AP Physics C: E&M Cheat Sheet

Constants and Common Values

Coulomb constant: $k = \dfrac{1}{4\pi\varepsilon_0} = 8.99\times10^{9}\ \text{N}\cdot\text{m}^2/\text{C}^2$
Permittivity of free space: $\varepsilon_0 = 8.85\times10^{-12}\ \text{C}^2/(\text{N}\cdot\text{m}^2) = 8.85\times10^{-12}\ \text{F/m}$
Permeability of free space: $\mu_0 = 4\pi\times10^{-7}\ \text{T}\cdot\text{m/A}$ , with $\dfrac{\mu_0}{2\pi} = 2\times10^{-7}\ \text{T}\cdot\text{m/A}$
Elementary charge: $e = 1.602\times10^{-19}\ \text{C}$
Electron mass: $m_e = 9.11\times10^{-31}\ \text{kg}$
Electron volt: $1\ \text{eV} = e(1\ \text{V}) \approx 1.602\times10^{-19}\ \text{J}$
Charge is quantized: $q = ne,\ n\in\mathbb{Z}$ , and conserved in any isolated system

Electric Fields and Forces

Coulomb’s law and point-charge field

Force between point charges: $\vec{F} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r^2}\hat{r} = k\dfrac{q_1 q_2}{r^2}\hat{r}$
Field of a point charge: $\vec{E} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r^2}\hat{r}$
Force from a field: $\vec{F} = q\vec{E}$
Superposition (add vectors): $\vec{E}_{\text{net}} = \displaystyle\sum_i \vec{E}_i$
Like charges repel, opposite charges attract; the field points away from positive charge, toward negative

Continuous charge distributions

Densities: $\lambda = \dfrac{dq}{dL}$ , $\sigma = \dfrac{dq}{dA}$ , $\rho = \dfrac{dq}{dV}$
Field by integration: $\vec{E} = \dfrac{1}{4\pi\varepsilon_0}\displaystyle\int \dfrac{dq}{r^2}\hat{r}$
On-axis ring: $E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Qx}{(x^2+R^2)^{3/2}}$
On-axis disk: $E = \dfrac{\sigma}{2\varepsilon_0}\left(1 - \dfrac{x}{\sqrt{x^2+R^2}}\right)$
Exploit symmetry: keep only non-cancelling components, then integrate

Electric dipole

Dipole moment: $\vec{p} = q\vec{d}$ (points from $-q$ to $+q$ )
Far field on axis: $E \approx \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}$ (falls as $1/r^3$ )
Torque in a uniform field: $\vec{\tau} = \vec{p}\times\vec{E},\quad \tau = pE\sin\phi$
Energy in a field: $U = -\vec{p}\cdot\vec{E}$ (minimized when aligned)

Charge in a uniform field

Constant acceleration: $\vec{a} = \dfrac{q\vec{E}}{m}$ , then apply kinematics (electrical analog of projectile motion)

Gauss’s Law and Symmetries

Flux and the law

Electric flux: $\Phi_E = \displaystyle\int \vec{E}\cdot d\vec{A} = EA\cos\theta$ (uniform, flat)
Gauss’s law: $\displaystyle\oint \vec{E}\cdot d\vec{A} = \dfrac{Q_{\text{enc}}}{\varepsilon_0}$
Always true; computationally useful only when symmetry lets $E$ come out of the integral
Choose a Gaussian surface where $\vec{E}$ is constant and parallel or perpendicular to $d\vec{A}$

Standard results

Infinite line ( $\lambda$ ): $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$ (coaxial cylinder)
Infinite sheet ( $\sigma$ ): $E = \dfrac{\sigma}{2\varepsilon_0}$ (pillbox; independent of distance)
Uniform solid sphere, charge $Q$ $Q$ , radius $R$ $R$ :
- Outside ( $r\ge R$ ): $E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r^2}$
- Inside ( $r<R$ ): $E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{Qr}{R^3}$ (grows linearly)
Spherical shell: outside acts like a point charge; inside $E = 0$

Electric Potential and Energy

Energy and potential

PE of two point charges: $U = k\dfrac{Qq}{r}$
PE of a system: $U = k\displaystyle\sum_{i<j}\dfrac{q_iq_j}{r_{ij}}$ (each pair once)
Potential of a point charge: $V = \dfrac{kQ}{r}$
Potential is a scalar — add with signs: $V = k\displaystyle\sum_i \dfrac{q_i}{r_i}$
Potential of a distribution: $V = k\displaystyle\int \dfrac{dq}{r}$
Energy of a charge in a potential: $U = qV$ , so $\Delta U = q\,\Delta V$
Work by the field: $W_{\text{field}} = -q\,\Delta V$ ; by an external agent: $W_{\text{ext}} = +q\,\Delta V$

Relating potential and field

Differential form: $\vec{E} = -\nabla V$ , and in 1D $E_x = -\dfrac{dV}{dx}$
Integral form: $V_b - V_a = -\displaystyle\int_a^b \vec{E}\cdot d\vec{r}$
Uniform-field plates: $\Delta V = -\vec{E}\cdot\vec{d}$ , magnitude $\lvert\Delta V\rvert = Ed$
$\vec{E}$ points from high to low potential and is perpendicular to equipotentials

On-axis distribution potentials

Ring: $V = \dfrac{kQ}{\sqrt{x^2+R^2}}$
Disk: $V = \dfrac{\sigma}{2\varepsilon_0}\left(\sqrt{x^2+R^2} - x\right)$
Solid sphere interior: $V(r) = \dfrac{kQ}{2R}\left(3 - \dfrac{r^2}{R^2}\right)$ for $r\le R$
$V$ is continuous everywhere, even where $E$ has a kink; inside a shell $V$ is constant (not zero)

Conductors and Capacitors

Conductors in electrostatic equilibrium

Field inside conducting material is zero; excess charge lives on the outer surface
Field just outside the surface: $E = \dfrac{\sigma}{\varepsilon_0}$ (perpendicular to the surface)
Charge density and field are largest where curvature is sharpest

Capacitance

Definition: $C = \dfrac{Q}{\Delta V}$ (depends on geometry and material, not on $Q$ or $\Delta V$ )
Parallel-plate: $C = \dfrac{\varepsilon_0 A}{d}$ , with field $E = \dfrac{\sigma}{\varepsilon_0} = \dfrac{Q}{\varepsilon_0 A}$
Cylindrical (coaxial): $C = \dfrac{2\pi\varepsilon_0 L}{\ln(b/a)}$
Spherical: $C = 4\pi\varepsilon_0\dfrac{ab}{b-a}$ ; isolated sphere: $C = 4\pi\varepsilon_0 a$
Method for any symmetric capacitor: Gauss for $E$ , integrate for $\Delta V$ , then $C = Q/\Delta V$

Combinations

Parallel (same voltage, charges add): $C_{\text{eq}} = C_1 + C_2 + \cdots$
Series (same charge, voltages add): $\dfrac{1}{C_{\text{eq}}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \cdots$

Energy and dielectrics

Stored energy: $U = \tfrac{1}{2}Q\,\Delta V = \dfrac{Q^2}{2C} = \tfrac{1}{2}C(\Delta V)^2$
Electric energy density: $u_E = \tfrac{1}{2}\varepsilon_0 E^2$
Force between plates: $F = \dfrac{Q^2}{2\varepsilon_0 A} = \tfrac{1}{2}QE$ (a plate cannot push on itself)
Dielectric fills gap: $C = \kappa C_0 = \dfrac{\kappa\varepsilon_0 A}{d}$
Disconnected ( $Q$ fixed): inserting dielectric drops $\Delta V$ and energy
Connected ( $\Delta V$ fixed): inserting dielectric raises $Q$ and energy

Circuits

Current, resistance, power

Current: $I = \dfrac{dQ}{dt}$ ; drift form: $I = nqAv_d$
Ohm’s law (ohmic only): $\Delta V = IR$
Resistance of a wire: $R = \dfrac{\rho L}{A}$
Microscopic Ohm’s law: $\vec{J} = \sigma_c\vec{E}$
Power: $P = I\,\Delta V = I^2 R = \dfrac{(\Delta V)^2}{R}$

Networks

Resistors in series (same current): $R_{\text{eq}} = R_1 + R_2 + \cdots$
Resistors in parallel (same voltage): $\dfrac{1}{R_{\text{eq}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots$
Junction rule (charge): $\displaystyle\sum I_{\text{in}} = \sum I_{\text{out}}$
Loop rule (energy): $\displaystyle\sum \Delta V = 0$
Resistor in current direction: $-IR$ ; battery $-$ to $+$ : $+\mathcal{E}$
Real battery terminal voltage: $V_{\text{terminal}} = \mathcal{E} - Ir$
Maximum power to a load when $R = r$

RC circuits

Time constant: $\tau = RC$
Charging loop equation: $R\dfrac{dQ}{dt} + \dfrac{Q}{C} = \mathcal{E}$
Charging: $Q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right)$ , $V_C = \mathcal{E}\left(1 - e^{-t/RC}\right)$ , $I(t) = \dfrac{\mathcal{E}}{R}e^{-t/RC}$
Discharging: $Q(t) = Q_0 e^{-t/RC}$ , $I(t) = -\dfrac{V_0}{R}e^{-t/RC}$
Discharge half-life: $t_{1/2} = RC\ln 2 \approx 0.693\,RC$
Limits: capacitor acts like a wire at $t = 0$ , like an open branch as $t\to\infty$
Ideal meters: ammeter (zero resistance, in series), voltmeter (infinite resistance, in parallel)

Magnetic Forces and Fields

Force on charges and currents

Lorentz force: $\vec{F} = q\vec{E} + q\vec{v}\times\vec{B}$
Magnetic force magnitude: $F = \lvert q\rvert vB\sin\theta$ (does no work; changes direction only)
Force on a wire: $\vec{F} = I\vec{L}\times\vec{B}$ , $F = ILB\sin\theta$
Circular motion: $\lvert q\rvert vB = \dfrac{mv^2}{r}\Rightarrow r = \dfrac{mv}{\lvert q\rvert B}$
Period (speed-independent): $T = \dfrac{2\pi m}{\lvert q\rvert B}$ , cyclotron frequency $f = \dfrac{\lvert q\rvert B}{2\pi m}$
Velocity selector (crossed fields): $v = \dfrac{E}{B}$
Force per length between parallel wires: $\dfrac{F}{L} = \dfrac{\mu_0 I_1 I_2}{2\pi d}$ (same direction attract)

Current loop as a dipole

Magnetic moment: $\vec{\mu} = NI\vec{A}$
Torque: $\vec{\tau} = \vec{\mu}\times\vec{B}$ , $\tau = NIAB\sin\theta$
Dipole energy: $U = -\vec{\mu}\cdot\vec{B} = -\mu B\cos\theta$ (lowest when aligned)

Fields from currents

Biot–Savart law: $d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec{\ell}\times\hat{r}}{r^2}$
Straight wire: $B = \dfrac{\mu_0 I}{2\pi r}$ (right-hand rule for direction)
On axis of a loop: $B = \dfrac{\mu_0 I R^2}{2(x^2+R^2)^{3/2}}$
Center of a loop ( $N$ turns): $B = \dfrac{\mu_0 NI}{2R}$
Inside a solenoid: $B = \mu_0 nI$ , where $n = N/L$
Inside a toroid: $B = \dfrac{\mu_0 NI}{2\pi r}$

Ampère’s law

$\displaystyle\oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}}$
Best with high symmetry (straight wires, solenoids, toroids); choose $\vec{B}$ parallel or perpendicular to $d\vec{\ell}$
Inside a uniform-current wire ( $r<a$ ): $B = \dfrac{\mu_0 I r}{2\pi a^2}$ (grows linearly)
Magnetic flux: $\Phi_B = \displaystyle\int \vec{B}\cdot d\vec{A} = BA\cos\theta$

Electromagnetic Induction

Faraday’s and Lenz’s laws

Faraday’s law: $\mathcal{E} = -\dfrac{d\Phi_B}{dt}$ ; for $N$ turns: $\mathcal{E} = -N\dfrac{d\Phi_B}{dt}$
Flux changes via changing $B$ , area, or orientation
Lenz’s law: induced current opposes the change in $\Phi_B$ (energy conservation)

Motional emf and generators

Sliding rod: $\mathcal{E} = B\ell v$ ; general: $\mathcal{E} = \displaystyle\oint (\vec{v}\times\vec{B})\cdot d\vec{\ell}$
Induced current: $I = \dfrac{\mathcal{E}}{R}$ ; power balance $P_{\text{mech}} = P_{\text{elec}} = I^2 R$
Rotating loop (AC generator): $\mathcal{E} = NBA\omega\sin(\omega t)$ , peak $\mathcal{E}_0 = NBA\omega$
Induced (nonconservative) E field: $\displaystyle\oint \vec{E}\cdot d\vec{\ell} = -\dfrac{d\Phi_B}{dt}$

Inductance and energy

Definition: $N\Phi_B = LI$
Back emf: $\mathcal{E}_L = -L\dfrac{dI}{dt}$ (opposes changes in current, not current itself)
Solenoid inductance: $L = \mu_0 n^2 A\ell$
Stored energy: $U_B = \tfrac{1}{2}LI^2$
Magnetic energy density: $u_B = \dfrac{B^2}{2\mu_0}$ (mirrors $u_E = \tfrac{1}{2}\varepsilon_0 E^2$ )

LR and LC circuits

LR loop: $\mathcal{E} - IR - L\dfrac{dI}{dt} = 0$ , time constant $\tau = \dfrac{L}{R}$
LR charging: $I(t) = \dfrac{\mathcal{E}}{R}\left(1 - e^{-Rt/L}\right)$ ; decay: $I(t) = I_0 e^{-Rt/L}$
Inductor blocks instantaneous current jumps; acts like a wire at steady state
LC oscillation: $\dfrac{d^2Q}{dt^2} + \dfrac{1}{LC}Q = 0$ , energy $\tfrac{1}{2}\dfrac{Q^2}{C} + \tfrac{1}{2}LI^2 = \text{const}$
LC frequency: $\omega = \dfrac{1}{\sqrt{LC}}$ , period $T = 2\pi\sqrt{LC}$

Ampère–Maxwell law

Displacement-current extension: $\displaystyle\oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}} + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$
Changing electric fields produce magnetic fields, completing the path to electromagnetic waves

Most Common AP Physics C: E&M Mistakes

Adding electric fields as scalars instead of vectors (potential adds as a scalar; field does not)
Forgetting that inside a conductor $E = 0$ , but $V$ is a nonzero constant
Misplacing the right-hand rule sign, especially for negative charges and Lenz’s law
Treating $E = \sigma/\varepsilon_0$ (conductor surface) and $E = \sigma/2\varepsilon_0$ (isolated sheet) as the same
Forgetting the $N$ factor in $\mathcal{E} = -N\,d\Phi_B/dt$ and $N\Phi_B = LI$
Mixing up capacitor and resistor combination rules (series capacitors add reciprocals)
Ignoring the capacitor-as-wire / inductor-as-wire limits when reading off $t=0$ and $t\to\infty$ states
Dropping units or leaving microfarads, nanocoulombs, and kilohms unconverted

Fast Problem-Solving Checklist

Identify the unit: field/force, flux, potential, capacitor, circuit, magnetic force, or induction.
Check for symmetry first — it decides between Gauss/Ampère and a Biot–Savart/Coulomb integral.
Decide scalar vs. vector: potential and energy add as scalars; fields and forces add as vectors.
For circuits, reduce series/parallel groups, then apply Ohm, Kirchhoff, and the $t=0$ / $t\to\infty$ limits.
Track units and convert prefixes before plugging in numbers.
Check sign and magnitude: does the field point the right way, and does energy go where it should?