AP Physics 2 cheat sheet

First law: ΔU = Q − W, where W is work done by the gas; sign of W flips if the convention W-on-gas is used.

Work done by a gas at constant pressure: W = PΔV; on a PV diagram, work equals the area under the process curve.

Isothermal process: ΔU = 0 for an ideal gas, so Q = W = nRT ln(V_f/V_i).

Adiabatic process: Q = 0, so ΔU = −W; for an ideal gas PV^γ = constant with γ = C_P/C_V.

Ideal gas law: PV = nRT; average translational kinetic energy per molecule = (3/2)k_BT.

Internal energy of a monatomic ideal gas: U = (3/2)nRT; for diatomic near room temperature U = (5/2)nRT.

Second law: entropy of an isolated system never decreases; ΔS = Q/T only for a reversible process at constant T.

Carnot efficiency: e = 1 − T_C/T_H with temperatures in kelvin; no real engine between the same reservoirs exceeds it.

Heat conduction rate: P = kAΔT/L; thermal resistance adds in series for layered slabs.

Coulomb's law: F = kq₁q₂/r², with k ≈ 9.0 × 10⁹ N·m²/C²; force is repulsive for like signs.

Electric field of a point charge: E = kq/r², directed away from positive charge and toward negative charge.

Force on a charge in a field: F = qE; field lines point from positive to negative and never cross.

Potential from a point charge: V = kq/r; potential is a scalar, so contributions add algebraically.

Potential difference relates to field: ΔV = −∫E·dr; for a uniform field, ΔV = −Ed.

Electric potential energy of two charges: U = kq₁q₂/r; a positive charge accelerates toward lower potential.

Field inside a conductor in electrostatic equilibrium is zero; excess charge resides on the outer surface.

Gauss's law: Φ = Q_enclosed/ε₀; symmetric charge distributions (sphere, line, sheet) allow direct field calculation.

Capacitance: C = Q/ΔV; parallel plate C = ε₀A/d, and stored energy U = (1/2)CΔV².

Ohm's law: V = IR; resistance of a wire R = ρL/A, so R rises with length and falls with cross-sectional area.

Resistors in series add directly; in parallel the reciprocals add, so equivalent resistance is less than the smallest branch.

Kirchhoff's junction rule: sum of currents into a node equals sum out; loop rule: net potential change around a closed loop is zero.

Power dissipated by a resistor: P = IV = I²R = V²/R.

Capacitors in parallel add directly; in series the reciprocals add, opposite to resistors.

RC charging: q(t) = Cε(1 − e^(−t/RC)); the time constant τ = RC gives 63% of final charge in one τ.

An ideal ammeter has near-zero resistance and is placed in series; an ideal voltmeter has near-infinite resistance and is in parallel.

Terminal voltage of a real battery: V = ε − Ir, where r is internal resistance.

Magnetic force on a moving charge: F = qv × B, magnitude qvB sinθ; force is zero when v is parallel to B.

Force on a current-carrying wire: F = IL × B; direction from the right-hand rule.

A charge moving perpendicular to a uniform B follows a circle of radius r = mv/(qB); period is independent of speed.

Long straight wire field: B = μ₀I/(2πr); inside a solenoid B = μ₀nI.

Faraday's law: induced emf = −N dΦ_B/dt; flux Φ_B = BA cosθ.

Lenz's law: induced current opposes the change in flux that produced it.

Motional emf for a rod of length L moving at speed v perpendicular to B: emf = BLv.

A changing electric field or capacitor charging current produces a magnetic field (Maxwell's addition to Ampère's law).

Transformers: V_s/V_p = N_s/N_p; ideal power is conserved, so I_s/I_p = N_p/N_s.

Law of reflection: angle of incidence equals angle of reflection, both measured from the normal.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; light bends toward the normal entering a higher-index medium.

Index of refraction: n = c/v; frequency is unchanged on refraction, wavelength becomes λ/n.

Total internal reflection occurs only going from higher to lower n, when θ exceeds sinθ_c = n₂/n₁.

Thin lens and mirror equation: 1/f = 1/d_o + 1/d_i; magnification m = −d_i/d_o = h_i/h_o.

Converging lens or concave mirror: f > 0; diverging lens or convex mirror: f < 0.

Real images form where rays actually converge, can be projected, and have positive d_i for a single lens.

Plane mirror image is virtual, upright, the same size, and as far behind the mirror as the object is in front.

Wave speed: v = fλ; for a stretched string v = √(F_T/μ), and for sound in air v ≈ 343 m/s at room temperature.

Double-slit constructive condition: d sinθ = mλ; destructive: d sinθ = (m + 1/2)λ.

Thin-film interference depends on both path difference and phase shifts on reflection; a phase shift of π occurs reflecting off a higher-n medium.

Single-slit minima: a sinθ = mλ for m ≠ 0; central maximum is twice as wide as the side maxima.

Diffraction grating maxima: d sinθ = mλ, with sharper peaks as the number of slits increases.

Standing waves on a string fixed at both ends: λ_n = 2L/n; open pipe: λ_n = 2L/n; pipe closed at one end: λ_n = 4L/n with odd n.

Doppler shift: f' = f(v ± v_o)/(v ∓ v_s), with signs chosen so approach raises the observed frequency.

Sound intensity level: β = 10 log(I/I₀) dB, with I₀ = 1 × 10⁻¹² W/m².

Beat frequency equals the absolute difference of the two source frequencies.

Photon energy: E = hf = hc/λ, with hc ≈ 1240 eV·nm; photon momentum p = h/λ.

Photoelectric effect: K_max = hf − φ; no emission occurs below the threshold frequency regardless of intensity.

de Broglie wavelength: λ = h/p = h/(mv); larger for slower, lighter particles.

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon energy equals the level difference ΔE.

Mass-energy equivalence: E = mc²; mass defect in a nucleus appears as binding energy.

Radioactive decay: N = N₀e^(−λt), half-life t₁/₂ = ln2/λ ≈ 0.693/λ.

Nuclear reactions conserve charge and nucleon number; alpha decay reduces mass number by 4 and atomic number by 2.

Uncertainty principle: ΔxΔp ≥ ħ/2, limiting simultaneous precision of position and momentum.

Half-life and activity: activity A = λN, and A falls by half each half-life.

First law: ΔU = Q − W, where W is work done by the gas; sign of W flips if the convention W-on-gas is used.

Work done by a gas at constant pressure: W = PΔV; on a PV diagram, work equals the area under the process curve.

Isothermal process: ΔU = 0 for an ideal gas, so Q = W = nRT ln(V_f/V_i).

Adiabatic process: Q = 0, so ΔU = −W; for an ideal gas PV^γ = constant with γ = C_P/C_V.

Ideal gas law: PV = nRT; average translational kinetic energy per molecule = (3/2)k_BT.

Internal energy of a monatomic ideal gas: U = (3/2)nRT; for diatomic near room temperature U = (5/2)nRT.

Second law: entropy of an isolated system never decreases; ΔS = Q/T only for a reversible process at constant T.

Carnot efficiency: e = 1 − T_C/T_H with temperatures in kelvin; no real engine between the same reservoirs exceeds it.

Heat conduction rate: P = kAΔT/L; thermal resistance adds in series for layered slabs.

Coulomb's law: F = kq₁q₂/r², with k ≈ 9.0 × 10⁹ N·m²/C²; force is repulsive for like signs.

Electric field of a point charge: E = kq/r², directed away from positive charge and toward negative charge.

Force on a charge in a field: F = qE; field lines point from positive to negative and never cross.

Potential from a point charge: V = kq/r; potential is a scalar, so contributions add algebraically.

Potential difference relates to field: ΔV = −∫E·dr; for a uniform field, ΔV = −Ed.

Electric potential energy of two charges: U = kq₁q₂/r; a positive charge accelerates toward lower potential.

Field inside a conductor in electrostatic equilibrium is zero; excess charge resides on the outer surface.

Gauss's law: Φ = Q_enclosed/ε₀; symmetric charge distributions (sphere, line, sheet) allow direct field calculation.

Capacitance: C = Q/ΔV; parallel plate C = ε₀A/d, and stored energy U = (1/2)CΔV².

Ohm's law: V = IR; resistance of a wire R = ρL/A, so R rises with length and falls with cross-sectional area.

Resistors in series add directly; in parallel the reciprocals add, so equivalent resistance is less than the smallest branch.

Kirchhoff's junction rule: sum of currents into a node equals sum out; loop rule: net potential change around a closed loop is zero.

Power dissipated by a resistor: P = IV = I²R = V²/R.

Capacitors in parallel add directly; in series the reciprocals add, opposite to resistors.

RC charging: q(t) = Cε(1 − e^(−t/RC)); the time constant τ = RC gives 63% of final charge in one τ.

An ideal ammeter has near-zero resistance and is placed in series; an ideal voltmeter has near-infinite resistance and is in parallel.

Terminal voltage of a real battery: V = ε − Ir, where r is internal resistance.

Magnetic force on a moving charge: F = qv × B, magnitude qvB sinθ; force is zero when v is parallel to B.

Force on a current-carrying wire: F = IL × B; direction from the right-hand rule.

A charge moving perpendicular to a uniform B follows a circle of radius r = mv/(qB); period is independent of speed.

Long straight wire field: B = μ₀I/(2πr); inside a solenoid B = μ₀nI.

Faraday's law: induced emf = −N dΦ_B/dt; flux Φ_B = BA cosθ.

Lenz's law: induced current opposes the change in flux that produced it.

Motional emf for a rod of length L moving at speed v perpendicular to B: emf = BLv.

A changing electric field or capacitor charging current produces a magnetic field (Maxwell's addition to Ampère's law).

Transformers: V_s/V_p = N_s/N_p; ideal power is conserved, so I_s/I_p = N_p/N_s.

Law of reflection: angle of incidence equals angle of reflection, both measured from the normal.

Snell's law: n₁ sinθ₁ = n₂ sinθ₂; light bends toward the normal entering a higher-index medium.

Index of refraction: n = c/v; frequency is unchanged on refraction, wavelength becomes λ/n.

Total internal reflection occurs only going from higher to lower n, when θ exceeds sinθ_c = n₂/n₁.

Thin lens and mirror equation: 1/f = 1/d_o + 1/d_i; magnification m = −d_i/d_o = h_i/h_o.

Converging lens or concave mirror: f > 0; diverging lens or convex mirror: f < 0.

Real images form where rays actually converge, can be projected, and have positive d_i for a single lens.

Plane mirror image is virtual, upright, the same size, and as far behind the mirror as the object is in front.

Wave speed: v = fλ; for a stretched string v = √(F_T/μ), and for sound in air v ≈ 343 m/s at room temperature.

Double-slit constructive condition: d sinθ = mλ; destructive: d sinθ = (m + 1/2)λ.

Thin-film interference depends on both path difference and phase shifts on reflection; a phase shift of π occurs reflecting off a higher-n medium.

Single-slit minima: a sinθ = mλ for m ≠ 0; central maximum is twice as wide as the side maxima.

Diffraction grating maxima: d sinθ = mλ, with sharper peaks as the number of slits increases.

Standing waves on a string fixed at both ends: λ_n = 2L/n; open pipe: λ_n = 2L/n; pipe closed at one end: λ_n = 4L/n with odd n.

Doppler shift: f' = f(v ± v_o)/(v ∓ v_s), with signs chosen so approach raises the observed frequency.

Sound intensity level: β = 10 log(I/I₀) dB, with I₀ = 1 × 10⁻¹² W/m².

Beat frequency equals the absolute difference of the two source frequencies.

Photon energy: E = hf = hc/λ, with hc ≈ 1240 eV·nm; photon momentum p = h/λ.

Photoelectric effect: K_max = hf − φ; no emission occurs below the threshold frequency regardless of intensity.

de Broglie wavelength: λ = h/p = h/(mv); larger for slower, lighter particles.

Bohr model: E_n = −13.6 eV/n² for hydrogen; photon energy equals the level difference ΔE.

Mass-energy equivalence: E = mc²; mass defect in a nucleus appears as binding energy.

Radioactive decay: N = N₀e^(−λt), half-life t₁/₂ = ln2/λ ≈ 0.693/λ.

Nuclear reactions conserve charge and nucleon number; alpha decay reduces mass number by 4 and atomic number by 2.

Uncertainty principle: ΔxΔp ≥ ħ/2, limiting simultaneous precision of position and momentum.

Half-life and activity: activity A = λN, and A falls by half each half-life.