Conductors, Capacitors, and Dielectrics

What this unit covers

The topics below follow the published Physics C: E&M course framework for Unit 3. This unit is worth 11–17% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Conductors & Boundary Conditions13 min · 3 objectivesJustify the equilibrium properties of conductors: E = 0 inside, charge on the surface · Derive the surface field E = σ/ε₀ with a Gaussian pillbox and state the boundary conditions · Analyze cavities and induced charges using Gauss’s law region by region
• Capacitance13 min · 3 objectivesDefine capacitance C = Q/V and explain why it depends only on geometry · Derive C = ε₀A/d for a parallel-plate capacitor starting from Gauss’s law · Apply the assume-Q → find-E → integrate-for-V → divide recipe to other geometries
• Energy Stored in a Capacitor12 min · 3 objectivesDerive U = Q²/2C = ½CV² by integrating dU = (q/C) dq · Show that the energy resides in the field, with density u = ½ε₀E² · Track energy when a charged capacitor’s geometry is changed at fixed Q or fixed V
• Dielectrics14 min · 3 objectivesExplain polarization and why a dielectric multiplies capacitance by κ · Analyze inserting a dielectric with the battery disconnected (Q fixed) · Analyze inserting a dielectric with the battery connected (V fixed)

Formulas in Unit 3

Every term in Unit 3

All 19 terms we publish for Conductors, Capacitors, and Dielectrics, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Capacitors in series and parallel

Parallel add directly; series combine reciprocally — the reverse of the resistor rules.

Energy stored in a capacitor

U = ½QV = ½CV² = Q²/2C. Which form is convenient depends on whether charge or voltage is held constant.

Capacitance definition

C = Q/V, depending only on geometry and the dielectric, not on the charge stored or voltage applied.

Parallel plate capacitance

C = κε₀A/d. Larger area or smaller separation raises capacitance.

Cylindrical and spherical capacitors

Found by computing V from the field via Gauss's law, then dividing Q by that V — the general method for any geometry.

Energy density of an electric field

u = ½ε₀E². The energy resides in the field itself, not on the plates.

Dielectric constant

κ > 1. Inserting a dielectric polarizes it, reducing the internal field and raising capacitance by the factor κ.

Dielectric at constant charge vs constant voltage

Disconnected from the battery, Q is fixed so V falls and U falls. Connected, V is fixed so Q rises and U rises.

RC circuit charging

q(t) = Cε(1 − e^(−t/RC)) and i(t) = (ε/R)e^(−t/RC). The capacitor is a wire at t = 0 and a break at t = ∞.

RC circuit discharging

q(t) = Q₀e^(−t/RC). After one time constant, about 37% of the charge remains.

Time constant

τ = RC, with units of seconds. Larger resistance or capacitance means slower charging.

General method for capacitance

Assume charge Q, find E from Gauss's law, integrate to get V, then form C = Q/V. Works for any geometry.

Capacitance of a spherical capacitor

C = 4πε₀(ab)/(b − a) for concentric shells of radii a and b. As b → ∞ it reduces to that of an isolated sphere.

Force between capacitor plates

Attractive, since the plates carry opposite charge. Found from F = −dU/dx at constant charge.

Partially filled dielectric

Treat the filled and empty regions as capacitors in series or parallel depending on whether the dielectric is layered across or along the gap.

Why energy changes when a dielectric is inserted

At constant charge the field weakens so U falls and the dielectric is pulled in; at constant voltage the battery supplies charge and U rises.

Charge redistribution between capacitors

Connecting a charged capacitor to an uncharged one conserves charge but loses energy to resistance and radiation — the total U always falls.

Capacitor in a circuit at steady state

No current flows through a fully charged capacitor, so it can be treated as an open branch when analyzing the rest of the circuit.

RC time constant reasoning

After τ the quantity has moved about 63% of the way to its final value; after 5τ it is within 1%.

What examiners penalize here

• The capacitance derivation is a guaranteed FRQ pattern — commit the recipe: (1) assume ±Q, (2) Gauss’s law for E, (3) V = ∫→E·d→l between the conductors, (4) C = Q/V. Show the Q canceling; that line is what proves C is geometric. Key results: C = ε₀A/d (plates), C = 4πε₀R (sphere).
• Dielectric quick table — battery **disconnected** (Q fixed): C ↑ ×κ, V ↓ ÷κ, E ↓ ÷κ, U ↓ ÷κ. Battery **connected** (V fixed): C ↑ ×κ, Q ↑ ×κ, E unchanged, U ↑ ×κ. In both cases the slab is pulled inward. Reproduce this table from C = κC₀ plus “what’s held fixed” rather than memorizing blindly.

Practice Physics C: E&M

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 6 units of AP Physics C: E & M

1. Unit 1 · Electric Charges, Fields, and Gauss’s Law
2. Unit 2 · Electric Potential
3. Unit 3 · Conductors, Capacitors, and Dielectrics
4. Unit 4 · Electric Circuits
5. Unit 5 · Magnetic Fields
6. Unit 6 · Electromagnetic Induction