AP Physics C: E&M Unit 10 ReviewConductors and Capacitors

AP Physics C: E&M Unit 10 is about what happens when charge meets matter. Conductors let charge move freely, so excess charge races to the surface and kills the field inside. Capacitors exploit that behavior to store charge and energy in the field between two plates, and dielectrics let you tune how much they store. The single biggest idea is that the electric field inside a conductor in electrostatic equilibrium is always zero, and almost everything else in the unit follows from it. Unit 10 makes up 10-15% of the AP exam.

What this unit covers

Conductors in electrostatic equilibrium

• An ideal conductor is a material where electrons move freely. Give it excess charge and the charges repel each other until they spread out as far as possible, which means they end up entirely on the surface.
• A negatively charged conductor has extra electrons on its surface. A positively charged conductor has a surface that is deficient in electrons. Either way, the interior is neutral.
• At equilibrium, the field inside the conducting material is zero. If it weren't, free electrons would still be moving, and you wouldn't be at equilibrium yet. This argument shows up constantly in justification questions.
• Because E = 0 inside, the entire conductor is one equipotential. The surface is an equipotential surface, and the field just outside must be perpendicular to it (any parallel component would push surface charges sideways).
• Charge density is not uniform on irregular shapes. Charge piles up where the surface curves sharply, so the field is strongest near points and edges.
• Gauss's law plus E = 0 inside gives you a powerful tool. Any Gaussian surface drawn inside conducting material encloses zero net charge, which is how you figure out induced charges on the inner and outer surfaces of conducting shells.

Moving charge around: contact, grounding, and induction

• When two conductors touch, charge flows until both surfaces sit at the same electric potential. For two identical spheres, that means the charge splits evenly. For different sizes, you solve for the charge split by setting the potentials equal.
• Ground is an idealized reference at zero electric potential. It can absorb or supply unlimited charge without its own potential changing. Think of it as an infinite charge reservoir.
• Charging by induction is the classic sequence. Bring a charged object near a conductor, ground the conductor so charge flows to or from ground, disconnect the ground wire, then remove the charged object. The conductor is left with a net charge opposite in sign to the inducing object, and it never touched anything charged.
• A grounded conductor near a charge always feels attraction, because the induced charge is opposite in sign and closer than any like charge.

Capacitors: storing charge and energy

• A parallel-plate capacitor is two separated conducting plates holding equal and opposite charges, +Q and -Q. The separated charge creates a potential difference between the plates.
• Capacitance is the ratio C = Q/ΔV. It tells you how much charge the capacitor stores per volt. Crucially, C depends only on the physical construction of the capacitor (geometry and the material between the plates), never on Q or ΔV themselves.
• For parallel plates, C = κε₀A/d. Bigger plates store more; closer plates store more. This is the formula you reason with when an exam question changes the geometry mid-problem.
• A charged capacitor stores energy in its electric field, U = ½ Q ΔV = ½ C(ΔV)² = Q²/2C. Which form you use depends on what stays constant when something changes (Q is fixed if the capacitor is isolated; ΔV is fixed if it stays connected to a battery).
• You should be able to derive capacitance from scratch for other geometries (spherical and cylindrical capacitors) by finding E with Gauss's law, integrating to get ΔV, then taking C = Q/ΔV. This is one of the signature calculus tasks of the course.

Dielectrics: insulators that polarize

• In a dielectric, charges are not free to move like in a conductor. Instead, the material polarizes in an external field. Molecular charges shift slightly, creating an internal field that points opposite to the external one.
• The result is a weaker net field inside the dielectric, a smaller ΔV for the same charge, and therefore a larger capacitance.
• The dielectric constant κ relates the material's permittivity to vacuum, κ = ε/ε₀, with κ ≥ 1. Inserting a dielectric multiplies the capacitance by κ.
• The before-and-after logic matters more than the formula. Insert a dielectric into an isolated capacitor and Q stays fixed while ΔV drops and stored energy decreases (the dielectric gets pulled in). Insert it while connected to a battery and ΔV stays fixed while Q and stored energy increase.

Unit 10, Conductors and Capacitors at a glance

Why Unit 10, Conductors and Capacitors matters in AP Physics E&M

This unit is the bridge between abstract field theory and real circuits. Units 8 and 9 built fields and potential as mathematical objects; Unit 10 is where you watch matter respond to them, and where the course's "fields store energy" theme becomes concrete and calculable.

• The conductor equilibrium argument (E = 0 inside, so the surface is an equipotential) is one of the most reused chains of reasoning in the whole course. It appears in shell problems, shielding questions, and capacitor derivations.
• Capacitance derivations are the course's showcase for combining Gauss's law, the potential integral, and a definition into one multi-step calculus argument. That skill is exactly what the hardest free-response parts demand.
• Energy storage in capacitors, U = ½C(ΔV)², is the electric twin of the inductor's U = ½LI² later on. Seeing the parallel early makes induction much easier.
• Dielectric before-and-after reasoning trains the "what is held constant?" habit you need for every circuit modification problem.

How this unit connects across the course

• Backward to Electric Charges, Fields, and Gauss's Law (Unit 8): Gauss's law is your main tool here. E = σ/ε₀ just outside a conductor and every capacitance derivation start with a Gaussian surface from Unit 8.
• Backward to Electric Potential (Unit 9): capacitance is literally defined through potential difference, C = Q/ΔV, and "conductors are equipotentials" only makes sense once you own the field-potential relationship V = -∫E·dr.
• Forward to Electric Circuits (Unit 11): capacitors become circuit elements there. RC charging and discharging, time constants, and series/parallel capacitor networks all assume you already know what C means physically.
• Forward to Electromagnetic Induction (Unit 13): the energy a capacitor stores in its electric field mirrors the energy an inductor stores in its magnetic field, and LC oscillations trade energy between exactly these two reservoirs.

Key equations and processes

• C = Q/ΔV defines capacitance. It is the charge stored per volt of potential difference, set entirely by the capacitor's construction.
• C = κε₀A/d gives parallel-plate capacitance. Use it to predict how C changes when you alter area, separation, or the filling material.
• U = ½ Q ΔV = ½ C(ΔV)² = Q²/2C gives stored energy. Pick the form whose variables stay constant in your scenario.
• κ = ε/ε₀ defines the dielectric constant, the factor by which a material's permittivity exceeds vacuum's.
• E = σ/ε₀ gives the field just outside a conductor's surface, perpendicular to the surface, where σ is the local surface charge density.
• The capacitance derivation process: assume charge ±Q, find E between the conductors with Gauss's law, compute ΔV = -∫E·dr, then form C = Q/ΔV. Practice this for plates, concentric spheres, and coaxial cylinders.
• The induction-charging process: bring a charged object near, ground the conductor, remove the ground connection, then remove the object. Track where electrons flow at each step.
• The interior-Gaussian-surface process: since E = 0 inside conducting material, any Gaussian surface inside it encloses zero net charge, which pins down induced charges on inner and outer shell surfaces.

Unit 10, Conductors and Capacitors on the AP exam

Unit 10 carries 10-15% of the exam, in both multiple-choice and free-response questions. Expect a few recurring jobs:

• Derive. Free-response questions love asking for the capacitance of a non-parallel-plate geometry. You show the full Gauss's law to ΔV to C = Q/ΔV chain, with calculus, and earn points for each step.
• Justify with physics. Conductor questions often ask you to explain why the field inside is zero, why charge sits on the surface, or what induced charges appear on a shell. The expected answer is the equilibrium argument, not a memorized fact.
• Compare before and after. Dielectric and geometry-change questions hinge on identifying what stays constant (Q for an isolated capacitor, ΔV for one connected to a battery) and reasoning out how E, ΔV, C, and U respond. Ranking and "increases/decreases/stays the same" formats are common in multiple choice.
• Work with representations. You may sketch charge distributions on conductors, draw field lines that meet conducting surfaces perpendicularly, or interpret graphs of stored quantities. Translation-between-representations free-response questions draw on exactly these skills.
• Experimental reasoning. A lab-design prompt might ask how to measure capacitance or test how C depends on plate separation, using the linear relationship between measurable quantities.

Essential questions

• Why does excess charge on a conductor always end up on the surface, and why must the field inside be zero at equilibrium?
• What does it physically mean for an object to "have capacitance," and why does capacitance depend only on geometry and material, never on the charge it holds?
• Where is the energy in a charged capacitor actually stored, and what happens to that energy when the capacitor's configuration changes?
• How does a material that cannot conduct charge still manage to change an electric field inside it?

Key terms to know

• Ideal conductor: a material in which electrons move completely freely, so it reaches electrostatic equilibrium with zero internal field.
• Electrostatic equilibrium: the state where no net charge motion occurs, requiring E = 0 inside the conductor and all excess charge on the surface.
• Equipotential surface: a surface where every point has the same electric potential; a conductor's surface is always one at equilibrium.
• Surface charge density (σ): charge per unit area on a surface, largest where a conductor curves sharply.
• Ground: an idealized zero-potential reference that can absorb or supply unlimited charge without changing its potential.
• Charging by induction: giving a conductor net charge without contact, by grounding it in the presence of an external field and then disconnecting ground.
• Capacitor: two separated conductors holding equal and opposite charges, creating a potential difference between them.
• Capacitance (C): the ratio of stored charge to potential difference, C = Q/ΔV, fixed by the capacitor's physical construction.
• Dielectric: an insulating material that polarizes in an external field instead of conducting, weakening the field inside it.
• Polarization: the slight shifting of bound charges in a dielectric that creates an internal field opposing the external one.
• Dielectric constant (κ): the ratio ε/ε₀, the factor by which a dielectric raises capacitance compared to vacuum.
• Permittivity (ε): a material property measuring how it responds to and reduces electric fields within it.
• Electrostatic shielding: the cancellation of external fields inside a conducting enclosure because surface charges rearrange to cancel them.

Common mix-ups

• "E = 0 inside a conductor" applies to the conducting material itself, not necessarily to a cavity inside it. A charge placed in a cavity creates a field in the cavity and induces charge on the cavity wall, even though E is still zero within the metal.
• Capacitance does not depend on Q or ΔV. Doubling the charge on a capacitor doubles ΔV and leaves C unchanged. C = Q/ΔV is a definition, not a cause-and-effect statement.
• Inserting a dielectric does not always increase stored energy. Connected to a battery, U increases because Q grows at fixed ΔV. Isolated, U decreases because ΔV drops at fixed Q. Always decide what is held constant first.
• The field just outside a conductor is σ/ε₀, but the field from a single charged sheet is σ/2ε₀. The conductor's value is larger because all the field lines emerge from one side of the surface.