AP Inter Physics (BIEAP) cheat sheet

SI base units: metre (length), kilogram (mass), second (time), ampere (current), kelvin (temperature), mole (amount), candela (luminous intensity).

Dimensional formula of force is [MLT^-2]; of energy and work [ML^2T^-2]; of pressure and stress [ML^-1T^-2]; of power [ML^2T^-3].

Dimensional analysis checks equations and converts units but cannot determine dimensionless constants or verify equations containing trigonometric, exponential or logarithmic terms.

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

Error propagation: for Z = A + B or A - B, absolute errors add; for Z = AB or A/B, relative (fractional) errors add.

If Z = A^n B^m / C^p, the maximum relative error is |n|(ΔA/A) + |m|(ΔB/B) + |p|(ΔC/C).

Screw gauge pitch = distance moved on main scale per full rotation; least count = pitch / number of circular scale divisions.

Vernier least count = 1 main scale division - 1 vernier scale division; typical value 0.1 mm for 10 vernier divisions.

Order-of-magnitude estimates and the principle of homogeneity of dimensions require every term in a valid equation to have identical dimensions.

Equations of motion for constant acceleration: v = u + at; s = ut + ½at²; v² = u² + 2as; s = (u+v)t/2.

Displacement is a vector with sign; distance is a scalar path length, so distance ≥ |displacement| always.

Relative velocity: v_AB = v_A - v_B; for river-boat problems, the boat must head upstream to cross straight across.

Projectile (no air resistance): time of flight T = 2u sinθ/g; range R = u² sin2θ/g; max height H = u² sin²θ/(2g); maximum range at θ = 45°.

Uniform circular motion: speed constant, velocity changes direction; centripetal acceleration a = v²/r = ω²r directed toward the centre.

Angular relations: v = ωr, θ = ωt, ω = 2π/T = 2πf, and tangential acceleration a_t = rα.

Graph reading: slope of x-t graph = velocity; slope of v-t graph = acceleration; area under v-t graph = displacement.

For motion under gravity alone, take g ≈ 9.8 m/s² downward and treat upward as positive with a = -g.

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; in component form ΣFx = max and ΣFy = may.

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

Linear momentum p = mv; impulse J = FΔt = Δp, so impulse equals the change in momentum.

Friction: static friction satisfies fs ≤ μsN; kinetic friction fk = μkN and is independent of contact area.

On an incline of angle θ, the component of weight along the plane is mg sinθ and the normal force is mg cosθ.

Banking of roads: tanθ = v²/(rg); for a banked curve with friction, the safe speed range widens with μ.

In a lift: apparent weight N = m(g + a) when accelerating upward and N = m(g - a) when accelerating downward; N = 0 in free fall.

Momentum conservation applies when net external force is zero, e.g. collisions, recoil of a gun, and explosion problems.

Work W = F·s cosθ; work is zero when force is perpendicular to displacement (e.g. centripetal force) and negative when θ > 90°.

Work-energy theorem: net work done on a body equals its change in kinetic energy, W_net = ½mv² - ½mu².

Kinetic energy KE = ½mv² = p²/(2m); doubling speed quadruples KE at constant mass.

Gravitational potential energy near Earth: U = mgh; elastic PE of a spring: U = ½kx².

Power P = W/t = F·v; SI unit watt; 1 horsepower ≈ 746 W.

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

In a perfectly elastic collision, both momentum and kinetic energy are conserved; in a perfectly inelastic collision, bodies stick together and KE loss is maximum.

Coefficient of restitution e = relative velocity of separation / relative velocity of approach; e = 1 elastic, e = 0 perfectly inelastic.

Efficiency = useful output energy / total input energy, expressed as a fraction or percentage.

SHM condition: acceleration a = -ω²x, directed opposite to displacement and proportional to it.

SHM displacement x = A sin(ωt + φ); velocity v = ω√(A² - x²); maximum speed = ωA at the mean position.

Period of a simple pendulum T = 2π√(L/g), independent of mass and (for small angles) of amplitude.

Period of a spring-mass system T = 2π√(m/k); the same for a body attached to a vertical spring.

Total energy in SHM E = ½mω²A² = ½kA², constant; KE and PE interchange and are equal at x = A/√2.

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

Speed of sound in air increases with temperature and is independent of pressure at constant temperature; v ≈ 331 + 0.6T m/s.

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 = nv/(4L).

Beat frequency = |f1 - f2|; Doppler shift for a source moving toward a stationary observer f' = f v/(v - v_s).

Laws of reflection: angle of incidence equals angle of reflection, and the incident ray, reflected ray and normal are coplanar.

Snell's law: n1 sin i = n2 sin r; refractive index n = c/v = real depth / apparent depth.

Critical angle for total internal reflection: sin C = 1/n (from denser to rarer medium); TIR requires i > C.

Mirror formula 1/f = 1/v + 1/u and lens formula 1/f = 1/v - 1/u, with sign convention: distances measured from the pole/optical centre, positive in the direction of incident light.

Lens maker's formula: 1/f = (n - 1)(1/R1 - 1/R2); power P = 1/f (in metres), unit dioptre.

Magnification m = v/u for mirrors and m = v/u for lenses (with sign convention), also m = h'/h.

Prism: deviation δ = i1 + i2 - A; at minimum deviation n = sin[(A + δm)/2] / sin(A/2).

Simple microscope M = 1 + D/f; compound microscope M ≈ (L/f_o)(D/f_e); telescope M = f_o/f_e with length f_o + f_e.

Dispersion occurs because refractive index depends on wavelength: violet bends most, red least; angular dispersion = (n_v - n_r)A.

Coulomb's law: F = (1/4πε0) q1q2/r², with 1/4πε0 ≈ 9 × 10^9 N·m²/C²; force acts along the line joining charges.

Electric field E = F/q0; for a point charge E = kq/r², directed radially outward for positive charge.

Electric flux Φ = E·A cosθ; Gauss's law: Φ = q_enclosed/ε0.

Field of an infinite line charge E = λ/(2πε0r); of an infinite sheet E = σ/(2ε0); inside a conductor E = 0.

Electric potential V = kq/r; potential energy U = kq1q2/r; work done in moving a charge W = q(V_B - V_A).

Relation E = -dV/dr; field points from high to low potential; equipotential surfaces are perpendicular to field lines.

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

Capacitors in series 1/C = 1/C1 + 1/C2; in parallel C = C1 + C2; energy stored U = ½CV² = Q²/(2C).

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; resistance R = ρL/A.

Resistivity ρ depends on material and temperature, not on the dimensions of the conductor; for metals ρ increases with temperature.

Series: R = R1 + R2 + ... and current is the same; parallel: 1/R = 1/R1 + 1/R2 + ... and voltage is the same.

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

Kirchhoff's laws: junction rule (ΣI = 0, charge conservation) and loop rule (ΣV = 0, energy conservation).

Wheatstone bridge is balanced when P/Q = R/S, giving zero galvanometer current; the metre bridge applies the same condition.

EMF and internal resistance: terminal voltage V = ε - Ir; maximum power transfer to load when R = r.

Heating effect: power P = I²R = V²/R = VI; electrical energy consumed in kWh where 1 kWh = 3.6 × 10^6 J.

Potentiometer compares EMFs by balance length: ε1/ε2 = l1/l2, and measures EMF without drawing current at balance.

Force on a moving charge F = qvB sinθ; direction given by Fleming's left-hand rule; a charge moving parallel to B feels no force.

Force on a current-carrying conductor F = BIL sinθ; torque on a current loop τ = NIAB sinθ.

Biot-Savart law dB = (μ0/4π)(I dl sinθ)/r²; field at the centre of a circular loop B = μ0I/(2R).

Field of a long straight wire B = μ0I/(2πr); inside a long solenoid B = μ0nI.

Ampere's circuital law: ∮B·dl = μ0I_enclosed.

Faraday's law: induced EMF ε = -dΦ/dt; Lenz's law states the induced current opposes the change in flux producing it.

Motional EMF for a rod moving in a field: ε = Bvl; induced current I = Bvl/R.

Self-inductance EMF ε = -L dI/dt; energy stored in an inductor U = ½LI².

Mutual inductance M relates flux linkage in one coil to current in the other: Φ2 = MI1; transformers use mutual induction with V_s/V_p = N_s/N_p.

Photon energy E = hf = hc/λ; h = 6.63 × 10^-34 J·s; 1 eV = 1.6 × 10^-19 J.

Photoelectric effect: emission occurs only if incident frequency exceeds the threshold frequency f0, regardless of intensity.

Einstein's photoelectric equation: hf = φ0 + K_max, where φ0 = hf0 is the work function.

Stopping potential V0 satisfies eV0 = K_max; maximum photocurrent depends on intensity, not frequency.

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

de Broglie wavelength λ = h/p = h/(mv) = h/√(2mK) for a particle of kinetic energy K.

de Broglie wavelength of an electron accelerated through V volts is approximately λ = 12.27/√V Å.

Davisson-Germer experiment confirmed electron diffraction, verifying the wave nature of matter.

Photon flux and intensity: intensity = number of photons per unit area per unit time × hf.

Bohr model postulates: angular momentum quantised as mvr = nh/2π; electrons orbit without radiating in stationary states.

Bohr radius r_n = 0.53 n² Å; energy levels E_n = -13.6/n² eV for hydrogen.

Spectral series of hydrogen: Lyman (ultraviolet), Balmer (visible), Paschen, Brackett and Pfund (infrared).

Energy of emitted photon equals the difference of levels: hf = E2 - E1; Rydberg formula 1/λ = R(1/n1² - 1/n2²).

Nuclear notation: A = Z + N; isotopes have same Z, isobars have same A, isotones have same N.

Nuclear radius R = R0 A^(1/3), with R0 ≈ 1.2 fm; density is nearly independent of A.

Mass defect Δm and binding energy BE = Δm c²; 1 u ≈ 931.5 MeV.

Radioactive decay law N = N0 e^(-λt); half-life T½ = 0.693/λ; activity A = λN.

Nuclear reactions conserve charge, mass number, energy and momentum; in fission and fusion, BE per nucleon increases and energy is released.

Intrinsic semiconductors: number of electrons equals number of holes; conductivity increases with temperature, giving a negative temperature coefficient.

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

p-n junction: diffusion creates a depletion layer and barrier potential (≈0.7 V for Si, ≈0.3 V for Ge); forward bias narrows the layer.

Diode I-V characteristic: negligible current below the knee voltage; reverse saturation current is small and nearly constant until breakdown.

Half-wave rectifier output frequency equals input frequency; full-wave rectifier output ripple frequency is twice the input frequency.

Zener diode operates in reverse breakdown to provide voltage regulation; it needs a series resistor to limit current.

Transistor in CE configuration has current gain β = Ic/Ib and voltage gain; α = Ic/Ie and β = α/(1 - α).

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

Communication: modulation superimposes a signal on a carrier; bandwidth of AM = 2f_m; skywave propagation uses ionospheric reflection for short-wave radio.