AP Physics C: E&M · 2027 exam
Equations and reference
Every E&M equation, constant, calculus rule and convention you'll have on test day, plus the Mechanics equations the sheet also carries. Same content as the official sheet, dark mode, printable.
Section 1
Electricity
Electrostatics
\(|\vec{F}_E| = \frac{1}{4\pi\varepsilon_0}\frac{|q_1q_2|}{r^2} = k\frac{|q_1q_2|}{r^2}\)
\(\vec{E} = \frac{\vec{F}_E}{q}\)
\(\vec{E} = \frac{1}{4\pi\varepsilon_0}\int\frac{dq}{r^2}\,\hat{r}\)
\(\Phi_E = \oint \vec{E}\cdot d\vec{A} = \frac{q_{\mathrm{enc}}}{\varepsilon_0}\)
\(q_{\mathrm{enc}} = \int \rho(r)\,dV\)
\(E_x = -\frac{dV}{dx}\)
\(\Delta V = -\int \vec{E}\cdot d\vec{r}\)
\(V = \frac{1}{4\pi\varepsilon_0}\sum_i \frac{q_i}{r_i}\)
\(V = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r}\)
\(U_E = qV = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}\)
Symbols
\(A\) = area
\(C\) = capacitance
\(d\) = distance
\(E\) = electric field
\(F\) = force
\(I\) = current
\(J\) = current density
\(\ell\) = length
\(P\) = power
\(q\) = charge
\(Q\) = charge
\(r\) = radius, distance, or position
\(R\) = resistance
\(t\) = time
\(U\) = potential energy
\(V\) = electric potential or volume
\(\varepsilon\) = electric permittivity
\(\rho\) = resistivity or charge density
\(\kappa\) = dielectric constant
\(\tau\) = time constant
\(\Phi\) = flux
Capacitors and Circuits
\(\Delta V = \frac{Q}{C}\)
\(C = \frac{\kappa\varepsilon_0 A}{d}\)
\(C_p = \sum_i C_i\)
\(\frac{1}{C_s} = \sum_i \frac{1}{C_i}\)
\(U_C = \frac{1}{2}Q\Delta V = \frac{1}{2}C(\Delta V)^2\)
\(I = \frac{dQ}{dt}\)
\(I = N e v_d A\)
\(I = \int \vec{J}\cdot d\vec{A}\)
\(\vec{E} = \rho\vec{J}\)
\(R = \frac{\rho\ell}{A}\)
\(I = \frac{\Delta V}{R}\)
\(R_s = \sum_i R_i\)
\(\frac{1}{R_p} = \sum_i \frac{1}{R_i}\)
\(P = I\Delta V\)
\(\tau_{RC} = RC\)
Symbols
\(N\) = number of charge carriers per unit volume
\(e\) = elementary charge
\(v_d\) = drift speed
\(C_p,\ R_p\) = parallel combination
\(C_s,\ R_s\) = series combination
\(\varepsilon_0\) = vacuum permittivity
\(k = 1/4\pi\varepsilon_0\) = Coulomb constant
Section 2
Magnetism
Magnetic Fields and Forces
\(\vec{F}_M = q\vec{v}\times\vec{B}\)
\(\vec{F}_M = \int I\,d\vec{\ell}\times\vec{B}\)
\(d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{\ell}\times\hat{r}}{r^2}\)
\(\oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{\mathrm{enc}}\)
\(B_{\mathrm{wire}} = \frac{\mu_0 I}{2\pi r}\)
\(B_{\mathrm{sol}} = \mu_0 n I\)
Symbols
\(A\) = area
\(B\) = magnetic field
\(C\) = capacitance
\(F\) = force
\(I\) = current
\(\ell\) = length
\(L\) = inductance
\(n\) = number of loops per unit length
\(N\) = number of loops
\(q\) = charge
\(r\) = radius, distance, or position
\(R\) = resistance
\(t\) = time
\(U\) = potential energy
\(v\) = velocity or speed
\(\varepsilon\) = emf
\(\mu\) = magnetic permeability
\(\tau\) = time constant
\(\Phi\) = flux
\(\omega\) = angular frequency
Induction, Inductors, Oscillation
\(\Phi_B = \int \vec{B}\cdot d\vec{A}\)
\(\varepsilon = \oint \vec{E}\cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}\)
\(\varepsilon = -N\frac{d\Phi_B}{dt}\)
\(\varepsilon = -L\frac{dI}{dt}\)
\(U_L = \frac{1}{2}LI^2\)
\(\tau_{LR} = \frac{L}{R}\)
\(\omega_{LC} = \frac{1}{\sqrt{LC}}\)
Right-hand rules
Force on a moving charge: fingers along \(\vec{v}\), curl toward \(\vec{B}\), thumb is \(\vec{F}\) on a POSITIVE charge; reverse for negative.
Field of a current: thumb along \(I\), fingers curl in the sense of \(\vec{B}\) around the wire.
Lenz's law: the induced current opposes the CHANGE in flux, not the flux.
Section 3
Constants and Conversion Factors
Coulomb constant
\(k = \frac{1}{4\pi\varepsilon_0} = 9.0\times10^{9}\ \mathrm{N\cdot m^2/C^2}\)
Vacuum permittivity
\(\varepsilon_0 = 8.85\times10^{-12}\ \mathrm{C^2/(N\cdot m^2)}\)
Vacuum permeability
\(\mu_0 = 4\pi\times10^{-7}\ \mathrm{(T\cdot m)/A}\)
Elementary charge
\(e = 1.60\times10^{-19}\ \mathrm{C}\)
\(1\ \mathrm{eV} = 1.60\times10^{-19}\ \mathrm{J}\)
Particle masses
\(m_p = 1.67\times10^{-27}\ \mathrm{kg}\)
\(m_n = 1.67\times10^{-27}\ \mathrm{kg}\)
\(m_e = 9.11\times10^{-31}\ \mathrm{kg}\)
Speed of light
\(c = 3.00\times10^{8}\ \mathrm{m/s}\)
Unified atomic mass unit
\(1\ \mathrm{u} = 1.66\times10^{-27}\ \mathrm{kg} = 931\ \mathrm{MeV}/c^2\)
Universal gravitational constant
\(G = 6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}\)
Gravity at Earth's surface
\(g = 9.8\ \mathrm{m/s^2} = 9.8\ \mathrm{N/kg}\)
Section 4
Mechanics (also on the sheet)
Translational Mechanics
\(v_x = v_{x0} + a_x t\)
\(x = x_0 + v_{x0}t + \frac{1}{2}a_xt^2\)
\(v_x^2 = v_{x0}^2 + 2a_x(x - x_0)\)
\(\Delta x = \int v_x(t)\,dt\)
\(\Delta v_x = \int a_x(t)\,dt\)
\(\vec{a}_{\mathrm{sys}} = \frac{\sum \vec{F}}{m_{\mathrm{sys}}} = \frac{\vec{F}_{\mathrm{net}}}{m_{\mathrm{sys}}}\)
\(|\vec{F}_g| = G\frac{m_1m_2}{r^2}\)
\(\vec{F}_s = -k\Delta\vec{x}\)
\(a_c = \frac{v^2}{r} = r\omega^2\)
\(K = \frac{1}{2}mv^2\)
\(W = \int_a^b \vec{F}\cdot d\vec{r}\)
\(\Delta U = -\int_a^b \vec{F}_{\mathrm{cf}}(r)\cdot d\vec{r}\)
\(F_x = -\frac{dU(x)}{dx}\)
\(P_{\mathrm{inst}} = \frac{dW}{dt}\)
\(\vec{p} = m\vec{v}\)
\(\vec{F}_{\mathrm{net}} = \frac{d\vec{p}}{dt}\)
Rotation and Oscillation
\(\omega = \frac{d\theta}{dt},\quad \alpha = \frac{d\omega}{dt}\)
\(v = r\omega,\quad a_T = r\alpha\)
\(\vec{\tau} = \vec{r}\times\vec{F}\)
\(I = \int r^2\,dm\)
\(\alpha_{\mathrm{sys}} = \frac{\tau_{\mathrm{net}}}{I_{\mathrm{sys}}}\)
\(K_{\mathrm{rot}} = \frac{1}{2}I\omega^2\)
\(\vec{L} = \vec{r}\times\vec{p} = I\vec{\omega}\)
\(T = \frac{2\pi}{\omega} = \frac{1}{f}\)
\(T_s = 2\pi\sqrt{\frac{m}{k}}\)
\(x = x_{\max}\cos(\omega t + \phi)\)
Where Mechanics reappears in E&M
A charge in a uniform \(\vec{B}\): \(qvB = \frac{mv^2}{r}\), so \(r = \frac{mv}{qB}\) and \(T = \frac{2\pi m}{qB}\).
An LC circuit is a harmonic oscillator: \(L\frac{d^2q}{dt^2} + \frac{q}{C} = 0\), \(\omega = 1/\sqrt{LC}\), energy trades between \(\frac{q^2}{2C}\) and \(\frac{1}{2}LI^2\).
Work by the field on a charge: \(W = -\Delta U = -q\Delta V\).
Section 5
Geometry and Trigonometry
Plane Figures
Rectangle: \(A = bh\)
Triangle: \(A = \frac{1}{2}bh\)
Circle: \(A = \pi r^2\)
Circle: \(C = 2\pi r\)
Arc length: \(s = r\theta\)
Solids
Rect. solid: \(V = \ell wh\)
Cylinder: \(V = \pi r^2\ell\)
Cylinder: \(S = 2\pi r\ell + 2\pi r^2\)
Sphere: \(V = \frac{4}{3}\pi r^3\)
Sphere: \(S = 4\pi r^2\)
Symbols
\(A\) = area
\(b\) = base
\(C\) = circumference
\(h\) = height
\(\ell\) = length
\(r\) = radius
\(s\) = arc length
\(S\) = surface area
\(V\) = volume
\(w\) = width
\(\theta\) = angle
Right Triangle
\(a^2 + b^2 = c^2\)
\(\sin\theta = \frac{a}{c}\)
\(\cos\theta = \frac{b}{c}\)
\(\tan\theta = \frac{a}{b}\)
Section 6
Vectors and Calculus
Vectors
\(\vec{A}\cdot\vec{B} = AB\cos\theta\)
\(|\vec{A}\times\vec{B}| = AB\sin\theta\)
\(\vec{r} = (A\hat{\imath} + B\hat{\jmath} + C\hat{k})\)
\(\vec{C} = \vec{A} + \vec{B}\)
\(\vec{C} = (A_x + B_x)\hat{\imath} + (A_y + B_y)\hat{\jmath}\)
Calculus
\(\frac{df}{dx} = \frac{df}{du}\frac{du}{dx}\)
\(\frac{d}{dx}(x^n) = nx^{n-1}\)
\(\frac{d}{dx}(e^{ax}) = ae^{ax}\)
\(\frac{d}{dx}(\ln ax) = \frac{1}{x}\)
\(\frac{d}{dx}[\sin(ax)] = a\cos(ax)\)
\(\frac{d}{dx}[\cos(ax)] = -a\sin(ax)\)
\(\int x^n\,dx = \frac{1}{n+1}x^{n+1},\ n \neq -1\)
\(\int e^{ax}\,dx = \frac{1}{a}e^{ax}\)
\(\int \frac{dx}{x+a} = \ln|x+a|\)
\(\int \cos(ax)\,dx = \frac{1}{a}\sin(ax)\)
\(\int \sin(ax)\,dx = -\frac{1}{a}\cos(ax)\)
Identities
\(\log(a\cdot b^x) = \log a + x\log b\)
\(\sin^2\theta + \cos^2\theta = 1\)
\(\sin(2\theta) = 2\sin\theta\cos\theta\)
\(\frac{\sin\theta}{\cos\theta} = \tan\theta\)
Section 7
Prefixes, Unit Symbols, and Trig Values
| Factor | Prefix | Symbol |
|---|---|---|
| \(10^{12}\) | tera | T |
| \(10^9\) | giga | G |
| \(10^6\) | mega | M |
| \(10^3\) | kilo | k |
| \(10^{-2}\) | centi | c |
| \(10^{-3}\) | milli | m |
| \(10^{-6}\) | micro | \(\mu\) |
| \(10^{-9}\) | nano | n |
| \(10^{-12}\) | pico | p |
| Quantity | Symbol | Quantity | Symbol |
|---|---|---|---|
| ampere | A | meter | m |
| coulomb | C | newton | N |
| electron volt | eV | ohm | \(\Omega\) |
| farad | F | second | s |
| henry | H | tesla | T |
| hertz | Hz | volt | V |
| joule | J | watt | W |
| kilogram | kg |
| \(\theta\) | \(0^\circ\) | \(30^\circ\) | \(37^\circ\) | \(45^\circ\) | \(53^\circ\) | \(60^\circ\) | \(90^\circ\) |
|---|---|---|---|---|---|---|---|
| \(\sin\theta\) | 0 | \(1/2\) | \(3/5\) | \(\sqrt{2}/2\) | \(4/5\) | \(\sqrt{3}/2\) | 1 |
| \(\cos\theta\) | 1 | \(\sqrt{3}/2\) | \(4/5\) | \(\sqrt{2}/2\) | \(3/5\) | \(1/2\) | 0 |
| \(\tan\theta\) | 0 | \(\sqrt{3}/3\) | \(3/4\) | 1 | \(4/3\) | \(\sqrt{3}\) | \(\infty\) |
Exam conventions
- Use an inertial frame of reference unless another frame is stated.
- Ignore air resistance unless it is stated.
- Treat springs and strings as ideal unless stated.
- The electric potential is zero at an infinite distance from an isolated point charge.
- The direction of current is the direction in which positive charges would drift.
- All batteries, wires, and meters are ideal unless otherwise stated.
Based on the AP Physics C: Electricity and Magnetism 2026 Exam Reference Information.
Reformatted as a study sheet, not an official College Board publication.