Assam Higher Secondary Physics cheat sheet

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

Electric field E = F/q₀ is a vector; field of a point charge E = kq/r² points radially outward for positive q and inward for negative q.

Electric dipole moment p = q·(2a), directed from −q to +q; torque in uniform field τ = pE sinθ, potential energy U = −pE cosθ.

Axial field of a short dipole E = 2kp/r³; equatorial field E = kp/r³, antiparallel to p.

Gauss's law: ∮E·dS = q_enclosed/ε₀; field inside a uniformly charged spherical shell is zero, outside it behaves as a point charge.

Field of an infinite line charge E = λ/(2πε₀r); of an infinite charged sheet E = σ/(2ε₀), independent of distance.

Potential V = kq/r is a scalar; E = −dV/dr; work done in moving a charge on an equipotential surface is zero.

Capacitance C = Q/V; parallel plate C = ε₀A/d, with dielectric C = Kε₀A/d; series 1/C = Σ1/Cᵢ, parallel C = ΣCᵢ.

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

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

Current density J = I/A = σE; resistivity ρ = m/(ne²τ) and conductivity σ = 1/ρ.

Temperature dependence: for metals ρ rises with T (ρ = ρ₀[1+αΔT]); for semiconductors and electrolytes ρ falls with T.

Series resistance R = ΣRᵢ; parallel 1/R = Σ1/Rᵢ; n equal resistors r in parallel give r/n.

Kirchhoff's rules: junction rule ΣI = 0 (charge conservation); loop rule ΣΔV = 0 (energy conservation).

Wheatstone bridge is balanced when P/Q = R/S; then no current flows through the galvanometer.

EMF and internal resistance: I = ε/(R+r); terminal voltage V = ε − Ir while discharging.

Power P = VI = I²R = V²/R; heat H = I²Rt (Joule's law).

Cell combination: n cells in series give I = nε/(R+nr); m in parallel give I = ε/(R + r/m); maximum power transfer when R = r.

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

Field at the centre of a circular loop B = μ₀I/2R; on the axis B = μ₀IR²/2(R²+x²)^{3/2}.

Long straight wire B = μ₀I/(2πr); inside a solenoid B = μ₀nI; in a toroid B = μ₀NI/(2πr).

Force on a moving charge F = qvB sinθ, perpendicular to both v and B; radius of circular path r = mv/(qB), period independent of speed.

Force on a current-carrying 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 symmetry to find B inside and outside conductors.

Torque on a current loop τ = NIAB sinθ = MB sinθ, with magnetic moment M = NIA; potential energy U = −MB cosθ.

Moving-coil galvanometer: current sensitivity = NAB/k, voltage sensitivity = NAB/(kR); conversion to ammeter uses a shunt, to voltmeter a series resistance.

Faraday's law: emf ε = −dΦ/dt; for N turns ε = −N dΦ/dt; Lenz's law fixes the direction opposing the change in flux.

Motional emf in a rod of length l moving with speed v perpendicular to B: ε = Bvl; induced charge q = ΔΦ/R independent of time taken.

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

Mutual inductance ε₂ = −M dI₁/dt; for two coaxial coils M = μ₀n₁n₂Al; coupling coefficient k ≤ 1.

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: Z = √(R² + (X_L − X_C)²), tanφ = (X_L − X_C)/R; resonance at ω₀ = 1/√(LC), with Z = R minimum and current maximum.

Q factor = (1/R)√(L/C); larger Q means sharper resonance.

RMS values I_rms = I₀/√2, V_rms = V₀/√2; average power P = V_rms I_rms cosφ, and the wattless component is V_rms I_rms sinφ.

Transformer: V_s/V_p = N_s/N_p = I_p/I_s for an ideal transformer; a step-up transformer raises voltage but lowers current.

Displacement current I_d = ε₀ dΦ_E/dt completes Ampere's law and explains the magnetic field between capacitor plates.

EM waves are transverse, with E ⊥ B ⊥ direction of propagation; speed c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s in vacuum, v = c/n in a medium.

Ratio E₀/B₀ = c; intensity I = ½ε₀E₀²c; momentum p = U/c and radiation pressure = I/c (absorbing) or 2I/c (reflecting).

Order of the spectrum by increasing wavelength: gamma, X-ray, UV, visible, IR, microwave, radio.

Mirror formula 1/v + 1/u = 1/f, with f = R/2; magnification m = −v/u; sign convention: distances measured from the pole.

Refraction: n₁ sin i = n₂ sin r; critical angle sin C = 1/n; total internal reflection occurs only when light passes 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; power P = 1/f (in metres), in dioptre.

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

Single-slit diffraction: minima at a sinθ = nλ; resolving power of a telescope = D/1.22λ; Brewster's law tan i_p = n.

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

Photoelectric effect: threshold frequency ν₀ = φ/h; maximum kinetic energy K_max = hν − φ = eV₀, where V₀ is the stopping potential.

Photoelectric current depends on intensity; K_max depends only on frequency, not on intensity.

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

For a particle of kinetic energy K: λ = h/√(2mK); heavier particles have shorter wavelengths at the same energy.

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

Heisenberg uncertainty principle: Δx·Δp ≥ h/4π; it rules out simultaneous exact position and momentum.

Work function values are of order a few eV for metals; alkali metals have the lowest work functions.

Bohr postulates: angular momentum L = nh/2π; electrons occupy stationary orbits without radiating.

Bohr radius r_n = 0.53 n²/Z Å; energy E_n = −13.6 Z²/n² eV; orbital speed v_n ∝ Z/n.

Rydberg formula 1/λ = R(1/n₁² − 1/n₂²), R = 1.097×10⁷ m⁻¹; Lyman series lies in the UV, Balmer in the visible.

Ionisation energy of hydrogen = 13.6 eV; energy of a photon emitted equals the difference of level energies.

Nuclear radius R = R₀A^{1/3}, R₀ ≈ 1.2 fm; nuclear density is roughly constant, about 2.3×10¹⁷ kg/m³.

Mass defect Δm gives 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 T½ = 0.693/λ; activity A = λN, unit becquerel.

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

Fission of heavy nuclei and fusion of light nuclei both release energy because the products are more tightly bound.

In an intrinsic semiconductor n_e = n_h = n_i; conduction rises with temperature because more electron–hole pairs are generated.

Doping: pentavalent donors give n-type (electrons majority), trivalent acceptors give p-type (holes majority); the crystal stays electrically neutral.

In a p-n junction, diffusion creates a depletion layer and a barrier potential (about 0.7 V for Si, 0.3 V for Ge).

Forward bias narrows the depletion layer and lowers resistance; reverse bias widens it and gives only a small leakage current.

A diode's ideal behaviour: conducts with near-zero drop in forward bias, blocks in reverse; used for rectification.

Half-wave rectifier output frequency equals the input frequency; a full-wave rectifier doubles it.

Zener diode works in reverse breakdown to give a stable reference voltage for regulation.

Transistor in CE configuration: current gain β = I_C/I_B, α = I_C/I_E, and β = α/(1−α); I_E = I_B + I_C.

Logic gates: AND gives 1 only for all inputs 1; OR gives 1 for any input 1; NOT inverts; NAND and NOR are universal gates.

SI base units: metre, kilogram, second, ampere, kelvin, mole, candela; all other units derive from these.

Dimensional analysis checks equations and converts units, but cannot determine dimensionless constants or verify equations containing sums.

Significant figures: multiplication and division keep the least number of significant figures; addition and subtraction keep the least decimal places.

Error propagation: for z = a ± b, Δz = Δa + Δb; for z = ab or a/b, relative errors add: Δz/z = Δa/a + Δb/b.

Equations of motion with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as.

Projectile motion: horizontal velocity constant; time of flight T = 2u sinθ/g; range R = u² sin2θ/g, maximum at 45°.

Relative velocity adds as vectors: v_AB = v_A − v_B.

Scalars have magnitude only; vectors add by the triangle or parallelogram law, and a vector's components are independent.

Newton's first law defines inertia; a body stays at rest or in uniform motion unless a net external force acts.

Newton's second law: F_net = ma = dp/dt; it is a vector equation applied along each axis separately.

Newton's third law: action and reaction are equal, opposite, and act on different bodies, so they never cancel on one body.

Friction: limiting value f_max = μ_s N; kinetic friction f_k = μ_k N, usually with μ_k < μ_s.

Banking of roads: tanθ = v²/(rg); with friction, the safe maximum speed increases.

Circular motion needs a centripetal force F = mv²/r directed toward the centre; the speed is tangential.

Impulse J = FΔt = Δp; the impulse–momentum theorem follows directly from Newton's second law.

Apparent weight in a lift: N = m(g+a) when accelerating upward, m(g−a) when accelerating downward, zero in free fall.

Work W = F·s = Fs cosθ; work is zero when force is perpendicular to displacement, as in uniform circular motion.

Work–energy theorem: net work done equals the change in kinetic energy, W_net = ½mv² − ½mu².

Kinetic energy K = ½mv² = p²/2m; momentum p = √(2mK).

Potential energy of a spring U = ½kx²; the force is F = −kx (Hooke's law).

Conservation of mechanical energy holds when only conservative forces act; friction converts mechanical energy into heat.

Power P = W/t = F·v; the unit watt equals one joule per second.

Elastic collision between equal masses: velocities are exchanged; in perfectly inelastic collisions momentum is conserved but kinetic energy is not.

Gravitational potential energy near the Earth U = mgh; the general form is U = −GMm/r.

Torque τ = r × F, magnitude rF sinθ; angular momentum L = Iω and τ = dL/dt.

Moment of inertia: ring about axis MR², disc ½MR², solid sphere ⅖MR², rod about centre ML²/12; parallel axis I = I_cm + Md².

Rotational kinetic energy K = ½Iω²; rolling without slipping combines ½mv² and ½Iω², with v = ωR.

For a body rolling down an incline, acceleration a = g sinθ/(1 + I/mR²), so a solid sphere beats a disc and a ring.

Angular momentum is conserved when net external torque is zero, as in a spinning skater pulling in the arms.

Newton's law of gravitation F = Gm₁m₂/r², with G = 6.67×10⁻¹¹ N·m²/kg².

Orbital speed v = √(GM/r); escape speed v_e = √(2GM/R) = √2 v_orbital ≈ 11.2 km/s for Earth.

Kepler's third law T² ∝ r³; gravitational field inside a uniform shell is zero.

Stress = force/area; strain = fractional change; Young's modulus Y = (F/A)/(ΔL/L), valid in the elastic limit.

Hooke's law region ends at the proportional limit; beyond the yield point the material deforms plastically.

Pressure in a fluid at depth h is P = P₀ + ρgh; Pascal's law underlies hydraulic machines.

Bernoulli's equation P + ½ρv² + ρgh = constant holds for steady, non-viscous, incompressible flow.

Surface tension acts along the surface; excess pressure inside a soap bubble is 4T/r, inside a liquid drop 2T/r.

First law of thermodynamics ΔQ = ΔU + ΔW; for an isothermal process ΔU = 0, for an adiabatic process ΔQ = 0.

Efficiency of a Carnot engine η = 1 − T₂/T₁, with temperatures in kelvin; no engine can exceed it.

Specific heats: C_p − C_v = R for an ideal gas; γ = C_p/C_v is 1.67 for monatomic and 1.4 for diatomic gases.

Newton's law of cooling: rate of cooling ∝ temperature difference from surroundings; Stefan's law P = σAeT⁴.

SHM: acceleration a = −ω²x, with ω = 2π/T; displacement x = A sin(ωt + φ).

Velocity in SHM v = ω√(A² − x²), maximum ωA at the mean position; acceleration is maximum ω²A at the extremes.

Spring–mass T = 2π√(m/k); simple pendulum T = 2π√(L/g), independent of mass and (for small angles) amplitude.

Energy in SHM: total E = ½mω²A²; kinetic and potential energies each average ½E over a cycle.

Wave speed v = fλ; for a stretched string v = √(T/μ), where μ is mass per unit length.

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 nv/4L.

Beat frequency = |f₁ − f₂|.

Doppler effect for sound: apparent frequency rises as source and observer approach, falls as they recede; the formula differs for moving source and moving observer.

Progressive wave: y = A sin(ωt − kx), with k = 2π/λ; energy and momentum travel with the wave.