light positive.

Refraction: Snell's law n₁ sin i = n₂ sin r; refractive index n = c/v; critical angle sin C = 1/n for light going from denser to rarer medium.

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

Total internal reflection occurs only when light travels from denser to rarer medium and i > C; it underlies optical fibres, mirage, and brilliance of diamond.

Prism: deviation A + i = e + r₁ + r₂; at minimum deviation n = sin[(A+D_m)/2]/sin(A/2); dispersion arises because n depends on wavelength.

Combination of thin lenses in contact: 1/F = 1/f₁ + 1/f₂; powers add algebraically.

Simple microscope M = 1 + D/f (near point) or D/f (relaxed eye); compound microscope M ≈ (v₀/u₀)(D/f_e); telescope M = f₀/f_e with length f₀ + f_e.

Defects: myopia corrected by concave lens, hypermetropia by convex, presbyopia by bifocal, astigmatism by cylindrical lens.

Huygens' principle: every point on a wavefront is a secondary source; the envelope of secondary wavelets gives the new wavefront; it explains reflection and refraction.

Young's double slit: path difference Δ = d sinθ; bright fringe when Δ = nλ, dark when Δ = (2n−1)λ/2.

Fringe width β = λD/d; it increases with wavelength and screen distance and decreases with slit separation.

Coherent sources are needed for sustained interference; the intensity of two coherent waves is I = I₁ + I₂ + 2√(I₁I₂)cosφ.

Single-slit diffraction: minima at a sinθ = nλ; central maximum width is twice that of the secondary maxima and the intensity falls off rapidly.

Diffraction limit of resolution: the Rayleigh criterion sets the smallest resolvable angular separation ≈ 1.22λ/a for a circular aperture.

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

Malus's law I = I₀ cos²θ for light passing through an analyser after a polariser; unpolarised light through one polariser gives I₀/2.

Doppler effect for light: apparent wavelength shifts with relative motion of source and observer; red shift for a receding source.

Photoelectric effect: emission occurs only above a threshold frequency ν₀; photoelectrons are emitted instantly, and saturation current depends on intensity, not frequency.

Einstein's equation hν = hν₀ + ½mv²_max; stopping potential eV₀ = hν − hν₀; slope of V₀ vs ν is h/e, independent of the metal.

Photon energy E = hν = hc/λ; h = 6.63×10⁻³⁴ J·s; 1 eV = 1.6×10⁻¹⁹ J; photon momentum p = h/λ.

de Broglie wavelength λ = h/p = h/(mv); for an electron accelerated through V volts, λ ≈ 12.27/√V Å.

Photons have zero rest mass, travel at c, and carry energy and momentum; intensity sets photon number, frequency sets photon energy.

Davisson–Germer experiment confirmed the wave nature of electrons by electron diffraction from a nickel crystal.

Heisenberg uncertainty: Δx·Δp ≥ h/4π; it limits simultaneous knowledge of position and momentum, not measurement precision of one alone.

Rutherford scattering: most alpha particles pass undeviated, a few deflect strongly, proving a tiny dense positive nucleus with most space empty.

Bohr model: angular momentum mvr = nh/2π; radius r_n = 0.53 n² Å for hydrogen; energy E_n = −13.6/n² eV.

Spectral lines: 1/λ = R(1/n₁² − 1/n₂²), R = 1.097×10⁷ m⁻¹; Lyman series is in the UV, Balmer in the visible, Paschen and beyond in the infrared.

Ionisation energy of hydrogen is 13.6 eV; the first excitation energy is 10.2 eV; the ionisation potential of a hydrogen atom is 13.6 V.

Nuclear notation ᴬ_ZX: mass number A = protons + neutrons; isotopes have same Z, isobars same A, isotones same neutron number.

Mass defect Δm and binding energy BE = Δm c²; 1 u = 931.5 MeV; BE per nucleon peaks near iron (≈8.8 MeV) and explains fission and fusion energy release.

Radioactive decay N = N₀e^(−λt); half-life T½ = 0.693/λ; activity A = λN, in becquerel; after n half-lives the fraction remaining is (1/2)ⁿ.

α-decay reduces Z by 2 and A by 4; β⁻ decay raises Z by 1 with A unchanged and emits an antineutrino; γ emission changes neither Z nor A.

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

Energy bands: conductors have overlapping bands, insulators a large gap (>3 eV), semiconductors a small gap (Si ≈ 1.1 eV, Ge ≈ 0.7 eV).

n-type doping uses pentavalent donors (P, As) giving electrons as majority carriers; p-type uses trivalent acceptors (B, Al) giving holes.

In a p-n junction, diffusion creates a depletion layer with an electric field opposing further diffusion; the barrier potential is ≈0.7 V for Si, ≈0.3 V for Ge.

Forward bias narrows the depletion layer and lowers the barrier, giving large current above the knee voltage; reverse bias widens it, giving only a small leakage current.

Half-wave rectifier gives one output pulse per input cycle; full-wave (centre-tap or bridge) gives two; a filter capacitor smooths the ripple.

Zener diode works in reverse breakdown as a voltage regulator; the breakdown voltage is set by doping and is nearly constant over the operating current range.

Transistor as amplifier: in CE configuration current gain β = I_C/I_B, voltage gain A_v = −β(R_C/r_in); the phase reverses between input and output.

Logic gates: AND (Y = A·B), OR (Y = A+B), NOT (Y = Ā); NAND and NOR are universal gates from which all others can be built.

Energy of photon emitted by an LED equals the band gap; solar cells use a large-area p-n junction to convert light into emf.

Vernier callipers: least count = 1 main-scale division / number of vernier divisions (commonly 0.1 mm); zero error is added with the opposite sign to the reading.

Screw gauge: least count = pitch / number of circular-scale divisions (commonly 0.01 mm); note and correct zero error before measuring.

Metre bridge: unknown resistance R = S·l/(100−l) where l is the balance length from the end joined to R; interchange R and S to remove end errors.

Ohm's law experiment: plot V against I; the slope gives resistance; the graph should be a straight line through the origin for an ohmic conductor.

Potentiometer compares emfs by ε₁/ε₂ = l₁/l₂ and measures internal resistance r = R(l₁/l₂ − 1).

Focal length of a convex mirror or concave lens is found using a auxiliary convex lens or a plane mirror, since both form virtual images.

Refractive index of a prism: measure angle A and minimum deviation D_m, then n = sin[(A+D_m)/2]/sin(A/2).

Galvanometer figure of merit k = I/θ from the deflection method; converting to an ammeter uses a shunt, to a voltmeter a series multiplier.

In resonance tube or sonometer experiments, end correction and frequency relations (f ∝ √T, f ∝ 1/L) are checked by plotting graphs, not by single readings.