Kerala Plus Two Physics (DHSE) cheat sheet

Coulomb's law: F = (1/4πε₀)·q₁q₂/r², with 1/4πε₀ = 9×10⁹ N·m²·C⁻²; force acts along the line joining charges and is attractive for unlike signs.

Electric field of a point charge E = q/4πε₀r²; field lines start on positive charge and end on negative charge, never crossing.

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

Field inside a uniformly charged spherical shell is zero; outside it behaves as a point charge at the centre.

Potential V = W/q is a scalar; E = −dV/dr, so field points from high to low potential.

Capacitance C = Q/V; parallel plate C = ε₀A/d, and with dielectric slab C = Kε₀A/d.

Series capacitors: 1/C = Σ1/Cᵢ (same charge); parallel: C = ΣCᵢ (same potential difference).

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

Dipole moment p = q·2a; torque in uniform field τ = pE sinθ, potential energy U = −pE cosθ.

Ohm's law V = IR holds for metallic conductors at constant temperature; I–V graph is a straight line through the origin.

Drift velocity v_d = I/(nAe); current density J = nev_d = σE.

Resistivity ρ = m/(ne²τ); it increases with temperature for metals and decreases for semiconductors.

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

Kirchhoff's junction rule: ΣI = 0 at a node; loop rule: ΣΔV = 0 around any closed loop.

Wheatstone bridge is balanced when P/Q = R/S, giving zero galvanometer deflection.

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

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

Potentiometer compares EMFs by balance length: ε₁/ε₂ = l₁/l₂.

Biot–Savart law: dB = (μ₀/4π)·I dl sinθ/r²; direction from the right-hand rule.

Straight wire: B = μ₀I/2πr; centre of a circular loop: B = μ₀I/2R; solenoid: B = μ₀nI.

Force on a moving charge F = qvB sinθ; on a current element F = BIL sinθ, direction by Fleming's left-hand rule.

Radius of circular motion in a magnetic field r = mv/qB; period T = 2πm/qB independent of speed.

Force per unit length between parallel currents: F/L = μ₀I₁I₂/2πd; same direction currents attract.

Torque on a current loop τ = NIAB sinθ; magnetic moment m = NIA.

Ampere's circuital law: ∮B·dl = μ₀I_enclosed; used for solenoid and toroid fields.

Bar magnet: axial field B = μ₀2M/4πr³, equatorial B = μ₀M/4πr³; tangent galvanometer deflection B = B_H tanθ.

Diamagnetic susceptibility is small negative, paramagnetic small positive, ferromagnetic large positive with hysteresis.

Faraday's law: ε = −dΦ/dt for a single turn; for N turns ε = −N dΦ/dt.

Magnetic flux Φ = BA cosθ; Lenz's law fixes the induced current direction opposing the flux change.

Self-inductance: ε = −L dI/dt; solenoid L = μ₀n²Al. Mutual inductance ε₂ = −M dI₁/dt.

Energy stored in an inductor U = ½LI².

AC through resistor: I and V in phase; through inductor V leads by 90°; through capacitor V lags by 90°.

Impedance Z = √(R² + (X_L − X_C)²) with X_L = ωL, X_C = 1/ωC.

Resonance at ω = 1/√(LC); at resonance Z = R and current is maximum.

Power in AC: P = V_rms I_rms cosφ; power factor cosφ = R/Z.

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

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

EM wave speed c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s in vacuum; in a medium v = 1/√(με).

In an EM wave E and B are perpendicular to each other and to the propagation direction; E₀/B₀ = c.

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

Mirror formula 1/v + 1/u = 1/f; lens formula 1/v − 1/u = 1/f; magnification m = v/u (mirror) or v/u with sign convention.

Refraction: n₁ sin i = n₂ sin r; critical angle sin C = 1/n for denser-to-rarer travel.

Lens maker's formula 1/f = (n − 1)(1/R₁ − 1/R₂); power P = 1/f in dioptres.

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

Diffraction grating: d sinθ = nλ; resolving power of a telescope = D/1.22λ.

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

Photoelectric equation: hν = φ₀ + K_max, with threshold frequency ν₀ = φ₀/h.

Stopping potential eV₀ = K_max; K_max depends on frequency, not on intensity.

Photocurrent saturation depends on intensity; emission is instantaneous, contradicting wave theory.

de Broglie wavelength λ = h/p = h/mv = h/√(2mK).

For an electron accelerated through V volts, λ ≈ 12.27/√V Å.

Photon momentum p = h/λ = E/c; radiation pressure follows from this momentum transfer.

Davisson–Germer experiment confirmed electron diffraction, verifying matter waves.

Bohr model: mvr = nh/2π and radius r_n = 0.53 n²/Z Å.

Energy levels E_n = −13.6 Z²/n² eV; ionisation energy of hydrogen is 13.6 eV.

Spectral series of hydrogen: Lyman (UV), Balmer (visible), Paschen, Brackett, Pfund (IR).

Rydberg formula 1/λ = R(1/n₁² − 1/n₂²), R = 1.097×10⁷ m⁻¹.

Nuclear radius R = R₀A^(1/3), R₀ ≈ 1.2 fm; density is independent of mass number.

Binding energy = Δm·c²; 1 u = 931.5 MeV; binding energy per nucleon peaks near A ≈ 56.

Radioactive decay N = N₀e^(−λt); half-life T½ = 0.693/λ; activity A = λN.

Alpha decay reduces A by 4 and Z by 2; beta-minus decay raises Z by 1 with no change in A.

Mass–energy: E = mc²; fission and fusion release energy because binding energy per nucleon increases.

Intrinsic semiconductors: n_e = n_h; conductivity rises with temperature because carrier density increases.

Doping: pentavalent donors give n-type (majority electrons), trivalent acceptors give p-type (majority holes).

p–n junction: diffusion creates a depletion layer and barrier potential (~0.7 V Si, ~0.3 V Ge).

Forward bias lowers the barrier and gives large current; reverse bias widens the depletion layer and gives small saturation current.

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

Zener diode works in reverse breakdown as a voltage regulator.

Transistor in CE configuration: current gain β = I_C/I_B, voltage gain A_v = β·R_L/R_in.

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

Modulation: AM varies amplitude, FM varies frequency; sky wave propagation uses ionospheric reflection for short waves.

SI base units: metre, kilogram, second, ampere, kelvin, mole, candela.

Dimensional analysis checks equations and converts units but cannot find dimensionless constants.

Errors: absolute, relative and percentage; for Z = A^p B^q/C^r, ΔZ/Z = p ΔA/A + q ΔB/B + r ΔC/C.

Equations of motion: v = u + at, s = ut + ½at², v² = u² + 2as (uniform acceleration only).

Projectile: time of flight 2u sinθ/g, range u² sin2θ/g, maximum height u² sin²θ/2g; range is maximum at 45°.

Relative velocity v_AB = v_A − v_B; river-boat problems use vector addition.

Uniform circular motion: a = v²/r = ω²r, directed towards the centre.

Scalars obey ordinary addition; vectors obey the triangle/parallelogram law and can be resolved into components.

Newton's second law F = dp/dt = ma for constant mass; the first law defines inertial frames.

Friction: f_s ≤ μ_sN (static), f_k = μ_kN (kinetic); kinetic friction is less than limiting static friction.

Banking of roads: tanθ = v²/rg; maximum safe speed v = √(μrg) on a level road.

Centripetal force is provided by friction, tension, gravity or normal reaction depending on the situation.

Impulse J = FΔt = change in momentum.

Action and reaction are equal and opposite but act on different bodies, so they never cancel.

Apparent weight in a lift: N = m(g ± a), plus for upward acceleration, minus for downward.

Momentum conservation applies when net external force is zero, e.g. collisions, recoil, explosions.

Work W = F·s cosθ; work is zero when force is perpendicular to displacement.

Work–energy theorem: W_net = ΔK = ½mv² − ½mu².

Potential energy of a spring U = ½kx²; Hooke's law F = −kx.

Conservative forces have path-independent work and a defined potential energy; friction is non-conservative.

Power P = dW/dt = F·v; unit watt, 1 hp ≈ 746 W.

Collisions: momentum always conserved; kinetic energy conserved only in elastic collisions.

Coefficient of restitution e = relative velocity of separation / relative velocity of approach.

For a perfectly inelastic collision of equal masses, final kinetic energy is half the initial value.

Newton's law: F = Gm₁m₂/r², with G = 6.67×10⁻¹¹ N·m²·kg⁻².

Acceleration due to gravity g = GM/R²; it decreases with altitude and depth.

Variation with depth: g_d = g(1 − d/R); with height (small h): g_h ≈ g(1 − 2h/R).

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

Kepler's third law: T² ∝ r³, i.e. T² = 4π²r³/GM.

Gravitational potential V = −GM/r; potential energy U = −GMm/r.

Satellite energy: total E = −GMm/2r, with K = −E and U = 2E.

Weightlessness in orbit arises because the satellite and its contents fall with the same acceleration.

Stress = force/area; strain = fractional change; Young's modulus Y = (F/A)/(ΔL/L).

Elastic potential energy per unit volume = ½ × stress × strain.

Pressure in a fluid at depth h: P = P₀ + ρgh; Archimedes' upthrust equals the weight of displaced fluid.

Equation of continuity A₁v₁ = A₂v₂; Bernoulli's equation P + ½ρv² + ρgh = constant.

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

First law of thermodynamics ΔQ = ΔU + ΔW; work done by a gas W = PΔV at constant pressure.

Isothermal process: PV = constant; adiabatic: PV^γ = constant, with Q = 0.

Carnot efficiency η = 1 − T₂/T₁, always less than 1 for finite temperature difference.

Specific heats: C_p − C_v = R for an ideal gas; γ = C_p/C_v.

SHM: a = −ω²x; displacement x = A sin(ωt + φ), velocity leads displacement by π/2.

Time period of a spring T = 2π√(m/k); of a simple pendulum T = 2π√(l/g).

Energy in SHM: E = ½kA²; kinetic and potential energies interchange, total constant.

Wave equation y = A sin(ωt − kx), with v = ω/k = νλ.

Speed of a transverse wave on a string v = √(T/μ).

Speed of sound in air increases with temperature: v ∝ √T.

Standing waves on a string: f = nv/2L; in a closed pipe f = (2n−1)v/4L; open pipe f = nv/2L.

Beats frequency = |ν₁ − ν₂|; Doppler shift for a receding source ν′ = ν v/(v + v_s).

Resonance occurs when driving frequency equals natural frequency, giving maximum amplitude.

Coulomb's law: F = (1/4πε₀)·q₁q₂/r², with 1/4πε₀ = 9×10⁹ N·m²·C⁻²; force acts along the line joining charges and is attractive for unlike signs.

Electric field of a point charge E = q/4πε₀r²; field lines start on positive charge and end on negative charge, never crossing.

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

Field inside a uniformly charged spherical shell is zero; outside it behaves as a point charge at the centre.

Potential V = W/q is a scalar; E = −dV/dr, so field points from high to low potential.

Capacitance C = Q/V; parallel plate C = ε₀A/d, and with dielectric slab C = Kε₀A/d.

Series capacitors: 1/C = Σ1/Cᵢ (same charge); parallel: C = ΣCᵢ (same potential difference).

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

Dipole moment p = q·2a; torque in uniform field τ = pE sinθ, potential energy U = −pE cosθ.