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AP Physics C Electricity and Magnetism Laws cheat sheet - grade 11-12

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Physics Grade 11-12

AP Physics C Electricity and Magnetism Laws Cheat Sheet

A printable reference covering Gauss's law, electric potential, capacitance, circuits, magnetic fields, induction, and Maxwell's equations for grades 11-12.

TopicsElectricityMagnetismElectromagnetismCircuitsPhysicsStudy GuideAP Physics
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AP Physics C Electricity and Magnetism connects electric charge, fields, circuits, magnetism, and changing fields using calculus-based laws. This cheat sheet helps students organize the major laws that appear across free-response and multiple-choice problems. It is especially useful for choosing the correct integral law, sign convention, or circuit relationship under time pressure.

The core ideas are that charges create electric fields, currents create magnetic fields, and changing fields can induce circulation in the other field. Gauss's law, Ampere's law, Faraday's law, and Maxwell's correction describe field behavior with symmetry and calculus. Circuits use charge conservation, energy conservation, and component laws such as V=IRV = IR and Q=CVQ = CV.

Key Facts

  • Coulomb's law gives the electric force between point charges as F⃗=kq1q2r2r^\vec{F} = k\frac{q_1 q_2}{r^2}\hat{r}, where k=14πϵ0k = \frac{1}{4\pi \epsilon_0}.
  • 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=kqr2r^\vec{E} = k\frac{q}{r^2}\hat{r}.
  • Gauss's law states ∮E⃗⋅dA⃗=Qencϵ0\oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}, which is most useful when symmetry makes EE constant on the Gaussian surface.
  • Electric potential difference is related to electric field by ΔV=−∫abE⃗⋅dℓ⃗\Delta V = -\int_a^b \vec{E}\cdot d\vec{\ell}, and electric 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} when filled with dielectric constant κ\kappa.
  • Kirchhoff's junction rule is ∑Iin=∑Iout\sum I_{\text{in}} = \sum I_{\text{out}}, and Kirchhoff's loop rule is ∑ΔV=0\sum \Delta V = 0 around any closed loop.
  • 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 of induction is 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 flux
Electric flux measures how much electric field passes through a surface and is calculated by ΦE=∫E⃗⋅dA⃗\Phi_E = \int \vec{E}\cdot d\vec{A}.
Gaussian surface
A Gaussian surface is an imaginary closed surface used with Gauss's law to relate electric flux to enclosed charge.
Electric potential
Electric potential is electric potential energy per unit charge, written as V=UqV = \frac{U}{q}.
Capacitance
Capacitance is a measure of how much charge a capacitor stores per potential difference, defined by C=QΔVC = \frac{Q}{\Delta V}.
Magnetic flux
Magnetic flux measures how much magnetic field passes through a surface and is given by ΦB=intB⃗⋅dA⃗\Phi_B = \\int \vec{B}\cdot d\vec{A}.
Induced emf
Induced emf is the voltage produced by changing magnetic flux, described by E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}.

Common Mistakes to Avoid

  • Using Gauss's law without symmetry is wrong because ∮E⃗⋅dA⃗=Qencϵ0\oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} is always true, but it only easily gives EE when the field is constant or has a simple direction on the surface.
  • Forgetting the negative sign in ΔV=−∫E⃗⋅dℓ⃗\Delta V = -\int \vec{E}\cdot d\vec{\ell} is wrong because electric field points in the direction of decreasing electric potential for a positive test charge.
  • Treating F⃗B=qv⃗×B⃗\vec{F}_B = q\vec{v}\times \vec{B} like ordinary multiplication is wrong because the magnetic force depends on direction and has magnitude FB=∣q∣vBsin⁡θF_B = |q|vB\sin \theta.
  • Adding capacitors like resistors is wrong because capacitors in parallel add as Ceq=C1+C2+⋯C_{\text{eq}} = C_1 + C_2 + \cdots, while capacitors in series satisfy 1Ceq=1C1+1C2+⋯\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots.
  • Ignoring displacement current in Ampere-Maxwell law is wrong because changing electric flux contributes according to ∮B⃗⋅dℓ⃗=μ0Ienc+μ0ϵ0dΦEdt\oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}} + \mu_0\epsilon_0\frac{d\Phi_E}{dt}.

Practice Questions

  1. 1 A point charge of +3.0 μC+3.0\ \mu\text{C} is located at the center of a spherical Gaussian surface of radius 0.20 m0.20\ \text{m}. What is the electric flux through the surface?
  2. 2 A parallel-plate capacitor has plate area 2.0×10−3 m22.0\times 10^{-3}\ \text{m}^2, plate separation 1.0×10−3 m1.0\times 10^{-3}\ \text{m}, and dielectric constant κ=4.0\kappa = 4.0. Find its capacitance.
  3. 3 A wire segment of length 0.50 m0.50\ \text{m} carries current 3.0 A3.0\ \text{A} perpendicular to a uniform magnetic field of magnitude 0.20 T0.20\ \text{T}. What is the magnetic force magnitude?
  4. 4 Why can Gauss's law determine the electric field easily for an infinite charged plane but not for an irregularly shaped charged object?

Understanding AP Physics C Electricity and Magnetism Laws

A major skill in this course is deciding what model fits the geometry. A single charge, a long charged wire, a flat sheet, and a spherical conductor produce fields with different patterns. Symmetry is not just a shortcut.

It tells you which directions are possible for a field and which components must cancel. For example, at the center of a symmetric charge arrangement, contributions can cancel even when each individual charge creates a strong field.

Draw the geometry before choosing an equation. Mark directions, distances, and any planes or axes of symmetry.

Electric potential is often easier to handle than electric field because potential is a scalar. It has no direction, so contributions from several charges add as ordinary signed numbers. This is useful when finding energy changes or motion caused by electric forces.

A positive charge naturally moves toward lower electric potential when electric potential energy decreases. An electron has negative charge, so its energy changes in the opposite way. Keep potential and potential energy separate.

Voltage belongs to locations in space or circuit points. Potential energy belongs to a particular charged object placed at that location.

Capacitors show how charge storage depends on physical design. Larger plates give more space for separated charge. Smaller plate spacing produces a stronger interaction between the two plates.

An insulating material between plates becomes polarized. Its internal charges shift slightly, reducing the electric field produced by the stored charge. This allows more charge to be stored for the same voltage.

In real devices, capacitors smooth power supplies, store energy for camera flashes, and help electronic circuits respond over a chosen time interval. In charging and discharging problems, track what changes with time. The current is usually greatest just after a switch is moved, then falls as the capacitor approaches its final state.

Circuit rules work because charge does not build up indefinitely at an ordinary junction and because energy changes around a complete path must balance. The signs cause many errors. Choose a direction for each current before solving.

A negative answer only means the real current goes opposite your chosen direction. When crossing a resistor in the direction of current, electric potential decreases. When crossing an ideal battery from its negative terminal to its positive terminal, electric potential increases.

Meters matter in lab situations. An ammeter is placed in series and has very low resistance. A voltmeter is placed in parallel and has very high resistance.

Magnetic problems require careful attention to direction and to what is changing. A magnetic field can bend a moving charge without changing its speed, since the magnetic force is perpendicular to the motion. That is why charged particles can travel in circles in a uniform field.

Use the right hand rule consistently, then reverse the result for a negative charge. 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 changes.

The induced current creates effects that oppose the change causing it. This opposition protects energy conservation. Maxwell's displacement current extends this idea to changing electric fields, which helps explain how electromagnetic waves can travel through empty space.