2027 exam · Equation sheet

AP® Physics C: Electricity and Magnetism equation sheet (2027), explained

The official Physics C: E&M reference sheet starts with a page of constants, unit symbols, prefixes, trig values and exam conventions. Then comes a two-column Electricity and Magnetism block, followed by the same Mechanics page and geometry, vectors and calculus page you get on the Physics C: Mechanics sheet. Below, we go through every E&M equation and every constant: what each symbol means, its unit, when to use it, the mistake students make most, and which topic teaches it. For the mechanics and math pages, we link to where they're explained on the Mechanics sheet page.

The official sheet: AP Physics C: Electricity and Magnetism Exam Reference Information (PDF, College Board) (opens in a new tab). Keep it open next to this page. We link to it instead of copying it, so you always see College Board's current version.

Checked against the 2027 version on October 5, 2026. The explanations are ours, not College Board's.

Using the sheet on exam day

The sheet is available for the whole exam, on paper and in the testing app. Learn where things sit before exam day: the E&M block is one page with two columns, and the left column runs from Coulomb's law down to power, while the right column runs from resistor combinations down to LC circuits.
Read the symbol lists next to each column. The sheet reuses letters: $V$ is electric potential or volume, $\rho$ is resistivity or charge density, $n$ is loops per unit length while $N$ is the total number of loops, and the same-looking $\varepsilon$ means electric permittivity in the left column but emf in the right one.
The conventions box on the first page is worth a look: potential is zero infinitely far from an isolated point charge, current points the way positive charges would drift, resistors and bulbs are ohmic, and capacitors are air-filled with $\kappa = 1.0$ unless a problem says otherwise.
The sheet won't tell you when an equation applies. Gauss's law and Ampère's law only give you a field quickly when there's symmetry, $B_{\text{sol}} = \mu_0 nI$ is for a long solenoid, and $C = \frac{\kappa\varepsilon_0 A}{d}$ is for parallel plates.
Use the prefix table before you plug in: μC, nC, pF and mH all need converting to C, F and H. The trig table also gives 37° and 53° (the 3-4-5 triangle), which turn up in flux and force problems with angles.

Constants and conversions

Coulomb constant: $\displaystyle k = \frac{1}{4\pi\varepsilon_0} = 9.0\times 10^{9}\ (\text{N}\cdot\text{m}^2)/\text{C}^2$; The shortcut number in Coulomb's law, point-charge fields and potentials. Writing $k$ or $\frac{1}{4\pi\varepsilon_0}$ means the same thing.
Vacuum permittivity: $\displaystyle \varepsilon_0 = 8.85\times 10^{-12}\ \text{C}^2/(\text{N}\cdot\text{m}^2)$; Shows up in Gauss's law, $C = \frac{\kappa\varepsilon_0 A}{d}$ and $\kappa = \frac{\varepsilon}{\varepsilon_0}$ . It sets how strong the field from a given charge is in empty space.
Vacuum permeability: $\displaystyle \mu_0 = 4\pi\times 10^{-7}\ (\text{T}\cdot\text{m})/\text{A}$; The magnetic partner of $\varepsilon_0$ : it's in the Biot–Savart law, Ampère's law and the solenoid field. Leave the $\pi$ in until the end, since it often cancels.
Proton mass: $\displaystyle m_p = 1.67\times 10^{-27}\ \text{kg}$; For a proton speeding up through a potential difference or circling in a magnetic field.
Neutron mass: $\displaystyle m_n = 1.67\times 10^{-27}\ \text{kg}$; Rare in E&M since neutrons have no charge, but it helps when you need the mass of a nucleus or an ion.
Electron mass: $\displaystyle m_e = 9.11\times 10^{-31}\ \text{kg}$; For an electron accelerated by a field or bent by a magnetic field. It's about 1,800 times lighter than a proton, so it speeds up far more for the same energy.
Elementary charge: $\displaystyle e = 1.60\times 10^{-19}\ \text{C}$; The size of the charge on one proton or electron. An electron's charge is $-e$ , so remember the minus sign in $\Delta U_E = q\Delta V$ and $\vec{F}_B = q(\vec{v}\times\vec{B})$ .
Electron volt: $\displaystyle 1\ \text{eV} = 1.60\times 10^{-19}\ \text{J}$; The energy one elementary charge gains moving through 1 V. Handy for particle energies: an electron through 200 V gains 200 eV.
Speed of light: $\displaystyle c = 3.00\times 10^{8}\ \text{m/s}$; Use it to check that a particle's speed is reasonable. The sheet doesn't print this, but $c$ also links $\varepsilon_0$ and $\mu_0$ : $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$ .
Unified atomic mass unit: $\displaystyle 1\ \text{u} = 1.66\times 10^{-27}\ \text{kg} = 931\ \text{MeV}/c^2$; Converts an ion's or nucleus's mass from u to kg before you use it in a force or energy equation.
Universal gravitational constant: $\displaystyle G = 6.67\times 10^{-11}\ \text{m}^3/(\text{kg}\cdot\text{s}^2) = 6.67\times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2$; Comes up when you compare gravity between two particles with the electric force between them; the electric force usually wins by a huge factor.
Acceleration due to gravity at Earth's surface: $\displaystyle g = 9.8\ \text{m/s}^2$; For a charged object that also has weight, like a charged drop held still between parallel plates where $qE = mg$ .
Gravitational field strength at Earth's surface: $\displaystyle g = 9.8\ \text{N/kg}$; The same number as above, written as force per kilogram. It's the gravity version of an electric field, which is force per coulomb.

Electric force and field

How charges push and pull, and the field that describes that push at every point.

Coulomb's law

\displaystyle \lvert \vec{F}_E \rvert = \frac{1}{4\pi\varepsilon_0}\frac{\lvert q_1 q_2 \rvert}{r^2} = k\frac{\lvert q_1 q_2 \rvert}{r^2}

What the symbols mean

$\lvert \vec{F}_E \rvert$: size of the electric force each charge puts on the other; Unit: N
$q_1, q_2$: the two point charges; Unit: C
$r$: distance between the charges, center to center; Unit: m
$\varepsilon_0$: vacuum permittivity; Unit: C²/(N·m²)
$k$: Coulomb constant, $\frac{1}{4\pi\varepsilon_0}$; Unit: N·m²/C²

Use it when: Two point charges (or charged spheres) and their separation are given, and you want the force between them. The absolute values give you only the size; like charges repel and opposite charges attract.

Watch out: Forgetting to square $r$ , or forgetting to turn μC into C. With charged spheres, $r$ is the distance between their centers, not between their surfaces.

Try it: A 2.0 μC charge and a −3.0 μC charge are 0.30 m apart. How big is the force between them?

Answer: $F = (9.0\times 10^{9})\frac{(2.0\times 10^{-6})(3.0\times 10^{-6})}{(0.30)^2} = 0.60$ N. The charges are opposite, so it's attractive.

Learn it: 8.1 Electric Charge and Electric Force

Electric field from force

\displaystyle \vec{E} = \frac{\vec{F}_E}{q}

What the symbols mean

$\vec{E}$: electric field at a point; Unit: N/C (same as V/m)
$\vec{F}_E$: electric force on a charge placed at that point; Unit: N
$q$: the charge feeling the force (the test charge); Unit: C

Use it when: You know the force on a charge and want the field, or you know the field and want the force, $\vec{F}_E = q\vec{E}$ .

Watch out: Forgetting that a negative charge feels a force opposite to the field. The field's direction is always the direction of the force on a positive charge.

Learn it: 8.3 Electric Fields

Electric field of a continuous charge

\displaystyle \vec{E} = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r^2}\hat{r}

What the symbols mean

$dq$: a tiny piece of the charge; Unit: C
$r$: distance from that piece to the point where you want the field; Unit: m
$\hat{r}$: unit vector pointing from the piece toward that point; Unit: none

Use it when: Charge is spread along a rod, ring or arc and you need the field at a point. Write $dq$ using a charge density, like $dq = \lambda\,dx$ , then integrate.

Watch out: Integrating the field's size without splitting it into components. Use symmetry to see which components cancel, and only integrate the one that's left.

Learn it: 8.4 Electric Fields of Charge Distributions

Flux and Gauss's law

How much field passes through a surface, and how that's tied to the charge inside a closed one.

Electric flux

\displaystyle \Phi_E = \int \vec{E}\cdot d\vec{A}

What the symbols mean

$\Phi_E$: electric flux through the surface; Unit: N·m²/C
$\vec{E}$: electric field at each bit of the surface; Unit: N/C
$d\vec{A}$: a tiny piece of area, as a vector pointing straight out of the surface; Unit: m²

Use it when: You want how much field passes through a surface. For a uniform field and a flat surface it becomes $\Phi_E = EA\cos\theta$ .

Watch out: Measuring $\theta$ from the surface instead of from the area vector, which sticks straight out of the surface. Field running along a surface gives zero flux.

Learn it: 8.5 Electric Flux

Gauss's law

\displaystyle \oint \vec{E}\cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}

What the symbols mean

$\oint$: integral over a closed surface (the Gaussian surface); Unit: none
$d\vec{A}$: a tiny piece of the closed surface, pointing outward; Unit: m²
$q_{\text{enc}}$: net charge inside the closed surface; Unit: C
$\varepsilon_0$: vacuum permittivity; Unit: C²/(N·m²)

Use it when: Charge has spherical, cylindrical or flat-sheet symmetry and you want the field. Pick a surface where $E$ is constant and straight through it, so the left side becomes $E$ times an area.

Watch out: Using all the charge instead of only the charge inside your Gaussian surface. Charges outside still change the field, but add nothing to the net flux.

Try it: A closed surface surrounds a 4.0 nC charge. What's the total electric flux through it?

Answer: $\Phi_E = \frac{q_{\text{enc}}}{\varepsilon_0} = \frac{4.0\times 10^{-9}}{8.85\times 10^{-12}} \approx 4.5\times 10^{2}$ N·m²/C, whatever the surface's shape.

Learn it: 8.6 Gauss’s Law

Total charge from a charge density

\displaystyle Q_{\text{total}} = \int \rho(r)\,dV

What the symbols mean

$Q_{\text{total}}$: total charge in the region; Unit: C
$\rho(r)$: volume charge density, which can change with distance r from the center (here ρ is charge density, not resistivity); Unit: C/m³
$dV$: a tiny piece of volume (here V is volume, not potential); Unit: m³

Use it when: A sphere or cylinder's charge density changes with $r$ and you need the enclosed charge for Gauss's law. For a sphere, use thin shells, $dV = 4\pi r^2\,dr$ .

Watch out: Multiplying $\rho$ by the whole volume when $\rho$ isn't constant. If it depends on $r$ , you have to integrate shell by shell.

Learn it: 8.4 Electric Fields of Charge Distributions · 8.6 Gauss’s Law

Electric potential and energy

Energy stored in arrangements of charge, and potential, which is that energy per coulomb.

Potential energy of two point charges

\displaystyle U_E = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r}

What the symbols mean

$U_E$: electric potential energy of the pair; Unit: J
$q_1, q_2$: the two charges, with their signs; Unit: C
$r$: distance between them; Unit: m

Use it when: You want the energy stored in a pair of charges, or the work needed to bring them together from far away. For several charges, add up $U_E$ for every pair.

Watch out: Dropping the signs. Unlike Coulomb's law there are no absolute values here: opposite charges give negative $U_E$ . The zero is when the charges are infinitely far apart.

Learn it: 9.1 Electric Potential Energy

Electric potential of a continuous charge

\displaystyle V = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r}

What the symbols mean

$V$: electric potential at a point (here V is potential, not volume); Unit: V (J/C)
$dq$: a tiny piece of the charge, with its sign; Unit: C
$r$: distance from that piece to the point; Unit: m

Use it when: Charge is spread over a rod, ring or disk and you want the potential at a point. For one point charge it's just $V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$ .

Watch out: Treating potential like a vector and canceling parts by symmetry. Potential is a plain number, so every piece adds; only the signs of the charges can cancel.

Learn it: 9.2 Electric Potential

Potential difference from the field

\displaystyle \Delta V = -\int_a^b \vec{E}\cdot d\vec{r}

What the symbols mean

$\Delta V$: change in potential going from point a to point b, $V_b - V_a$; Unit: V
$\vec{E}$: electric field along the path; Unit: N/C
$d\vec{r}$: a tiny step along the path from a to b; Unit: m

Use it when: You know the field (often from Gauss's law) and want the potential difference between two points, like the plates of a capacitor. In a uniform field it becomes $\lvert\Delta V\rvert = Ed$ .

Watch out: Losing the minus sign. Moving along the field always takes you to lower potential, so check that your sign agrees.

Learn it: 9.2 Electric Potential · 10.3 Capacitors

Field from the slope of potential

\displaystyle E_x = -\frac{dV}{dx}

What the symbols mean

$E_x$: x-component of the electric field; Unit: V/m (same as N/C)
$\frac{dV}{dx}$: how fast the potential changes with position; Unit: V/m

Use it when: You have $V$ as a function of $x$ , or a graph of $V$ against $x$ , and want the field. The field is the negative slope.

Watch out: Thinking $E = 0$ wherever $V = 0$ . The field depends on how steeply $V$ changes, not on the value of $V$ itself.

Learn it: 9.2 Electric Potential

Change in potential energy across a potential difference

\displaystyle \Delta U_E = q\Delta V

What the symbols mean

$\Delta U_E$: change in electric potential energy; Unit: J
$q$: the moving charge, with its sign; Unit: C
$\Delta V$: change in potential it moves through; Unit: V

Use it when: A charge moves between two points at different potentials and you want its energy change or its speed. With no other forces doing work, $\Delta K = -\Delta U_E$ .

Watch out: Dropping the sign of an electron's charge. Electrons gain kinetic energy moving toward higher potential, the opposite of positive charges.

Try it: An electron starts at rest at 0 V and moves to a point at +200 V. How does its potential energy change?

Answer: $\Delta U_E = q\Delta V = (-1.60\times 10^{-19})(200) = -3.2\times 10^{-17}$ J, so it gains $3.2\times 10^{-17}$ J (200 eV) of kinetic energy.

Learn it: 9.3 Conservation of Electric Energy

Capacitors and dielectrics

Devices that store charge and energy, and what filling them with an insulator does.

Capacitance

\displaystyle C = \frac{Q}{\Delta V}

What the symbols mean

$C$: capacitance; Unit: F (C/V)
$Q$: size of the charge on either plate; Unit: C
$\Delta V$: potential difference between the plates; Unit: V

Use it when: Any two of charge, voltage and capacitance are given and you want the third. To find $C$ for spherical or cylindrical capacitors, use Gauss's law for $E$ , integrate for $\Delta V$ , then divide.

Watch out: Thinking $C$ changes when you change $Q$ or $\Delta V$ . It depends only on the geometry and what's between the plates; $Q$ and $\Delta V$ rise together.

Learn it: 10.3 Capacitors

Parallel-plate capacitance

\displaystyle C = \frac{\kappa\varepsilon_0 A}{d}

What the symbols mean

$\kappa$: dielectric constant of the material between the plates (1.0 for air); Unit: none
$A$: area of one plate; Unit: m²
$d$: gap between the plates; Unit: m

Use it when: You're given plate area, gap and filling, and want the capacitance, or you're asked what happens when the plates move or a dielectric slides in.

Watch out: Not converting units: 1 cm² is $1\times 10^{-4}$ m² and 1 mm is $1\times 10^{-3}$ m. Small slips here throw the answer off by powers of ten.

Try it: Air-filled plates of area 0.010 m² sit 1.0 mm apart and are connected to a 12 V battery. Find the capacitance and the charge.

Answer: $C = \frac{(1.0)(8.85\times 10^{-12})(0.010)}{1.0\times 10^{-3}} \approx 8.9\times 10^{-11}$ F (about 89 pF), and $Q = C\Delta V \approx 1.1\times 10^{-9}$ C.

Learn it: 10.3 Capacitors · 10.4 Dielectrics

Energy stored in a capacitor

\displaystyle U_C = \frac{1}{2}Q\Delta V

What the symbols mean

$U_C$: energy stored in the capacitor; Unit: J
$Q$: charge on either plate; Unit: C
$\Delta V$: potential difference across it; Unit: V

Use it when: You want the energy a charged capacitor holds. Swap in $Q = C\Delta V$ to get the other forms, $\frac{Q^2}{2C}$ and $\frac{1}{2}C(\Delta V)^2$ .

Watch out: Forgetting the $\frac{1}{2}$ . The battery does $Q\Delta V$ of work, but only half ends up stored in the capacitor.

Learn it: 10.3 Capacitors

Dielectric constant

\displaystyle \kappa = \frac{\varepsilon}{\varepsilon_0}

What the symbols mean

$\kappa$: dielectric constant; Unit: none
$\varepsilon$: electric permittivity of the material (not emf here); Unit: C²/(N·m²)
$\varepsilon_0$: permittivity of empty space; Unit: C²/(N·m²)

Use it when: You're told a material's permittivity and need its dielectric constant, or the reverse. Filling the gap between a capacitor's plates with a dielectric multiplies its $C$ by $\kappa$ .

Watch out: Mixing this $\varepsilon$ up with emf. The sheet uses the same-looking letter for both: in this equation it's permittivity, a property of a material.

Learn it: 10.4 Dielectrics

Current and resistance

Charge on the move, what resists it, and the power it delivers.

Current

\displaystyle I = \frac{dq}{dt}

What the symbols mean

$I$: current; Unit: A (C/s)
$\frac{dq}{dt}$: rate charge flows past a point; Unit: C/s

Use it when: You know how charge changes with time and want the current, or you integrate a current to find the charge that flowed.

Watch out: Getting the direction wrong. Current points the way positive charges would drift, which is opposite to the electrons actually moving in a wire.

Learn it: 11.1 Electric Current

Current from current density

\displaystyle I = \int \vec{J}\cdot d\vec{A}

What the symbols mean

$\vec{J}$: current density, current per unit area; Unit: A/m²
$d\vec{A}$: a tiny piece of the wire's cross section; Unit: m²

Use it when: Current isn't spread evenly across a wire, so $J$ depends on $r$ , and you want the total current. Use rings of area $dA = 2\pi r\,dr$ .

Watch out: Multiplying $J$ by $\pi R^2$ when $J$ changes across the wire. That only works when $J$ is the same everywhere.

Learn it: 11.1 Electric Current

Field and current density in a material

\displaystyle \vec{E} = \rho\vec{J}

What the symbols mean

$\vec{E}$: electric field inside the conductor; Unit: V/m
$\rho$: resistivity of the material (here ρ is resistivity, not charge density); Unit: Ω·m
$\vec{J}$: current density; Unit: A/m²

Use it when: You want the field that drives current in a wire, or the current density a field produces. It's Ohm's law written for one point in a material.

Watch out: Reading $\rho$ as charge density. In this equation, and in $R = \frac{\rho\ell}{A}$ , it's resistivity.

Learn it: 11.3 Resistance, Resistivity, and Ohm’s Law

Resistance from resistivity

\displaystyle R = \frac{\rho\ell}{A}

What the symbols mean

$R$: resistance; Unit: Ω
$\rho$: resistivity of the material; Unit: Ω·m
$\ell$: length of the wire; Unit: m
$A$: cross-sectional area of the wire; Unit: m²

Use it when: You're given a wire's material, length and thickness and want its resistance, or asked how resistance changes if you stretch or thicken it.

Watch out: Using the diameter in $A = \pi r^2$ . Halve the diameter first, and convert mm to m.

Try it: A copper wire ( $\rho = 1.7\times 10^{-8}$ Ω·m) is 10 m long with a cross section of $1.0\times 10^{-6}$ m². Find its resistance.

Answer: $R = \frac{(1.7\times 10^{-8})(10)}{1.0\times 10^{-6}} = 0.17$ Ω.

Learn it: 11.3 Resistance, Resistivity, and Ohm’s Law

Ohm's law

\displaystyle I = \frac{\Delta V}{R}

What the symbols mean

$I$: current through the element; Unit: A
$\Delta V$: potential difference across that same element; Unit: V
$R$: its resistance; Unit: Ω

Use it when: Any two of current, voltage and resistance for one resistor (or a whole circuit's equivalent) are known and you want the third. The sheet assumes resistors and bulbs are ohmic.

Watch out: Using the battery's voltage across one resistor that shares the circuit with others. Use the voltage across that resistor only.

Learn it: 11.3 Resistance, Resistivity, and Ohm’s Law

Electric power

\displaystyle P = I\Delta V

What the symbols mean

$P$: rate energy is delivered or used; Unit: W
$I$: current through the element; Unit: A
$\Delta V$: potential difference across it; Unit: V

Use it when: You want the power a resistor uses, a battery supplies, or how bright a bulb is. For a resistor, combine with Ohm's law: $P = I^2R = \frac{(\Delta V)^2}{R}$ .

Watch out: Mixing the current through one element with the voltage across a different one. Both must belong to the same element.

Learn it: 11.4 Electric Power

Combining resistors and capacitors

Shrinking a circuit to one equivalent element, and how fast an RC circuit changes.

Resistors in series

\displaystyle R_{\text{eq},s} = \sum_i R_i

What the symbols mean

$R_{\text{eq},s}$: equivalent resistance of resistors in series; Unit: Ω
$R_i$: each resistor's resistance; Unit: Ω

Use it when: Resistors sit one after another on a single path, so they carry the same current. The total is always bigger than any one of them.

Watch out: Calling resistors in series just because they're drawn in a line. They're in series only if no branch splits off between them.

Learn it: 11.5 Compound Direct Current Circuits

Resistors in parallel

\displaystyle \frac{1}{R_{\text{eq},p}} = \sum_i \frac{1}{R_i}

What the symbols mean

$R_{\text{eq},p}$: equivalent resistance of resistors in parallel; Unit: Ω
$R_i$: each resistor's resistance; Unit: Ω

Use it when: Resistors connect across the same two points, so they share the same potential difference. The total is always smaller than the smallest one.

Watch out: Forgetting to flip at the end. The sum gives $\frac{1}{R_{\text{eq},p}}$ , so take the reciprocal before you report it.

Try it: A 6.0 Ω and a 3.0 Ω resistor are in parallel, and that pair is in series with a 4.0 Ω resistor across a 12 V battery. Find the current from the battery.

Answer: $\frac{1}{R_{\text{eq},p}} = \frac{1}{6.0} + \frac{1}{3.0}$ , so $R_{\text{eq},p} = 2.0$ Ω. The total is $2.0 + 4.0 = 6.0$ Ω, so $I = \frac{12}{6.0} = 2.0$ A.

Learn it: 11.5 Compound Direct Current Circuits

Capacitors in series

\displaystyle \frac{1}{C_{\text{eq},s}} = \sum_i \frac{1}{C_i}

What the symbols mean

$C_{\text{eq},s}$: equivalent capacitance of capacitors in series; Unit: F
$C_i$: each capacitor's capacitance; Unit: F

Use it when: Capacitors sit one after another on a single path; each holds the same charge. The total is smaller than the smallest one.

Watch out: Using the resistor rules. Capacitors combine the opposite way: series uses reciprocals, parallel just adds.

Learn it: 11.8 Resistor-Capacitor (RC) Circuits

Capacitors in parallel

\displaystyle C_{\text{eq},p} = \sum_i C_i

What the symbols mean

$C_{\text{eq},p}$: equivalent capacitance of capacitors in parallel; Unit: F
$C_i$: each capacitor's capacitance; Unit: F

Use it when: Capacitors connect across the same two points, so they share the same potential difference. Adding one is like making the plates bigger.

Watch out: Assuming parallel capacitors hold the same charge. They have the same voltage; the bigger capacitor holds more charge.

Learn it: 11.8 Resistor-Capacitor (RC) Circuits

Time constant of an RC circuit

\displaystyle \tau = R_{\text{eq}}C_{\text{eq}}

What the symbols mean

$\tau$: time constant (not torque here); Unit: s
$R_{\text{eq}}$: equivalent resistance the capacitor charges or discharges through; Unit: Ω
$C_{\text{eq}}$: equivalent capacitance; Unit: F

Use it when: You want how quickly a capacitor charges or discharges. After one $\tau$ , a charging capacitor has about 63% of its final charge, and a discharging one keeps about 37%.

Watch out: Using the wrong resistance. Use only the resistance in the path the charge actually flows through, which can be different for charging and discharging.

Learn it: 11.8 Resistor-Capacitor (RC) Circuits

Magnetic fields and forces

Forces on moving charges and currents, and the fields that currents make.

Gauss's law for magnetism

\displaystyle \oint \vec{B}\cdot d\vec{A} = 0

What the symbols mean

$\vec{B}$: magnetic field; Unit: T
$d\vec{A}$: a tiny piece of a closed surface, pointing outward; Unit: m²

Use it when: Explaining why magnetic field lines form closed loops and why there are no lone north or south poles: every line that leaves a closed surface comes back in.

Watch out: Applying the zero to an open surface. Only the net flux through a closed surface is zero; flux through a single loop of wire usually isn't.

Learn it: 12.1 Magnetic Fields

Magnetic force on a moving charge

\displaystyle \vec{F}_B = q\left(\vec{v}\times\vec{B}\right)

What the symbols mean

$\vec{F}_B$: magnetic force on the charge; Unit: N
$q$: the charge, with its sign; Unit: C
$\vec{v}$: velocity of the charge; Unit: m/s
$\vec{B}$: magnetic field; Unit: T

Use it when: A charge moves through a magnetic field and you want the force, its direction, or the radius of its circular path. The size is $qvB\sin\theta$ .

Watch out: Forgetting to flip the right-hand-rule answer for a negative charge like an electron.

Learn it: 12.2 Magnetism and Moving Charges

Biot–Savart law

\displaystyle d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\left(d\vec{\ell}\times\hat{r}\right)}{r^2}

What the symbols mean

$d\vec{B}$: field made by one tiny piece of wire; Unit: T
$\mu_0$: vacuum permeability; Unit: T·m/A
$I$: current in the wire; Unit: A
$d\vec{\ell}$: a tiny piece of wire, pointing the way the current flows; Unit: m
$\hat{r}$: unit vector from that piece to the point where you want the field; Unit: none
$r$: distance from the piece to that point; Unit: m

Use it when: You want the field from a current in a shape without enough symmetry for Ampère's law, like the center of a loop or arc, or a point near a short wire.

Watch out: Forgetting that the cross product is zero when $d\vec{\ell}$ and $\hat{r}$ point the same way. Pieces of wire lined up with the point add no field there.

Learn it: 12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law

Magnetic force on a current-carrying wire

\displaystyle \vec{F}_B = \int I\left(d\vec{\ell}\times\vec{B}\right)

What the symbols mean

$\vec{F}_B$: magnetic force on the wire; Unit: N
$I$: current in the wire; Unit: A
$d\vec{\ell}$: a tiny piece of wire, pointing the way the current flows; Unit: m
$\vec{B}$: magnetic field at that piece; Unit: T

Use it when: A current-carrying wire sits in a magnetic field and you want the force on it. For a straight wire in a uniform field, the size is $I\ell B\sin\theta$ .

Watch out: Using the whole wire's length when only part of it is in the field. Only the length inside the field feels a force.

Learn it: 12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law · 13.3 Induced Currents and Magnetic Forces

Ampère's law

\displaystyle \oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}}

What the symbols mean

$\oint$: integral around a closed path (the Amperian loop); Unit: none
$d\vec{\ell}$: a tiny step along the loop; Unit: m
$I_{\text{enc}}$: net current passing through the loop; Unit: A

Use it when: Current has a lot of symmetry (a long straight wire, a thick wire, a coaxial cable, a solenoid) and you want the field. Choose a loop where $B$ is constant along it.

Watch out: Counting all the current instead of just the part that passes through your loop. Inside a thick wire, only the current inside radius $r$ counts.

Learn it: 12.4 Ampère’s Law

Field inside a long solenoid

\displaystyle B_{\text{sol}} = \mu_0 nI

What the symbols mean

$B_{\text{sol}}$: field inside a long solenoid; Unit: T
$n$: number of loops per unit length; Unit: 1/m
$I$: current in the wire; Unit: A

Use it when: You want the field inside a long coil. It's nearly uniform inside and about zero just outside.

Watch out: Plugging in the total number of loops $N$ . Here $n$ is loops per meter, $n = \frac{N}{\ell}$ .

Try it: A long solenoid has 1,000 loops per meter and carries 2.0 A. What's the field inside?

Answer: $B = (4\pi\times 10^{-7})(1000)(2.0) \approx 2.5\times 10^{-3}$ T.

Learn it: 12.4 Ampère’s Law

Induction and inductance

Changing magnetic flux makes an emf, and coils that push back against changing current.

Magnetic flux

\displaystyle \Phi_B = \int \vec{B}\cdot d\vec{A}

What the symbols mean

$\Phi_B$: magnetic flux through the surface; Unit: T·m²
$\vec{B}$: magnetic field; Unit: T
$d\vec{A}$: a tiny piece of area, pointing straight out of the surface; Unit: m²

Use it when: You want how much field passes through a loop, the first step of every induction problem. When the field is the same everywhere and the loop lies flat, $\Phi_B = BA\cos\theta$ .

Watch out: Measuring $\theta$ from the plane of the loop instead of from the line straight out of it.

Learn it: 13.1 Magnetic Flux

Faraday's law

\displaystyle \varepsilon = \oint \vec{E}\cdot d\vec{\ell} = -\frac{d\Phi_B}{dt}

What the symbols mean

$\varepsilon$: induced emf around the loop (not permittivity here); Unit: V
$\vec{E}$: induced electric field along the loop; Unit: V/m
$d\vec{\ell}$: a tiny step around the loop; Unit: m
$\frac{d\Phi_B}{dt}$: rate the magnetic flux through the loop changes; Unit: T·m²/s (same as V)

Use it when: Flux through a loop changes because the field, the area, or the angle changes, and you want the emf or the induced current. The minus sign is Lenz's law: the emf opposes the change.

Watch out: Thinking a big steady flux makes an emf. Only a changing flux does; a strong field that stays put induces nothing.

Learn it: 13.2 Electromagnetic Induction

Emf in a coil of many loops

\displaystyle \lvert \varepsilon_{\text{sol}} \rvert = N\left\lvert \frac{d\Phi_B}{dt} \right\rvert

What the symbols mean

$\varepsilon_{\text{sol}}$: emf induced in the whole coil; Unit: V
$N$: total number of loops; Unit: none
$\Phi_B$: flux through one loop; Unit: T·m²

Use it when: Flux changes through a coil with many turns and you want the size of the emf. Each loop adds its own emf, so the total is $N$ times one loop's.

Watch out: Forgetting the factor $N$ , or using flux through the whole coil and then multiplying by $N$ again. $\Phi_B$ here is for one loop.

Try it: The flux through each loop of a 50-loop coil drops from 0.020 T·m² to zero in 0.10 s. What's the average emf?

Answer: $\lvert\varepsilon\rvert = N\frac{\Delta\Phi_B}{\Delta t} = 50\times\frac{0.020}{0.10} = 10$ V.

Learn it: 13.2 Electromagnetic Induction

Inductance of a solenoid

\displaystyle L_{\text{sol}} = \frac{\mu_{\text{core}}N^2 A}{\ell}

What the symbols mean

$L_{\text{sol}}$: inductance of the solenoid; Unit: H
$\mu_{\text{core}}$: magnetic permeability of the core inside the coil (μ₀ for an air or empty core); Unit: T·m/A
$N$: total number of loops; Unit: none
$A$: cross-sectional area of the coil; Unit: m²
$\ell$: length of the coil; Unit: m

Use it when: You're given a coil's size, turns and core and want its inductance, or asked how inductance changes if you double the turns or add an iron core.

Watch out: Forgetting to square $N$ , or using $n$ (loops per meter) here. This equation uses the total number of loops.

Learn it: 13.4 Inductance

Energy stored in an inductor

\displaystyle U_L = \frac{1}{2}LI^2

What the symbols mean

$U_L$: energy stored in the inductor's magnetic field; Unit: J
$L$: inductance; Unit: H
$I$: current through it; Unit: A

Use it when: You want the energy an inductor holds, or you're tracking energy moving between an inductor and a capacitor or resistor.

Watch out: Forgetting to square the current. Doubling the current makes four times the energy.

Learn it: 13.4 Inductance · 13.6 Circuits with Capacitors and Inductors (LC Circuits)

Self-induced emf

\displaystyle \varepsilon = -L\frac{dI}{dt}

What the symbols mean

$\varepsilon$: emf across the inductor; Unit: V
$L$: inductance; Unit: H
$\frac{dI}{dt}$: rate the current changes; Unit: A/s

Use it when: Current through an inductor is changing and you want the voltage across it, or you're writing the loop rule for an LR or LC circuit.

Watch out: Thinking a big current means a big emf. The emf depends on how fast the current changes; a steady current gives zero.

Learn it: 13.4 Inductance · 13.5 Circuits with Resistors and Inductors (LR Circuits)

Time constant of an LR circuit

\displaystyle \tau = \frac{L}{R_{\text{eq}}}

What the symbols mean

$\tau$: time constant; Unit: s
$L$: inductance; Unit: H
$R_{\text{eq}}$: equivalent resistance in the inductor's path; Unit: Ω

Use it when: You want how quickly the current in an LR circuit rises or falls after a switch flips. After one $\tau$ , a rising current is about 63% of its final value.

Watch out: Multiplying $L$ by $R$ the way you multiply $R$ by $C$ . For an LR circuit you divide: $\tau = \frac{L}{R_{\text{eq}}}$ .

Learn it: 13.5 Circuits with Resistors and Inductors (LR Circuits)

Angular frequency of an LC circuit

\displaystyle \omega_{LC} = \frac{1}{\sqrt{LC}}

What the symbols mean

$\omega_{LC}$: angular frequency of the charge and current oscillation; Unit: rad/s
$L$: inductance; Unit: H
$C$: capacitance; Unit: F

Use it when: A charged capacitor is connected to an inductor and you want how fast energy sloshes back and forth. The period is $T = \frac{2\pi}{\omega_{LC}}$ .

Watch out: Reporting $\omega$ as the frequency in Hz. Divide by $2\pi$ to get $f$ : for $L = 10$ mH and $C = 1.0$ μF, $\omega = 1.0\times 10^{4}$ rad/s but $f \approx 1.6\times 10^{3}$ Hz.

Learn it: 13.6 Circuits with Capacitors and Inductors (LC Circuits)

Pages shared with another sheet

Mechanics
The E&M booklet reprints the full Mechanics page from the Physics C: Mechanics sheet. You'll use it here for forces on charges, energy conservation, circular motion of charges in magnetic fields, Newton's second law for a sliding bar in a field, and the oscillation equations that match LC circuits.
Physics C: Mechanics equation sheet, explained →
Geometry, trigonometry, vectors, calculus and identities
The last page matches the Mechanics sheet too. Its dot and cross products are what you need for flux, $\vec{E}\cdot d\vec{A}$ and $\vec{B}\cdot d\vec{A}$ , and for magnetic force, $\vec{v}\times\vec{B}$ , and its derivative and integral rules cover RC and LR circuits.
Geometry, vectors and calculus, explained →

Not on the sheet: know these

The exam expects you to know these without being given them.

Kirchhoff's loop rule
$\displaystyle \sum \Delta V = 0$
The potential differences around any closed loop add to zero. It's how you write equations for unknown currents, and for the differential equations of RC and LR circuits.
Learn it: 11.6 Kirchhoff’s Loop Rule
Kirchhoff's junction rule
$\displaystyle \sum I_{\text{in}} = \sum I_{\text{out}}$
Current into a junction equals current out, since charge can't pile up. Together with the loop rule it solves multi-loop circuits.
Learn it: 11.7 Kirchhoff’s Junction Rule
Charging and discharging a capacitor
$\displaystyle q(t) = Q_{\text{max}}\left(1 - e^{-t/\tau}\right),\quad q(t) = Q_0 e^{-t/\tau}$
The sheet gives only $\tau = R_{\text{eq}}C_{\text{eq}}$ . You need these exponentials (and the matching current, which decays as $e^{-t/\tau}$ in both cases) to find charge or current at a given time.
Learn it: 11.8 Resistor-Capacitor (RC) Circuits
Current growing in an LR circuit
$\displaystyle I(t) = \frac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right)$
After a switch closes, current rises toward $\frac{\varepsilon}{R}$ with $\tau = \frac{L}{R}$ ; when the battery is removed it decays as $e^{-t/\tau}$ . You get it by solving the loop rule.
Learn it: 13.5 Circuits with Resistors and Inductors (LR Circuits)
Capacitors and inductors right after a switch and after a long time
An uncharged capacitor acts like a wire at first and like a break after a long time. An inductor does the opposite: a break at first (no sudden change in current), a plain wire after a long time. Many circuit questions only need these two snapshots.
Learn it: 11.8 Resistor-Capacitor (RC) Circuits · 13.5 Circuits with Resistors and Inductors (LR Circuits)
Field of a long straight wire
$\displaystyle B = \frac{\mu_0 I}{2\pi r}$
The sheet doesn't list it; you get it from Ampère's law with a circular loop. Field lines circle the wire, with direction from the right-hand rule.
Learn it: 12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law · 12.4 Ampère’s Law
Lenz's law
The induced current makes a magnetic field that opposes the change in flux. The sheet shows it only as a minus sign in Faraday's law; you need it to find the direction of induced current.
Learn it: 13.2 Electromagnetic Induction · 13.3 Induced Currents and Magnetic Forces
Motional emf
$\displaystyle \varepsilon = B\ell v$
A bar of length $\ell$ sliding at speed $v$ through a perpendicular field has this emf. It comes from Faraday's law, and it's the start of every sliding-bar-on-rails problem.
Learn it: 13.3 Induced Currents and Magnetic Forces
Conductors in electrostatic equilibrium
Inside a conductor the field is zero, any extra charge sits on its surface, the field just outside is perpendicular to the surface, and the whole conductor is at one potential. Gauss's law problems with conducting shells depend on this.
Learn it: 8.3 Electric Fields · 10.1 Electrostatics with Conductors · 10.2 Redistribution of Charge Between Conductors
Other forms of power and capacitor energy
$\displaystyle P = I^2R = \frac{(\Delta V)^2}{R},\quad U_C = \frac{Q^2}{2C} = \frac{1}{2}C(\Delta V)^2$
The sheet gives only $P = I\Delta V$ and $U_C = \frac{1}{2}Q\Delta V$ . These come from swapping in Ohm's law or $Q = C\Delta V$ , and save time when one quantity is missing.
Learn it: 10.3 Capacitors · 11.4 Electric Power