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AP Physics C E&M Formula Sheet cheat sheet - grade 11-12

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Physics Grade 11-12

AP Physics C E&M Formula Sheet Cheat Sheet

A printable reference covering electrostatics, Gauss’s law, capacitance, circuits, magnetism, induction, and Maxwell’s equations for grades 11-12.

TopicsElectromagnetismElectricityMagnetismCircuitsPhysicsStudy GuideAP Physics
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This AP Physics C E&M formula sheet covers the core equations students need for electrostatics, electric potential, capacitance, circuits, magnetism, and electromagnetic induction. It is designed to help grades 11-12 students connect calculus-based formulas with the physical situations they describe. A clear reference is useful because E&M problems often require choosing between field, potential, energy, flux, and circuit models.

Key Facts

  • Coulomb’s law gives the electric force between point charges as F⃗=14πϵ0q1q2r2r^\vec{F} = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}\hat{r}.
  • The electric field is force per unit positive test charge, so E⃗=F⃗q\vec{E} = \frac{\vec{F}}{q} and for a point charge E⃗=14πϵ0qr2r^\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}.
  • Gauss’s law relates electric flux to enclosed charge by ∮E⃗⋅dA⃗=Qencϵ0\oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}.
  • Electric potential difference is related to electric field by ΔV=−∫abE⃗⋅dℓ⃗\Delta V = -\int_a^b \vec{E}\cdot d\vec{\ell}, and potential energy is U=qVU = qV.
  • Capacitance is defined by C=QΔVC = \frac{Q}{\Delta V}, and a parallel-plate capacitor has C=κϵ0AdC = \kappa\epsilon_0\frac{A}{d}.
  • Circuit equations include Ohm’s law V=IRV = IR, power P=IV=I2R=V2RP = IV = I^2R = \frac{V^2}{R}, and capacitor charging q(t)=CE(1−e−t/RC)q(t) = C\mathcal{E}(1 - e^{-t/RC}).
  • The magnetic force on a moving charge is F⃗B=qv⃗×B⃗\vec{F}_B = q\vec{v}\times\vec{B}, and the force on a current-carrying wire is F⃗B=IL⃗×B⃗\vec{F}_B = I\vec{L}\times\vec{B}.
  • Faraday’s law gives induced emf as E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}, where magnetic flux is ΦB=∫B⃗⋅dA⃗\Phi_B = \int \vec{B}\cdot d\vec{A}.

Vocabulary

Electric field
The electric field E⃗\vec{E} is the force per unit positive test charge at a point in space.
Electric flux
Electric flux ΦE=∫E⃗⋅dA⃗\Phi_E = \int \vec{E}\cdot d\vec{A} measures how much electric field passes through a surface.
Electric potential
Electric potential VV is electric potential energy per unit charge, defined by V=UqV = \frac{U}{q}.
Capacitance
Capacitance CC is the ratio of stored charge to potential difference, given by C=QΔVC = \frac{Q}{\Delta V}.
Magnetic flux
Magnetic flux ΦB=∫B⃗⋅dA⃗\Phi_B = \int \vec{B}\cdot d\vec{A} measures the amount of magnetic field passing through a surface.
Induced emf
Induced emf E\mathcal{E} is the voltage produced by a changing magnetic flux, described by E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}.

Common Mistakes to Avoid

  • Using E=kqr2E = \frac{kq}{r^2} inside any charge distribution, which is wrong because that formula applies directly to a point charge or a spherically symmetric distribution outside the charge.
  • Forgetting the negative sign in ΔV=−∫E⃗⋅dℓ⃗\Delta V = -\int \vec{E}\cdot d\vec{\ell}, which is wrong because electric potential decreases in the direction of the electric field.
  • Treating C=QΔVC = \frac{Q}{\Delta V} as meaning capacitance changes whenever QQ changes, which is wrong because capacitance depends on geometry and dielectric material for a fixed capacitor.
  • Using F=qvBF = qvB without checking direction or angle, which is wrong because the full magnitude is F=∣q∣vBsin⁡θF = |q|vB\sin\theta and the direction comes from v⃗×B⃗\vec{v}\times\vec{B}.
  • Ignoring Lenz’s law in induction problems, which is wrong because the negative sign in E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt} determines the direction of the induced current.

Practice Questions

  1. 1 A charge of +3.0 μC+3.0\ \mu\text{C} is placed 0.20 m0.20\ \text{m} from a charge of −2.0 μC-2.0\ \mu\text{C}. Find the magnitude of the electric force using k=8.99×109 N⋅m2/C2k = 8.99\times10^9\ \text{N}\cdot\text{m}^2/\text{C}^2.
  2. 2 A parallel-plate capacitor has area A=0.015 m2A = 0.015\ \text{m}^2, plate separation d=1.0×10−3 md = 1.0\times10^{-3}\ \text{m}, and dielectric constant κ=2.5\kappa = 2.5. Find CC using C=κϵ0AdC = \kappa\epsilon_0\frac{A}{d}.
  3. 3 A circular loop with area 0.040 m20.040\ \text{m}^2 is perpendicular to a uniform magnetic field that changes from 0.20 T0.20\ \text{T} to 0.80 T0.80\ \text{T} in 0.30 s0.30\ \text{s}. Find the magnitude of the average induced emf.
  4. 4 Explain why a point on a conductor in electrostatic equilibrium can have E⃗=0\vec{E} = 0 inside the material while excess charge remains on the surface.

Understanding AP Physics C E&M Formula Sheet

A useful first step is to separate force, field, potential, and energy. Force describes what happens to one particular charge. Field describes the influence produced in the space around source charges, independent of the test charge chosen to detect it.

Potential is different because it tracks energy changes per unit charge. Electric fields have direction, while electric potential has no direction. This difference explains why potentials from several charges are often easier to add than fields.

A positive charge naturally moves toward lower potential energy, but whether it moves toward higher or lower electric potential depends on the sign of that charge. Keep charge sign visible throughout a problem. It controls directions, energy changes, and whether work is done by the field or by an outside agent.

Calculus becomes important when charge is spread over a line, surface, or volume. Break the distribution into tiny pieces, find the contribution from one piece, then add every contribution with an integral. The hardest part is usually not the integration.

It is choosing the geometry and directions correctly. Symmetry can remove much of the work. A spherical distribution, a very long charged wire, or a large flat sheet can have enough symmetry for Gauss’s law to find the field directly.

The Gaussian surface is an imaginary counting surface, not a physical object. It works best when the field has a constant size over part of that surface or is perpendicular to it. Enclosed charge matters for net flux, yet charges outside the surface can still change the field at points on the surface.

Capacitors store separated charge and energy in an electric field. Adding an insulating material between capacitor plates changes how the material responds to the field. Its molecules shift slightly, which weakens the internal field for a given amount of free charge.

This lets the capacitor hold more charge at the same voltage. In a charging circuit, current is largest at the start because the capacitor has little voltage across it. As charge builds up, the capacitor opposes further charging, so current falls smoothly rather than stopping suddenly.

The product of resistance and capacitance sets the time scale. After one time constant, the process is not complete, but it has made a predictable amount of progress.

Check units often. Resistance times capacitance has units of time, which helps catch mistakes.

Magnetic fields act differently from electric fields because a magnetic force on a moving charge is sideways to its motion. A magnetic field can bend a charged particle’s path without changing its speed or kinetic energy. This idea appears in mass spectrometers, particle detectors, electric motors, and the curved motion of charged particles in Earth’s magnetic field.

Induction depends on changing magnetic flux, not simply on the presence of a magnetic field. Flux can change when field strength changes, when loop area changes, or when the angle between a loop and field changes. The induced current creates effects that oppose the change that produced it.

This is Lenz’s law and it protects energy conservation. Maxwell’s equations bring these ideas together by showing that changing electric fields produce magnetic fields and changing magnetic fields produce electric fields. That feedback allows electromagnetic waves, including light, to travel through empty space.