AP Physics C: E & M — Cheatsheet
Formulas, exam-day tips, and key terms on one page.
Formulas & relationships
Coulomb’s law (magnitude)
F = k·q₁q₂ / r², with k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²
ε₀ = 8.85 × 10⁻¹² C²/(N·m²) is the permittivity of free space. The force falls off as 1/r², not 1/r.
Field from force / force from field
→E = →F / q₀ ⇔ →F = q→E
A positive charge feels a force along E; a negative charge feels a force opposite to E.
Field of a point charge
E = k·q / r² = q / (4πε₀ r²)
Directed radially outward for q > 0 and radially inward for q < 0. Like the force, it falls off as 1/r².
Field of a continuous distribution
→E = ∫ k·dq / r² (r̂), dq = λ dx = σ dA = ρ dV
A vector integral: resolve dE into components and integrate each. Symmetry often makes one component vanish.
Gauss’s law
Φ = ∮ →E·d→A = Q_enc / ε₀
The circle on the integral means a closed surface. The net flux out of any closed surface equals the charge enclosed divided by ε₀ — nothing else matters.
Potential energy from work
ΔU = U_b − U_a = −W_field = −∫ₐᵇ →F·d→l
The line integral is path-independent, so U depends only on the configuration, not on how the charges got there.
Potential energy of two point charges
U = k·q₁q₂ / r, k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²
Plug the charges in *with their signs*. Note the single power of r — energy goes as 1/r, force as 1/r².
Potential difference from the field
V_b − V_a = −∫ₐᵇ →E·d→l
Path-independent. Moving *with* the field, potential drops; moving against it, potential rises. For a uniform field along a straight displacement d: ΔV = −E·d.
Potential of point charges and distributions
V = k·q/r, V = Σ k·qᵢ/rᵢ, V = ∫ k·dq/r
All plain scalar sums — signed numbers, never components. This is why V is often far easier to compute than E.
Work and field from equipotentials
W_field = −qΔV = q(V_a − V_b), |E| ≈ |ΔV| / Δs
Δs is measured perpendicular to the surfaces. Along an equipotential, ΔV = 0 and the field does no work.
Field from potential (gradient)
Eₓ = −∂V/∂x, E_y = −∂V/∂y, E_z = −∂V/∂z; →E = −∇V
Units: 1 V/m = 1 N/C — the two field units are identical. For radial potentials, E_r = −dV/dr.
Field at a conductor surface
E = σ/ε₀ (just outside, ⊥ to surface); E = 0 (inside); ε₀ = 8.85 × 10⁻¹² F/m
σ is the local density, which varies over a non-spherical conductor — largest where the surface is most sharply curved.
Definition of capacitance
C = Q / V (1 F = 1 C/V)
Q is the magnitude of the charge on either plate; V is the potential difference between the plates.
Parallel-plate capacitor
C = ε₀A / d, with ε₀ = 8.85 × 10⁻¹² F/m
Valid when d is much smaller than the plate dimensions, so fringing fields at the edges are negligible.
Stored energy (three equivalent forms)
U = Q²/(2C) = ½CV² = ½QV
Choose the form whose variables are held constant in your scenario: Q²/2C when the capacitor is isolated, ½CV² when a battery pins the voltage.
Energy density of the electric field
u = ½ε₀E² (energy per unit volume)
General, not just for capacitors. At air’s breakdown field, 3 × 10⁶ V/m, u ≈ 40 J/m³ — why electric fields make poor bulk energy storage.
Capacitance with a dielectric
C = κC₀ = κε₀A/d, effective permittivity ε = κε₀
Filling the gap always *raises* capacitance: for the same Q, the weakened field means a smaller V = Ed, and C = Q/V grows by κ.
Current, resistance & Ohm’s law
I = dQ/dt ⇔ Q = ∫ I dt; R = ρL/A; V = IR
Integrate a varying current for charge — never multiply a changing current by total time. Doubling length doubles R; doubling radius quarters R, because A = πr² quadruples.
Electrical power
P = IV = I²R = V²/R
All three are the same statement. Pick the form whose two variables you actually know for the element in question.
Series and parallel resistors
Series: R_eq = R₁ + R₂ + … Parallel: 1/R_eq = 1/R₁ + 1/R₂ + …
Series resistors share the same current; parallel resistors share the same voltage. A parallel combination is always smaller than its smallest member.
Kirchhoff’s rules
Junction: Σ I_in = Σ I_out Loop: Σ ΔV = 0 around any closed loop
Sign convention: crossing a resistor with the assumed current is −IR (against it, +IR); crossing a battery from − to + terminal is +ε.
Charging capacitor
q(t) = Q_max(1 − e^(−t/RC)), i(t) = (ε/R)e^(−t/RC), Q_max = εC
Charge grows toward Q_max while current decays from ε/R — the two curves are mirror images in behavior.
Time constant
τ = RC (1 Ω × 1 F = 1 s)
After one τ the capacitor holds 1 − e⁻¹ ≈ 63% of full charge and the current has fallen to e⁻¹ ≈ 37% of its start. After 5τ the circuit is effectively settled.
Discharging capacitor
q(t) = Q₀e^(−t/RC), i(t) = I₀e^(−t/RC), I₀ = V₀/R
Half of the charge remains after t = RC·ln 2 ≈ 0.69 RC — the same half-life logic as radioactive decay.
Magnetic force on a charge
→F = q→v × →B, F = qvB·sinθ
F is always perpendicular to both v and B. Maximum force at θ = 90°; zero force for motion along the field.
Circular motion in a magnetic field
r = mv/(qB), T = 2πm/(qB)
Radius grows with momentum mv; the period depends only on m, q, and B — not on how fast the particle moves.
Biot–Savart law
d→B = (μ₀/4π) · I d→l × r̂ / r², μ₀ = 4π × 10⁻⁷ T·m/A
Superpose by integrating over the whole current path. |dl × r̂| = dl·sinφ, where φ is the angle between the element and the line to the point.
Key Biot–Savart results
Straight wire: B = μ₀I/(2πd) Center of ring: B = μ₀I/(2R) On axis: B = μ₀IR²/[2(x² + R²)^(3/2)]
Wire field lines are circles around the wire: point the right thumb along I and the fingers curl in the direction of B.
Force on a current-carrying wire
→F = I→L × →B, F = BIL·sinθ
Right-hand rule: fingers along I, curl toward B, thumb gives F. A wire parallel to B feels nothing.
Force between parallel wires
F/L = μ₀I₁I₂/(2πd)
Currents in the same direction attract; opposite directions repel. Newton’s third law holds: each wire feels the same magnitude of force.
Ampère’s law
∮ →B·d→l = μ₀ I_enc, μ₀ = 4π × 10⁻⁷ T·m/A
Only currents that pierce the surface bounded by the loop count toward I_enc. Curl the right-hand fingers along the traversal direction; the thumb gives the positive current direction.
Solenoid and toroid fields
Solenoid: B = μ₀nI (inside, uniform; ≈0 outside) Toroid: B = μ₀NI/(2πr) (inside the windings)
n = N/ℓ is turns per meter for the solenoid; N is the total turn count for the toroid, whose field falls as 1/r across its interior and vanishes outside.
Faraday’s law
ε = −N·dΦ_B/dt, Φ_B = ∫ →B·d→A
N is the number of turns, each contributing the same flux. The derivative — not the flux itself — drives the EMF: a huge steady flux induces nothing.
Motional EMF
|ε| = BLv
A conducting rod moving across field lines acts like a battery: the magnetic force piles positive charge at one end until an internal E field balances it.
Self-inductance
ε_L = −L·di/dt, L = NΦ_B/i
The back-EMF depends on how fast the current changes, not on how big it is. A steady current through an ideal inductor drops zero volts.
Solenoid inductance and stored energy
L = μ₀n²Aℓ = μ₀N²A/ℓ, U = ½Li², u = B²/(2μ₀)
U = ½Li² is the work done against the back-EMF to establish the current; u is that energy per unit volume, stored in the field itself.
LR circuit solutions
Rise: i(t) = (ε/R)(1 − e^(−t/τ)) Decay: i(t) = i₀e^(−t/τ) τ = L/R
Current is the quantity that changes smoothly in an LR circuit, just as charge is in an RC circuit. At t = τ the rising current reaches 63% of ε/R.
Maxwell’s equations (integral form)
∮→E·d→A = Q_enc/ε₀ ∮→B·d→A = 0 ∮→E·d→l = −dΦ_B/dt ∮→B·d→l = μ₀I_enc + μ₀ε₀·dΦ_E/dt
Two flux laws (what makes fields start and end) and two circulation laws (what makes fields curl). The final term is Maxwell’s displacement current.
On the exam
- On free-response problems, always resolve each Coulomb force into components before summing. Only add magnitudes directly when every force lies along the same line, as it did above. Otherwise sum Fₓ and Fᵧ separately, then recombine.
- A continuous-distribution FRP earns its points in the setup: state dq in terms of a density, draw dE and identify which component survives by symmetry, write correct integration limits, and only then integrate. Show the symmetry argument explicitly — graders reward it.
- For a Gaussian-surface FRP: (1) name the symmetry and pick a matching surface, (2) argue E is constant over the part carrying flux, (3) write ∮ E·dA = E·A, (4) find Q_enc — using ρ, σ, or λ times the enclosed volume/area/length — and (5) solve E·A = Q_enc/ε₀. Missing the Q_enc step is the usual lost point.
- Sign conventions to lock in for the exam: ΔV = −∫→E·d→l (potential drops along the field), U = qV and ΔU = qΔV (signs of q included), and W_field = −ΔU. A positive charge released from rest falls toward *lower* V; a negative charge toward *higher* V. Nearly every potential FRQ point hinges on one of these signs.
- Memorize the inverse pair and their graphical readings: V_b − V_a = −∫ₐᵇ →E·d→l (area under an Eₓ graph, negated) and Eₓ = −dV/dx (slope of a V graph, negated). AP free-response loves handing you one graph and demanding the other — check your minus sign at a point where you know which way the field must push a positive charge.
- The capacitance derivation is a guaranteed FRQ pattern — commit the recipe: (1) assume ±Q, (2) Gauss’s law for E, (3) V = ∫→E·d→l between the conductors, (4) C = Q/V. Show the Q cancelling; that line is what proves C is geometric. Key results: C = ε₀A/d (plates), C = 4πε₀R (sphere).
- Dielectric quick table — battery **disconnected** (Q fixed): C ↑ ×κ, V ↓ ÷κ, E ↓ ÷κ, U ↓ ÷κ. Battery **connected** (V fixed): C ↑ ×κ, Q ↑ ×κ, E unchanged, U ↑ ×κ. In both cases the slab is pulled inward. Reproduce this table from C = κC₀ plus “what’s held fixed” rather than memorizing blindly.
- Loop-rule bookkeeping wins or loses the multiloop FRQ. Commit to this convention: pick a travel direction around each loop; a resistor traversed *with* its assumed current contributes −IR and *against* it +IR; a battery crossed from − to + contributes +ε, from + to − contributes −ε. State the junction equation explicitly — graders award a point for it even before any algebra.
- Transient shortcut worth memorizing for the exam: **t = 0** → replace each uncharged capacitor with a plain wire; **t → ∞** → replace each capacitor with an open break, then find V_C from the resistors around it. Most multiple-choice RC questions are exactly these two substitutions, no exponentials required.
- Ampère’s-law FRQs award points for the argument, not just the answer: (1) name the symmetry and draw the Amperian loop, (2) justify that B is constant and parallel to dl on it, so ∮B·dl = B·(length), (3) compute I_enc — for distributed currents use the current density times the enclosed area, (4) solve. Skipping step 3’s enclosed-fraction logic is the most common lost point on thick-wire problems.
- Faraday FRQs almost always follow one script: (1) write Φ_B symbolically as B·A·cosθ or an integral, (2) identify which factor depends on t, (3) differentiate to get ε = −N dΦ/dt, (4) if asked for current, divide by resistance, and (5) give the direction from Lenz’s law with an explicit “opposes the increase/decrease of flux” sentence. Write the derivative before plugging in numbers.
- Know each Maxwell equation by *claim*, not just symbol: Gauss (E) — charges make diverging E fields; Gauss (B) — no monopoles, B lines close; Faraday — changing Φ_B makes circulating E; Ampère–Maxwell — currents and changing Φ_E make circulating B. Exam questions ask “which equation forbids magnetic monopoles?” or “which term did Maxwell add, and why?” far more often than they ask you to compute with them.
How to get a 5
- For Gauss's and Ampère's Law problems, explicitly state the symmetry and why you chose your specific Gaussian surface or Amperian loop.
- Be extremely careful with signs in Faraday's Law and Lenz's Law. Always double check if the induced field adds to or subtracts from the external field.
- When setting up differential equations for RC or LR circuits, ensure your initial conditions (e.g., I=0 or q=0 at t=0) guide your integration constants.
- Remember the Right-Hand Rule comes in three flavors: forces (qv × B), B-fields from wires (thumb I, fingers B), and solenoids/loops (fingers I, thumb B).
Key terms
Gauss's Law — ∮E·dA = q_enc / ε₀. Best used for high symmetry (spherical, cylindrical, planar).
Electric Potential Gradient — E = -∇V (in 1D, E_x = -dV/dx). Electric field points toward lower potential.
Capacitors in Series and Parallel — Series: 1/C_eq = 1/C₁ + 1/C₂. Parallel: C_eq = C₁ + C₂. (Opposite of resistors).
Energy Stored in a Capacitor — U_C = ½CV² = ½Q²/C = ½QV.
Biot-Savart Law — dB = (μ₀I/4π) (dl × r̂) / r². Used to find B from arbitrary current distributions.
Ampère's Law — ∮B·dl = μ₀I_enc. Best used for high symmetry (infinite wires, solenoids).
Magnetic Field of a Long Wire — B = μ₀I / (2πr). Derived from Ampère's Law.
Magnetic Field of a Solenoid — B = μ₀nI (where n = N/L, turns per unit length).
Faraday's Law of Induction — ε = -dΦ_B/dt = ∮E·dl. A changing magnetic flux creates an induced EMF.
Inductor Voltage — ε_L = -L(dI/dt). Opposes changes in current.
Energy Stored in an Inductor — U_L = ½LI².
Maxwell's Addition to Ampère's Law — Displacement current: I_d = ε₀(dΦ_E/dt). Accounts for changing electric fields producing magnetic fields.