Must-know sheet

Physics 2 must-know sheet

The real AP Physics 2 exam gives you an equation sheet with constants, exam conventions and the main equations (including the Physics 1 mechanics and fluids equations), and you can use a calculator on the whole exam. Start with the recent changes at the top. The rest covers what the equation sheet doesn't tell you: when each relationship applies, sign conventions and direction rules, the cases and shortcuts worth knowing cold, graph shapes and how to analyze lab data.

What's new: recent changes to the course and exam

Units 9, 10, 11, 12, 13, 14, 15

May 2027: 42 multiple-choice questions in 85 minutes: That's up from 40 questions in 80 minutes. The free-response section is now 95 minutes instead of 100, still for 4 questions. Each section is still half your score, with a calculator and the equation sheet on both.
Fall 2026: gamma decay counts as radioactive decay: College Board clarified that radioactive decay includes a nucleus dropping to a lower energy level without becoming a different nucleus. So a gamma-ray emission is a decay too, alongside alpha and beta decay (topic 15.7).
Fall 2026: the equation sheet's conventions were updated: The 2026 reference information updated some of the conventions, the assumptions the exam makes unless a question says otherwise. The current list is in "What the exam assumes" in the next section; check old practice tests against it. The symbols on this sheet match the 2026 equation sheet, like $k_B$ for Boltzmann's constant, N₀ for Avogadro's number and κ for the dielectric constant.
Since 2024–25: units are numbered 9 to 15: AP Physics 2 picks up where AP Physics 1 (Units 1–8) ends, so its first unit is Unit 9. Older books and videos number the same ideas differently.
Since 2024–25: fluids moved to AP Physics 1: Pressure, buoyancy and fluid flow are now Unit 8 of AP Physics 1, so Physics 2 starts with thermodynamics. Fluids equations still appear on the shared equation sheet.
Since 2024–25: mechanical waves and sound are in Physics 2: Wave pulses, periodic waves, sound, the Doppler effect, beats and standing waves moved in from Physics 1. They're in topics 14.1–14.6, mixed in with electromagnetic waves (14.4) and polarization (14.3), which were already in Physics 2. Physics 2 reviews made before 2024 skip the mechanical-wave and sound material, so study them here or from a Physics 1 wave lesson.
Since 2024–25: optics is split in two: Unit 13 covers rays: reflection, refraction, mirrors and lenses. Interference, diffraction and thin films, the wave side of light, are in Unit 14.
Since 2024–25: four fixed free-response types: In order: mathematical routines (10 points), translation between representations (12), experimental design and analysis (10) and qualitative/quantitative translation (8). Free-response questions from Physics 2 exams before May 2025 are still good practice, but their format and point totals differ.

Units, constants and exam conventions

Units 9, 10, 11, 12, 13, 14, 15

SI units before you plug in: Convert to meters, kilograms, seconds, kelvins and coulombs first. Watch the small ones: 1 cm² = 10⁻⁴ m², 1 L = 10⁻³ m³, 1 nm = 10⁻⁹ m, and 1 μF = 10⁻⁶ F.
Prefixes you'll see most: p, n, μ, m, k, M: pico = 10⁻¹², nano = 10⁻⁹, micro = 10⁻⁶, milli = 10⁻³, kilo = 10³, mega = 10⁶. Capacitors are often in μF or pF, wavelengths of light in nm, and nuclear energies in MeV.
Kelvins for anything thermal: T(K) = T(°C) + 273. Use kelvins in the ideal gas law, in $K_{\text{avg}} = \frac{3}{2}k_BT$ and for blackbodies. A temperature change ΔT is the same number in K and °C, so Q = mcΔT works with either.
Electric units: Field: N/C, which equals V/m. Potential: V = J/C. Current: A = C/s. Resistance: Ω = V/A. Resistivity: Ω·m. Capacitance: F = C/V. Power: W = J/s = V·A. The time constant RC comes out in seconds.
Magnetic and thermal units: Magnetic field: tesla, T = N/(A·m). Magnetic flux: weber, Wb = T·m². Pressure: Pa = N/m², and 1 atm ≈ 1.0 × 10⁵ Pa. Thermal conductivity: W/(m·K). Specific heat: J/(kg·K).
The electron volt: 1 eV = 1.60 × 10⁻¹⁹ J: It's the energy one elementary charge gains crossing 1 V. Divide joules by 1.60 × 10⁻¹⁹ to get eV. Atomic energies are a few eV and nuclear energies are millions of eV (MeV).
Shortcuts: hc = 1240 eV·nm and h = 4.14 × 10⁻¹⁵ eV·s: Use them to skip converting joules: a photon's energy in eV is $E = \frac{1240}{\lambda\text{ (in nm)}}$ , so 620 nm red light carries about 2.0 eV. Both come from h = 6.63 × 10⁻³⁴ J·s, c = 3.00 × 10⁸ m/s and 1 eV = 1.60 × 10⁻¹⁹ J.
Moles vs. atoms: N = nN₀, with Avogadro's number N₀ = 6.02 × 10²³ per mole. Use R = 8.31 J/(mol·K) with moles (n) and $k_B$ = 1.38 × 10⁻²³ J/K with numbers of atoms (N); $R = N_0 k_B$ .
Mass–energy: 1 u ≈ 931 MeV/c²: An atomic mass unit is 1.66 × 10⁻²⁷ kg, and its rest energy is about 931 MeV. So a mass change in u, times 931, gives the energy released in MeV.
What the exam assumes unless it says otherwise: The reference frame is inertial and friction is negligible; strings, springs, batteries, wires and meters are ideal; resistors and bulbs are ohmic; ideal gases are monatomic; potential is zero infinitely far from an isolated point charge; current is conventional current; capacitors are air-filled (κ = 1.0); and the small-angle approximation works for single and double slits.

Kinetic theory and the ideal gas law

Unit 9

Pressure comes from collisions: $P = \frac{F_{\perp}}{A}$ : the total perpendicular force from atoms hitting a surface, divided by its area. An atom bouncing straight off a wall elastically changes momentum by 2mv, so faster atoms or more frequent hits mean more pressure. Pressure exists throughout the gas, not just at the walls.
Temperature measures average kinetic energy: $K_{\text{avg}} = \frac{3}{2}k_BT$ for each atom, with T in kelvins. Doubling the kelvin temperature doubles the average kinetic energy. At the same temperature, every ideal gas has the same $K_{\text{avg}}$ , whatever its atoms' mass.
Root-mean-square speed: $v_{\text{rms}} = \sqrt{\dfrac{3k_BT}{m}}$ , where m is one atom's mass. Speed grows with $\sqrt{T}$ : quadruple T to double $v_{\text{rms}}$ . At the same temperature, lighter atoms move faster.
Maxwell–Boltzmann distribution (graph only): A graph of how many atoms have each speed: a hump with a long tail toward high speeds. When the gas gets hotter, the peak moves to a higher speed and the curve gets lower and wider, while the area under it (the number of atoms) stays the same. You need its shape, not its formula.
The ideal gas model: Atoms are tiny compared with the space they fill, move randomly, collide elastically and exert no forces on each other except during collisions. Real gases act most like this at low pressure and high temperature.
Ideal gas law: PV = nRT = Nk_BT: Use pascals, m³ and kelvins. For a fixed amount of gas, $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ , so you can predict changes without knowing n.
Proportional reasoning with PV = nRT: Constant T: P ∝ 1/V (halve the volume, double the pressure). Constant P: V ∝ T. Constant V: P ∝ T. Always use kelvins: going from 20 °C to 40 °C is only about a 7% rise in T, not double.
Absolute zero from a graph: Plot pressure against temperature in °C for a gas at constant volume and extend the straight line down to P = 0. It crosses at about −273 °C, which is 0 K.
Internal energy of an ideal monatomic gas: $U = \frac{3}{2}nRT = \frac{3}{2}Nk_BT = \frac{3}{2}PV$ . The atoms don't interact, so there's no potential energy and U depends only on temperature. Same temperature means same U, so ΔU = 0 for any process that ends where it started in T.

Heating, the first law, PV diagrams and entropy

Unit 9

Heating flows from hot to cold on its own: Energy moves by conduction, convection or radiation from the higher-temperature system to the lower one. At thermal equilibrium the temperatures match and there's no net energy transfer.
First law: ΔU = Q + W: Q is energy added by heating (positive in, negative out). W is work done ON the gas (positive when the gas is compressed, negative when it expands). Some books write ΔU = Q − W with W done BY the gas; it's the same physics with the sign flipped.
Work on a gas: W = −PΔV: At constant (or average) pressure. Expansion (ΔV > 0) makes W negative because the gas pushes outward and does work. The size of the work in any process is the area under the curve on a PV diagram.
Isobaric (constant pressure): A horizontal line on a PV diagram. W = −PΔV, and an expansion at constant pressure needs heating because both T and U go up.
Isovolumetric (constant volume): A vertical line on a PV diagram. W = 0, so ΔU = Q: all the energy from heating goes into internal energy, and P rises with T.
Isothermal (constant temperature): Follows an isotherm, a curve where PV is constant. ΔU = 0, so Q = −W: a gas expanding isothermally absorbs exactly as much energy by heating as it does in work. Isotherms farther from the origin are hotter.
Adiabatic (no heating, Q = 0): ΔU = W. Compressing the gas fast or in an insulated container raises its temperature; letting it expand cools it. On a PV diagram an adiabat is steeper than the isotherm through the same point.
Comparing temperatures on a PV diagram: T is proportional to PV, so the state with the larger product P × V is hotter. Moving up and to the right means higher T and higher U.
Cycles on a PV diagram: Over a full cycle the gas returns to its starting state, so ΔU = 0 and Q_net = −W_net. The enclosed area is the net work. Going clockwise, the gas does net work on its surroundings (a heat engine), so W on the gas is negative and net heating is positive.
Specific heat: Q = mcΔT: c is the energy to warm 1 kg by 1 K, in J/(kg·K); AP treats it as constant. Water's is large (about 4200 J/(kg·K)), so water warms slowly. When objects are mixed and isolated, the energy one loses equals the energy the other gains, and the final temperature lands between the two starting ones.
Conduction rate: Q/Δt = kAΔT/L: The rate (in watts) grows with thermal conductivity k, area A and temperature difference ΔT, and shrinks with thickness L. Doubling the thickness halves the rate. Metals have large k, which is why metal feels colder than wood at the same temperature.
Second law of thermodynamics (qualitative only): The total entropy of an isolated system never decreases, and stays the same only if every process is reversible. Entropy describes how spread out energy is, so energy spreads out on its own and the system heads toward equilibrium, where entropy is greatest.
Entropy facts to know: Entropy is a state function: it depends only on the current state, not on how the system got there. An isolated system's entropy never drops, but a closed system's entropy can drop if energy leaves it (a freezer cools food by pushing energy out).

Charge, Coulomb's law and electric fields

Unit 10

Charge comes in multiples of e: e = 1.60 × 10⁻¹⁹ C. A proton is +e, an electron is −e and a neutron is 0, so any object's charge is q = Ne for some whole number N. An object turns negative by gaining electrons and positive by losing them; in solids the protons stay put.
Conservation of charge: Charge is never created or destroyed, only moved. When two identical conducting spheres touch, they share the total equally: each ends with $\frac{q_1 + q_2}{2}$ .
Three ways to charge an object: Friction: rubbing moves electrons from one material to the other. Conduction: touching a charged object shares its charge (same sign). Induction: hold a charged rod near a conductor, ground the conductor, break the ground, then remove the rod; the conductor ends up with the opposite sign to the rod.
Polarization: why neutral things are attracted: A charged object pulls opposite charges in a nearby neutral object closer and pushes like charges away. The closer charges feel a stronger force, so the neutral object is attracted to a charged object of either sign.
Conductors vs. insulators; grounding: In conductors (metals) charges move freely; in insulators (plastic, glass, rubber) they don't. Grounding connects an object to Earth, a huge neutral reservoir that can supply or take away electrons.
Coulomb's law: $\lvert \vec{F}_E \rvert = \frac{k\lvert q_1q_2 \rvert}{r^2}$: k = 9.0 × 10⁹ N·m²/C². The force points along the line between the charges: like charges repel and opposite charges attract. It's an inverse square law, so doubling r cuts the force to one-fourth, and the two charges always feel equal and opposite forces, even if their sizes differ.
Electric force vs. gravity: Both follow inverse square laws, but for particles the electric force is far stronger (about 10³⁶ times stronger between two protons). Gravity still rules at large scales because big objects are almost exactly neutral. Gravity only attracts; electric forces can attract or repel.
Electric field: $\vec{E} = \frac{\vec{F}_E}{q}$: The force per coulomb a small test charge would feel, in N/C. It points away from positive charges and toward negative ones. A positive charge feels a force along $\vec{E}$ ; a negative charge feels a force opposite to it.
Field of a point charge: $\lvert \vec{E} \rvert = \frac{k\lvert q \rvert}{r^2}$: Also inverse square. Fields from several charges add as vectors, so break them into x- and y-components and look for symmetry that cancels parts. AP calculations use four or fewer point charges (more only when the setup is highly symmetric).
Reading field-line diagrams: Lines start on positive charges and end on negative ones, never cross, and point the way a positive charge would be pushed. Where lines are closer together, the field is stronger. Field vector maps show the same thing with arrows at points.
Conductors in electrostatic equilibrium: Extra charge sits on the outer surface, the field inside the metal is zero, and the field just outside is perpendicular to the surface. Outside a uniformly charged sphere, the field is the same as a point charge at its center. Inside an insulator, charge can be spread through it and the field may be nonzero.
A charge in a uniform field moves like a projectile: The force qE is constant, so the acceleration $a = \frac{qE}{m}$ is constant. Use the same kinematics as a ball thrown sideways near Earth: steady velocity across the field, constant acceleration along it. To hold a charged drop still, set qE = mg.

Electric potential energy, potential and capacitors

Unit 10

Potential energy of two point charges: $U_E = \frac{kq_1q_2}{r}$: Keep the signs: like charges give positive U, opposite charges negative U, and U = 0 when they're infinitely far apart. It equals the work an outside force must do to bring the charges together from infinity.
Systems of several charges: U is a scalar, so add the U of every pair: three charges have 3 pairs and four charges have 6. When like charges are released, U drops as they fly apart and their kinetic energy rises by the same amount.
Electric potential: $V = \frac{kq}{r}$: Potential is potential energy per coulomb, in volts (J/C). For several point charges, $V = k\sum_i \frac{q_i}{r_i}$ : add each one's kq/r with its sign, with no components, because potential is a scalar.
Zero field doesn't mean zero potential: Halfway between two equal positive charges, E = 0 but V = 2kq/r. Halfway between +q and −q, V = 0 but E points toward the negative charge. Find E and V separately.
Potential difference and energy: $\Delta U_E = q\Delta V$: With only electric forces, ΔK = −qΔV. A positive charge released from rest speeds up toward lower potential, and a negative charge (like an electron) speeds up toward higher potential. From rest, $\frac{1}{2}mv^2 = \lvert q\Delta V \rvert$ .
Field and potential: $E = \left\lvert \frac{\Delta V}{\Delta r} \right\rvert$: The field points toward lower potential, and its size is how fast V changes with distance. In a uniform field, $E = \frac{\Delta V}{d}$ , which is why V/m equals N/C.
Equipotential lines: Lines of equal potential always cross field lines at right angles, and the field has no component along them. Moving a charge along one takes no work. Where equipotentials are packed closer together, the field is stronger.
Conductors touching share one potential: Electrons shift until every connected conductor is at the same potential. A whole conductor in equilibrium is one equipotential.
Capacitance: $C = \frac{Q}{\Delta V}$: Q is the charge on either plate (one is +Q, the other −Q), and C is in farads. C depends only on the capacitor's shape and what's between the plates, not on the charge or voltage.
Parallel-plate capacitor: $C = \frac{\kappa\varepsilon_0 A}{d}$: Bigger plates mean more capacitance; a wider gap means less. ε₀ = 8.85 × 10⁻¹² C²/(N·m²), and κ is the dielectric constant (1 for air). AP only uses parallel-plate capacitors and ignores edge effects unless told otherwise.
The field between the plates is uniform: Away from the edges, $E_C = \frac{\Delta V}{d} = \frac{Q}{\kappa\varepsilon_0 A}$ everywhere between the plates, pointing from the + plate to the − plate. A charge placed there feels the same force anywhere in the gap.
Energy stored: $U_C = \frac{1}{2}Q\Delta V = \frac{1}{2}C(\Delta V)^2$: It equals the work done to separate the charges. Use $\frac{Q^2}{2C}$ when Q is the quantity that stays fixed.
Battery connected or disconnected?: Still connected: ΔV stays the same. Disconnected (isolated): Q stays the same. Example: pull the plates apart on an isolated capacitor, so C drops and Q stays, so ΔV and the stored energy both rise (you did work).
Adding a dielectric: The dielectric polarizes and makes a field opposing the plates' field, so C becomes κ times larger. With the battery connected, Q and U rise by κ; isolated, ΔV and U drop to 1/κ of their values.

Current, resistance, power and series–parallel circuits

Unit 11

Current: $I = \frac{\Delta q}{\Delta t}$: Measured in amperes (C/s). Conventional current points the way positive charge would move, from the + terminal around the outside of the circuit to the − terminal, even though electrons in wires actually move the other way. Current isn't used up: it's the same all the way around a single loop.
Closed, open and short circuits: Charge flows only around a closed loop; a break (open switch, missing bulb) stops current in that loop. A short circuit is a path with essentially no resistance, so there's no potential difference across it and anything wired in parallel with it gets no current.
Resistance of a wire: $R = \frac{\rho\ell}{A}$: Longer wires resist more and thicker wires less. Doubling the diameter quadruples A, so R drops to one-fourth. Resistivity ρ (Ω·m) is a property of the material, and for most conductors it rises with temperature.
Ohm's law: $I = \frac{\Delta V}{R}$: Ohmic elements keep the same R at every current, so a graph of I against ΔV is a straight line through the origin with slope 1/R (or ΔV against I has slope R). A real bulb's filament heats up and its resistance rises, so its I–ΔV graph curves, but the exam treats bulbs as ohmic unless told otherwise.
Power: $P = I\Delta V = I^2R = \frac{(\Delta V)^2}{R}$: The rate an element transfers energy, in watts; energy used is PΔt. A bulb's brightness goes up with its power, so ranking brightness means ranking power.
Brightness rules for bulbs: In series (same I), the bulb with more resistance is brighter, since P = I²R. In parallel (same ΔV), the bulb with less resistance is brighter, since P = (ΔV)²/R.
Series: same current, voltages add: $R_{\text{eq}} = R_1 + R_2 + \cdots$ , always more than the largest resistor. The battery's voltage splits among the series elements in proportion to their resistances.
Parallel: same voltage, currents add: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$ , always less than the smallest resistor. Two resistors: $R_{\text{eq}} = \frac{R_1R_2}{R_1 + R_2}$ ; n identical resistors: R/n. More paths mean less total resistance.
What changes when you add or remove an element: Adding a branch in parallel lowers R_eq, so the battery's total current rises; branches wired directly across an ideal battery keep the same ΔV and current. Adding a resistor in series raises R_eq, so the current in that loop drops. Re-find R_eq, then the total current, then work outward.
Internal resistance: $\Delta V_{\text{terminal}} = \varepsilon - Ir$: Model a real battery as an ideal emf ε in series with a small resistor r. With no current, the terminal voltage equals ε; the more current it supplies, the lower the terminal voltage. Current: $I = \frac{\varepsilon}{R + r}$ .
Meters: ammeter in series, voltmeter in parallel: An ideal ammeter has zero resistance and an ideal voltmeter has infinite resistance, so neither changes the circuit. Real meters do: a real ammeter adds resistance and lowers the current, and a real voltmeter lets some current through and lowers the reading (qualitative only).

Kirchhoff's rules and RC circuits

Unit 11

Loop rule: $\sum \Delta V = 0$ around any closed loop: It's conservation of energy: a charge going all the way around ends at the potential where it started. Going through a battery from − to + is +ε; going through a resistor in the direction of the current is −IR (opposite the current, +IR).
Junction rule: $\sum I_{\text{in}} = \sum I_{\text{out}}$: It's conservation of charge: charge doesn't pile up at a junction, so the current flowing in equals the current flowing out.
Solving a multi-loop circuit: Label a current in each branch with a guessed direction, write junction equations, then write loop equations until you have as many equations as unknowns. A negative answer just means that current flows the other way. Batteries of different voltages in parallel aren't tested.
Potential vs. position graphs: Walking around a loop, the potential jumps up across the battery (− to +) and steps down across each resistor, by IR, ending where it started. Ideal wires don't change the potential, so those parts of the graph are flat.
Capacitors combine the opposite way to resistors: Parallel: $C_{\text{eq}} = C_1 + C_2 + \cdots$ , with the same ΔV on each. Series: $\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$ , with the same charge Q on each, and C_eq smaller than the smallest capacitor.
Time constant: $\tau = R_{\text{eq}}C_{\text{eq}}$: In seconds (Ω·F = s), using the equivalent resistance and capacitance the capacitor charges or discharges through. It measures how fast a capacitor charges or discharges: bigger R or bigger C means slower. After one τ, a charging capacitor holds about 63% of its final charge, and a discharging one keeps about 37% of its starting charge.
Right after the switch closes: an uncharged capacitor acts like a wire: At that instant ΔV across it is zero, so redraw it as a plain wire to find the starting currents. Its branch carries the largest current it ever will.
After a long time: a capacitor acts like a break: Once it's fully charged, no current flows in its branch. Remove that branch, solve the resistor circuit, then the capacitor's ΔV equals the potential difference across whatever it's connected across, and Q = CΔV.
Charging and discharging curves (describe, don't calculate): Charging: Q and ΔV_C rise quickly at first, then level off; the current starts at its maximum and falls toward zero. Discharging: Q, ΔV_C and the current all drop toward zero, fast at first and then more slowly. AP doesn't ask for the exponential formulas, only the starting and long-time values and the shapes.

Magnetic fields, magnetic forces and induction

Unit 12

Magnets are always dipoles: Magnetic fields come from moving charges, like electrons moving in atoms. Every magnet has a north and a south pole; break one in half and you get two smaller magnets, never a lone pole. Like poles repel and opposite poles attract, and the field weakens with distance.
Magnetic field lines: They form closed loops: outside a bar magnet they leave the north end and return to the south end, and inside they run from south to north. A compass needle lines up with the field, its north end pointing along the field lines. Earth acts like a dipole whose magnetic south pole is near the geographic North Pole.
Magnetic materials: Ferromagnetic (iron, nickel, cobalt): magnetic domains line up and can stay lined up, making a permanent magnet. Paramagnetic (aluminum, titanium, magnesium): weakly pulled in, but the alignment disappears when the field is removed. Diamagnetic: every material has this weak effect, which lines up opposite the field.
Force on a moving charge: $F_B = \lvert q \rvert vB\sin\theta$: θ is the angle between v and B. The force is zero for a charge at rest or moving parallel to B (0° or 180°) and largest at 90°. AP calculates only 0°, 90° and 180°; other angles are qualitative.
Right-hand rule for the force: Point your fingers along v and bend them toward B; your thumb gives F on a positive charge. For a negative charge, such as an electron, the force is the opposite way. ⊙ means out of the page and ⊗ means into it.
Magnetic forces do no work: The force is always at right angles to the velocity, so it changes the direction of motion but never the speed or kinetic energy.
Circular motion in a uniform field: A charge moving at right angles to B moves in a circle: $\lvert q \rvert vB = \frac{mv^2}{r}$ , so $r = \frac{mv}{\lvert q \rvert B}$ . Faster or heavier particles make bigger circles; a stronger field makes tighter ones.
Electric and magnetic forces together: Each field exerts its own force, and they add as vectors. When E, B and v are all perpendicular and the forces balance (qE = qvB), the charge goes straight through at $v = \frac{E}{B}$ , whatever its charge or mass.
The Hall effect: Charges moving through a conductor in a magnetic field get pushed to one side, creating a small potential difference across the conductor. Which side becomes positive tells you the sign of the moving charges.
Field of a long straight wire: $B = \frac{\mu_0 I}{2\pi r}$: The field circles the wire: point your right thumb along the current and your fingers curl the way B goes. Double the distance and B halves. μ₀ = 4π × 10⁻⁷ T·m/A. At the center of a current loop, the field points along the loop's axis (curl your fingers with the current; your thumb gives B).
Force on a current-carrying wire: $F_B = I\ell B\sin\theta$: ℓ is the length of wire inside the field, and θ is the angle between the current and B. Use the same right-hand rule with your fingers along I. A wire parallel to the field feels no force.
Parallel wires: Currents in the same direction attract, and opposite currents repel. The forces on the two wires are equal and opposite (Newton's third law), even if the currents differ.
Magnetic flux: $\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta$: θ is the angle between B and the line perpendicular to the surface (the area vector). Flux is greatest when the field passes straight through the loop and zero when the field runs along the loop's surface. Units: Wb = T·m².
Faraday's law: $\lvert \varepsilon \rvert = \left\lvert \frac{\Delta\Phi_B}{\Delta t} \right\rvert$: The size of the induced emf is how fast the flux changes; Lenz's law gives its direction (it's the minus sign in $\varepsilon = -\frac{\Delta\Phi_B}{\Delta t}$ ). Only a changing flux induces an emf; a big but steady flux induces nothing. Flux changes when B, the loop's area or the loop's angle changes. A coil of N identical loops has N times the emf of one loop.
Lenz's law: the induced current opposes the change: 1) Find the direction of the outside field through the loop. 2) Decide whether the flux is increasing or decreasing. 3) The induced field points opposite the outside field if the flux is increasing, and the same way if it's decreasing. 4) Use the right-hand rule to turn that induced field into a current direction.
Moving rod on rails: ε = Bℓv: The circuit's area grows, so the flux grows. The current is $I = \frac{B\ell v}{R}$ , and the magnetic force on the rod, IℓB, pushes against its motion. Keeping the rod moving at steady speed takes a push whose power equals the I²R heating in the circuit.

Reflection, refraction, mirrors and lenses

Unit 13

Rays: A ray is a straight line perpendicular to the wavefronts, pointing the way light travels. Use rays when the wave nature of light doesn't matter; for diffraction and interference you need waves (Unit 14).
Law of reflection: θ_i = θ_r: Measure both angles from the normal (the line perpendicular to the surface), not from the surface. A smooth surface gives specular reflection in one direction; a rough surface gives diffuse reflection in many directions.
Index of refraction: $n = \frac{c}{v}$: n is 1 for vacuum (about 1 for air) and bigger in denser optical materials (about 1.33 for water, 1.5 for glass). Higher n means slower light. Entering a new material, the frequency stays the same, so the wavelength shrinks to $\frac{\lambda}{n}$ .
Snell's law: n₁ sin θ₁ = n₂ sin θ₂: Angles are from the normal. Going into a higher index, light bends toward the normal; going into a lower index, away from it. A ray hitting the surface along the normal goes straight through without bending.
Critical angle and total internal reflection: Only when light goes from higher n to lower n. At the critical angle, $\sin\theta_c = \frac{n_2}{n_1}$ , the refracted ray skims along the surface at 90°; beyond it, all the light reflects and none is transmitted. Optical fibers and sparkling diamonds use this.
Focal points: Converging: a concave mirror or a convex lens sends parallel rays through a real focal point. Diverging: a convex mirror or a concave lens spreads parallel rays as if they came from a focal point. For a spherical mirror, f is about half the radius of curvature; a plane mirror's focal point is infinitely far away.
Mirror/lens equation: $\frac{1}{s_o} + \frac{1}{s_i} = \frac{1}{f}$: The same equation works for mirrors and thin lenses. Signs: f is positive for converging and negative for diverging. s_i is positive for a real image (in front of a mirror, or on the far side of a lens) and negative for a virtual image.
Magnification: $\lvert M \rvert = \left\lvert \frac{h_i}{h_o} \right\rvert = \left\lvert \frac{s_i}{s_o} \right\rvert$: |M| > 1 means enlarged and |M| < 1 means reduced. For a single mirror or lens, a real image is inverted and a virtual image is upright. Many books write $M = -\frac{s_i}{s_o}$ , where a negative M means inverted.
Real vs. virtual images: A real image forms where light rays actually meet, so it can be projected on a screen. A virtual image forms where rays only seem to come from, like the image behind a plane mirror; a screen there shows nothing.
Converging mirror or lens: object outside f: Beyond 2f: real, inverted, smaller, between f and 2f (a camera or the eye). At 2f: real, inverted, same size, at 2f. Between f and 2f: real, inverted, larger, beyond 2f (a projector). At exactly f: no image, since the rays leave parallel.
Converging mirror or lens: object inside f: The image is virtual, upright and enlarged, on the same side as the object for a lens (behind the mirror for a mirror). That's how a magnifying glass or a makeup mirror works.
Diverging mirror or lens: always the same: For any real object the image is virtual, upright and smaller, located between the mirror or lens and its focal point. Store security mirrors and car side mirrors are convex for a wide view.
Plane mirror: The image is virtual, upright, the same size, and as far behind the mirror as the object is in front (s_i = −s_o).
Principal rays: Mirror: a ray parallel to the axis reflects through F (for a convex mirror, as if it came from F behind the mirror); a ray headed through F (or toward F, for a convex mirror) reflects parallel; a ray hitting the mirror's center reflects at the same angle on the other side of the axis. Lens: a ray parallel to the axis bends through the far F (for a diverging lens, as if it came from the near F); a ray through the center goes straight; a ray headed through the near F (or toward the far F, for a diverging lens) leaves parallel. Two rays locate the image; the third checks it.

Waves and sound

Unit 14

Waves carry energy, not matter: A pulse is a single disturbance; a periodic wave repeats. The medium only moves back and forth while the energy travels. Mechanical waves like sound need a medium, so sound can't cross a vacuum; electromagnetic waves don't need one.
Transverse vs. longitudinal: Transverse: the medium moves at right angles to the wave's direction (waves on a string, light). Longitudinal: the medium moves along the wave's direction, making compressions (high pressure) and rarefactions (low pressure); sound is longitudinal.
v = fλ and f = 1/T: Period T is the time for one cycle, frequency f is cycles per second (Hz), and wavelength λ is the distance between matching points such as crest to crest. In one medium the speed is fixed, so raising f shortens λ.
The medium sets the speed: Wave speed depends on the medium, not on the frequency or amplitude. On a string, $v_{\text{string}} = \sqrt{\frac{F_T}{m/\ell}}$ , where m/ℓ is the mass per length (kg/m), so tighter, lighter strings carry faster waves. Sound in air is about 343 m/s at room temperature and gets faster as the air warms.
Crossing into a new medium: frequency stays the same: The source sets f, so f never changes at a boundary. The speed changes, and λ changes in proportion: slower medium, shorter wavelength.
Amplitude, loudness and pitch: Amplitude is the maximum displacement from equilibrium (or the maximum pressure change, for sound) and doesn't depend on f. Bigger amplitude carries more energy and sounds louder; higher frequency sounds higher in pitch. Intensity is power per area, in W/m².
Reading wave graphs: Displacement vs. position is a snapshot of the whole wave at one instant: crest to crest is λ. Displacement vs. time follows one point of the medium: crest to crest is T. A sinusoidal wave can be written as $A\cos(2\pi ft)$ for one point over time, or $y(x) = A\cos\left(2\pi\frac{x}{\lambda}\right)$ for the whole wave at one instant.
Reflection at a boundary: Part of a wave reflects and part transmits. The reflected pulse is flipped upside down if the wave is entering a slower medium (light string into heavy string, or a fixed end) and stays upright if it's entering a faster one (heavy into light, or a free end). The transmitted pulse is never flipped.
Polarization: Only transverse waves can be polarized, so light can be and sound can't. A polarizing filter passes vibrations in one direction: unpolarized light through one filter keeps half its intensity, and two filters at right angles block it all. Reflected glare is partly polarized, which is why polarized sunglasses cut it.
Electromagnetic waves: Oscillating electric and magnetic fields at right angles to each other and to the direction of travel, so they're transverse. In a vacuum all of them travel at c = 3.00 × 10⁸ m/s.
The EM spectrum in order of decreasing wavelength: Radio, microwave, infrared, visible, ultraviolet, X-rays, gamma rays. Along that list, frequency and photon energy increase. Visible light from longest to shortest wavelength: red, orange, yellow, green, blue, violet (roughly 700 nm down to 400 nm; exact ranges aren't tested).
Doppler effect (qualitative only): When the source and observer move toward each other, the observed frequency is higher than the source's (higher pitch, shorter wavelength); when they move apart, it's lower. A bigger relative speed gives a bigger shift, and if they move together at the same velocity there's no shift.
Superposition: Overlapping waves pass through each other, and the displacement at each point is the sum of the individual displacements. Same-direction displacements interfere constructively; opposite ones interfere destructively. After overlapping, each pulse continues unchanged.
Beats: $f_{\text{beat}} = \lvert f_1 - f_2 \rvert$: Two sounds of slightly different frequency drift in and out of phase, so the loudness pulses at the difference frequency. Musicians tune by adjusting until the beats slow down and disappear.
Standing waves: nodes and antinodes: Waves reflecting back and forth in a confined region form a standing wave. Nodes never move, and antinodes move the most. Node to node is λ/2, and node to the next antinode is λ/4.
Strings fixed at both ends and pipes open at both ends: Both ends are the same kind (nodes for a string, antinodes for an open pipe), so $\lambda_n = \frac{2L}{n}$ and $f_n = \frac{nv}{2L}$ for n = 1, 2, 3, …. The fundamental (n = 1) has the longest wavelength, 2L.
Pipes closed at one end: odd harmonics only: A node at the closed end and an antinode at the open end, so $\lambda_n = \frac{4L}{n}$ and $f_n = \frac{nv}{4L}$ for n = 1, 3, 5, …. The fundamental's wavelength is 4L, so its frequency is half that of an open pipe of the same length.

Diffraction, interference and thin films

Unit 14

Diffraction: Waves spread out through an opening or around an edge, and the effect is biggest when the opening is about the size of the wavelength. That's why you can hear around a corner (sound wavelengths are about a meter) but can't see around it (light wavelengths are under a micrometer).
Path length difference ΔD decides the interference: If two waves from the same source travel distances that differ by a whole number of wavelengths (0, λ, 2λ, …), they arrive in step and interfere constructively. If they differ by a half-number (λ/2, 3λ/2, …), they cancel.
Single slit of width a: dark bands where ΔD = a sin θ = mλ: m = 1, 2, 3, … counts the dark bands out from the center. For small angles, $a\frac{y_{\min}}{L} \approx m\lambda$ , so the m-th dark band is $y_{\min} \approx \frac{m\lambda L}{a}$ from the center on a screen a distance L away. The central bright band is twice as wide as the others, and a narrower slit or longer wavelength spreads the pattern out.
Double slit with separation d: bright fringes where ΔD = d sin θ = mλ: m = 0, ±1, ±2, …, with the central bright fringe at m = 0; dark fringes fall halfway between. For small angles, $d\frac{y_{\max}}{L} \approx m\lambda$ , so the bright fringes are evenly spaced by $\Delta y \approx \frac{\lambda L}{d}$ : wider slit spacing squeezes the fringes together and longer wavelength or a farther screen spreads them apart.
Small-angle approximation: For angles under about 10°, sin θ ≈ tan θ = $\frac{y}{L}$ . The exam lets you use it for single and double slits. Don't automatically use it for diffraction gratings, where the angles are often large.
Real double-slit pattern: Evenly spaced double-slit fringes sit inside the wider envelope of single-slit diffraction, so the fringes fade away from the center. Young's double-slit experiment was key evidence that light is a wave.
Diffraction gratings: Many evenly spaced slits give much sharper, brighter maxima at the same angles, d sin θ = mλ, where d is 1 divided by the number of lines per meter. With white light the center is white and each higher order spreads into a rainbow, with red (longest λ) farthest out.
Phase change on reflection: Reflecting off a material with a higher index of refraction flips the wave by half a wavelength (180°). Reflecting off a lower index doesn't flip it, and refraction never changes the phase.
Thin film: count the flips first: With light hitting the film straight on, the second ray travels an extra 2t, where t is the thickness, and you measure in the film's wavelength, $\lambda_{\text{film}} = \frac{\lambda}{n_{\text{film}}}$ . If exactly one reflection flips, 2t = (m + ½)λ_film is bright and 2t = mλ_film is dark. If both or neither flip, those conditions swap.
Soap bubbles and oil films: Different thicknesses reinforce different colors, so a film of changing thickness shows bands of color. A soap film in air so thin that t is almost zero looks dark, because only one reflection flips and the two reflected waves cancel.
Antireflection coating: $t = \frac{\lambda}{4n_{\text{coating}}}$: The coating's index is between air's and the lens's, so both reflections flip and the extra trip of 2t = λ_film/2 makes them cancel. So the simplest coating is a quarter of a wavelength thick, measured in the coating. Calculations use light hitting the film straight on.

Photons, matter waves and atoms

Unit 15

Photon energy: $E = hf = \frac{hc}{\lambda}$: h = 6.63 × 10⁻³⁴ J·s. Photons are massless and neutral and travel at c in a vacuum. Shorter wavelength means more energy per photon, so ultraviolet photons carry more energy than red ones. Brighter light means more photons, not more energetic ones.
Photon momentum: $p = \frac{h}{\lambda} = \frac{E}{c}$: Even without mass, a photon carries momentum, which is what makes Compton scattering work.
De Broglie wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$: Moving particles act like waves, with longer wavelengths at smaller momentum. Electrons sent through a double slit build up an interference pattern. Wave behavior matters only when λ is about as big as the system, which is why you never notice a baseball's wavelength.
Wave–particle duality: which evidence shows what: Interference and diffraction show wave behavior (Young's experiment for light, electron diffraction for matter). The photoelectric effect and Compton scattering show particle behavior of light. Atomic spectra and blackbody radiation need quantized energy.
Inside an atom: A tiny positive nucleus of protons and neutrons holds almost all the mass, surrounded by electrons. The number of protons Z identifies the element; the mass number A (protons + neutrons) identifies the isotope, so neutrons = A − Z. An ion is an atom with a net charge.
Bohr model: The electric force on the electron provides the centripetal force: $\frac{ke^2}{r^2} = \frac{mv^2}{r}$ for hydrogen. Only orbits whose circumference fits a whole number of de Broglie wavelengths are allowed, so the atom has only certain energy levels. AP stops at energy levels: no orbitals or orbital shapes.
Energy-level diagrams: Levels are usually negative (the electron is bound), with 0 meaning the electron is just freed. The lowest level is the ground state. Hydrogen's levels are $E_n = -\frac{13.6\text{ eV}}{n^2}$ ; the exam gives you the levels it needs, and only single-electron atoms are used.
Photon emitted or absorbed: $hf = \lvert E_{\text{upper}} - E_{\text{lower}} \rvert$: An atom absorbs a photon only if its energy exactly matches a gap, jumping up; it emits a photon when it drops down. Bigger gap, higher frequency, shorter wavelength: $\lambda = \frac{hc}{\Delta E}$ .
Counting spectral lines: An atom excited to level n can emit photons for every downward jump between levels, which gives $\frac{n(n-1)}{2}$ possible lines (6 from level 4). Absorption from the ground state only shows lines that start at the ground state.
Emission and absorption spectra: A hot thin gas gives bright lines at its transition wavelengths; light passing through a cool gas loses those same wavelengths, leaving dark lines. Each element's set of lines is a fingerprint that identifies it.
Ionization energy: The energy needed to remove an electron completely, taking it to the 0 level. It's largest from the ground state: 13.6 eV for hydrogen.
Blackbody radiation: A blackbody absorbs all light that hits it and gives off a continuous spectrum that depends only on its temperature. Hotter means more radiation at every wavelength and a peak at a shorter wavelength. Classical physics couldn't explain the curve; Planck fixed it by assuming energy is given off in discrete packets (quanta).
Wien's law: $\lambda_{\max} = \frac{b}{T}$: Wien's constant b = 2.90 × 10⁻³ m·K, so doubling the kelvin temperature halves the peak wavelength. The Sun, at about 5800 K, peaks near 500 nm; cooler stars look red and hotter stars look blue-white.
Stefan–Boltzmann law: power ∝ AT⁴: $P = A\sigma T^4$ for a blackbody, with σ = 5.67 × 10⁻⁸ W/(m²·K⁴) and T in kelvins. Doubling the temperature multiplies the emitted power by 2⁴ = 16; doubling the surface area doubles it.
Photoelectric effect: $K_{\max} = hf - \phi$: The work function φ is the least energy to free an electron (the exam gives it). Below the threshold frequency $f_0 = \frac{\phi}{h}$ , no electrons come out, however bright the light. Brighter light above threshold frees more electrons (more current) but not faster ones; higher frequency makes them faster.
Stopping potential: $eV_{\text{stop}} = K_{\max}$: Raise the reverse voltage until the current just stops; that voltage measures the fastest electrons' kinetic energy. A graph of K_max (or eV_stop) against f is a straight line with slope h, a horizontal-axis intercept at the threshold frequency and a vertical-axis intercept of −φ.
Compton scattering: An X-ray photon hits a free electron and comes off with less energy and a longer wavelength; the electron recoils. Solve it as a 2-D collision: conserve energy ( $hf = hf' + K_e$ ) and momentum in x and y, using $p = \frac{h}{\lambda}$ for the photon. The wavelength shift is $\Delta\lambda = \frac{h}{m_ec}(1 - \cos\theta)$ : the bigger the scattering angle θ, the bigger the shift, largest when the photon bounces straight back.

Nuclear physics and radioactive decay

Unit 15

The strong force: An attractive force between nucleons (protons and neutrons) that works only over nuclear distances. It's what holds the nucleus together against the electric repulsion of its protons.
Conservation laws in nuclear reactions: Nucleon number (the top numbers, A) and charge (the bottom numbers, Z) must balance on both sides. Energy, including rest energy mc², and momentum are conserved too. In decays, lepton number is also conserved.
Mass–energy equivalence: E = mc²: When a reaction releases energy, the products have less total mass than what you started with, and the missing mass Δm becomes energy Δmc², as kinetic energy of the products or as photons. A mass change of 1 u is about 931 MeV.
Binding energy: The energy needed to pull a nucleus apart into separate protons and neutrons, equal to its mass defect times c². Fusing very light nuclei or splitting very heavy ones makes products that are more tightly bound per nucleon, which is where the released energy comes from.
Fusion vs. fission: Fusion joins small nuclei into a bigger one (it powers the Sun). Fission splits a large nucleus, like uranium-235, into smaller ones plus a few neutrons (it powers nuclear reactors). Fission can happen on its own or need an energy input, such as absorbing a neutron.
Radioactive decay is random: You can't predict when one nucleus will decay, but a large sample follows a predictable pattern. Decay is a nucleus changing into a different nucleus, or the same nucleus dropping to a lower energy level (gamma decay).
Half-life: $N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}$: After each half-life, half of the remaining undecayed nuclei are left: 1/2, 1/4, 1/8, … after 1, 2, 3 half-lives. A graph of N against t is a decaying curve that never quite reaches zero. Half-lives range from fractions of a second to billions of years, and you don't need to memorize any.
Decay constant: $\lambda = \frac{\ln 2}{t_{1/2}}$: $N = N_0e^{-\lambda t}$ is the same curve written with the decay constant λ (in 1/s); a bigger λ means faster decay and a shorter half-life. Use it to find how much is left after a time, or to find a sample's age from how much is left.
Alpha decay (α): The nucleus ejects a helium-4 nucleus, ⁴₂He: A drops by 4 and Z drops by 2, so it becomes a different element. Example: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He.
Beta-minus decay (β⁻): A neutron turns into a proton, emitting an electron (⁰₋₁e) and an antineutrino. A stays the same and Z goes up by 1. Example: ¹⁴₆C → ¹⁴₇N + ⁰₋₁e + antineutrino.
Beta-plus decay (β⁺): A proton turns into a neutron, emitting a positron (⁰₊₁e, same mass as an electron but positive charge) and a neutrino. A stays the same and Z goes down by 1.
Gamma decay (γ): An excited nucleus, often just after an alpha or beta decay, drops to a lower energy state by emitting a high-energy photon. A and Z don't change.
Neutrinos, antineutrinos and lepton number: Both have no charge and almost no mass and barely interact with matter. Count electrons and neutrinos as lepton number +1 and positrons and antineutrinos as −1, so β⁻ gives an electron with an antineutrino and β⁺ gives a positron with a neutrino. Types of neutrinos, anything else that tells neutrinos and antineutrinos apart, electron capture and neutron emission aren't in this course.
Balancing a decay equation: Make the top numbers (A) add up the same on both sides, then the bottom numbers (Z). The new Z tells you which element you have; look it up on the periodic table, since you're not expected to memorize decay chains.

Graphs, labs and free-response habits

Units 9, 10, 11, 12, 13, 14, 15

Linearize: make it y = (slope)x + b: Rearrange the equation so the quantity you measure is on one axis and something you can calculate from your data is on the other, giving a straight line. Then read the unknown from the slope or the intercept, with units.
Thermal and gas examples: P against T (in K) at constant volume: a line through the origin with slope nR/V. P against T in °C: the line hits P = 0 near −273 °C. P against 1/V at constant temperature: slope nRT. Q against ΔT: slope mc.
Electric examples: ΔV against I for a resistor: slope R. R against ℓ for a wire: slope ρ/A. Q against ΔV for a capacitor: slope C. C against 1/d for a parallel-plate capacitor: slope κε₀A. F against 1/r² for two charges: slope kq₁q₂.
Magnetism examples: B against I at a fixed distance from a wire: slope $\frac{\mu_0}{2\pi r}$ . B against 1/r: slope $\frac{\mu_0 I}{2\pi}$ . Force on a wire against current: slope ℓB, so the slope gives the field.
Optics and wave examples: sin θ₂ against sin θ₁: slope $\frac{n_1}{n_2}$ , so you can find an unknown index. 1/s_i against 1/s_o for a lens: slope −1 and intercept 1/f. Fringe spacing against L: slope λ/d. v² against tension for a string: slope ℓ/m (1 over the mass per length). λ against 1/f: slope v.
Modern physics examples: K_max (or eV_stop) against frequency: slope h, horizontal-axis intercept the threshold frequency, vertical-axis intercept −φ. ln N against t for a decaying sample: a straight line with slope −λ.
Designing an experiment: Name what you'll change (independent variable), what you'll measure (dependent variable) and what you'll keep the same. Say which equipment measures each quantity and how, take data over a wide range, and repeat trials. Common tools: meterstick, thermometer, pressure sensor, multimeter (as an ammeter or voltmeter), compass or magnetic field sensor, protractor and ray box, and a light sensor.
Drawing a best-fit line: Label each axis with the quantity and unit, and use most of the grid. Draw one straight line with the points spread evenly on both sides, and find its slope from two points on the line far apart, not from data points. A line that should pass through the origin but doesn't points to a systematic error.
Graph shapes worth knowing: Inverse square (Coulomb force, E of a point charge): drops steeply, to 1/4 at twice the distance. Inverse (V of a point charge, B of a wire): drops to 1/2 at twice the distance. Isotherm on a PV diagram: a curve with PV constant. RC charging: rises and levels off. Decay: falls by half every half-life.
'What happens if it doubles?' questions: Write the relationship, cancel what stays the same, then scale. Example: in $E = \frac{kq}{r^2}$ , tripling r cuts E to 1/9; in $f_n = \frac{nv}{2L}$ , quadrupling the string's tension doubles v and so doubles every harmonic's frequency.
Derivations and justifications: Start from a basic principle (Newton's second law, conservation of energy, charge or momentum, Kirchhoff's rules) and write it for this situation, using only the given quantities and constants. To justify a claim, name the law, connect it step by step to these specific objects and state the result; just saying “because of the right-hand rule” doesn't earn the point.
Consistency across representations: The translation-between-representations and qualitative/quantitative questions ask whether your diagram, graph, equation and words agree. Check limiting cases (what if R = 0, or the angle is 90°?) and make sure the trend in your equation matches the trend you described.
Exam format: Section I: 42 multiple-choice questions (4 choices each) in 85 minutes. Section II: 4 free-response questions in 95 minutes: mathematical routines (10 points), translation between representations (12), experimental design and analysis (10) and qualitative/quantitative translation (8). Each section is half the score, and you have a calculator and the equation sheet for both.