AP Physics C: Electricity & Magnetism Unit 10: Conductors and Capacitors 2027
Dedicated Unit 10 chapter hub for 4 current topics. Use it to move through the lesson sequence, connect formulas and labs, and hand off cleanly to the previous or next unit without mixing topic and unit search intent.
Unit 10 map: electrostatic equilibrium, charge storage, and dielectric constraints
Unit 10 converts the fields and potentials of Units 8–9 into material and device behavior. Conductors rearrange free charge until electrostatic boundary conditions are satisfied; capacitors use two-conductor geometry to store separated charge and field energy; dielectrics change the field response. The most important exam habit is to state what remains fixed—charge, voltage, geometry, or connection to ground/source—before predicting what changes.
All four Unit 10 topics
Electrostatics with Conductors
Use E=0 inside conducting material, equipotential surfaces, induced charge, shielding, curvature, and Gauss-law boundary reasoning.
Current lesson · 10.1 →Redistribution of Charge Between Conductors
Combine charge conservation with equal-potential equilibrium, grounding, and induction; keep approximations for separated spheres explicit.
Open 10.2 →Capacitors
Connect C=Q/ΔV with geometry, electric field, stored energy, and the parallel-plate, spherical, and coaxial capacitor models.
Open 10.3 →Dielectrics
Track polarization and how capacitance, field, charge, voltage, and energy respond differently under fixed-Q and fixed-V constraints.
Open 10.4 →Electrostatic conductor rules you must keep together
| Equilibrium statement | Consequence | Important qualifier |
|---|---|---|
| E=0 inside the conducting material | Free charges have stopped redistributing in the static model. | This is electrostatic equilibrium, not a current-carrying resistive conductor. |
| Conductor is equipotential | No tangential electric field exists along its surface. | The constant potential need not equal zero. |
| Excess free charge resides on surfaces | Surface density adjusts to geometry and nearby charges. | Sharp curvature can produce larger |σ|. |
| Field just outside is normal to the surface | In vacuum, E_n=σ/ε₀ for the local boundary model. | Direction follows the sign of σ and chosen normal. |
| Closed conductor can shield a cavity from external electrostatic fields | Boundary conditions force the interior electrostatic solution accordingly. | Charges placed inside cavities create a different induced-charge problem. |
Capacitor and dielectric model-selection table
| Question feature | Best starting relation | Constraint check |
|---|---|---|
| Geometry → capacitance | Use the appropriate parallel-plate, spherical, or coaxial model | Check symmetry and neglect of edge/end effects. |
| Known Q and C | |ΔV|=Q/C and U=Q²/(2C) | Capacitance belongs to geometry/material, not to the current stored charge. |
| Known ΔV and C | Q=C|ΔV| and U=½C(ΔV)² | If connected to a battery, charge can flow as C changes. |
| Dielectric inserted into isolated capacitor | Q fixed; C increases | Then |ΔV|, E, and U decrease for full insertion in the ideal model. |
| Dielectric inserted while battery remains connected | ΔV fixed; C increases | Then Q and capacitor stored energy increase; include battery energy exchange in a full system ledger. |
Calculus, energy, and geometry ladder
Field → potential → capacitance
For a capacitor geometry, find E from symmetry/Gauss when appropriate, integrate ΔV=−∫E·dℓ, then use C=Q/|ΔV|. This is the general route behind the spherical and coaxial formulas.
Energy from charging
Because V=Q/C for a linear capacitor, the work to build charge is U=∫₀^Q (q/C)dq=Q²/(2C)=½Q|ΔV|.
Field-energy connection
For a uniform linear dielectric, capacitor energy can be expressed through the field energy density u=½εE²; multiplying by field volume reproduces ½C(ΔV)² in the parallel-plate model.
Redistribution between conductors
When isolated conductors are connected, total free charge is conserved while charges move until the connected conductor reaches one common potential. Geometry controls the final division.
Three Unit 10 synthesis problems
Two conducting spheres connected by a wire
For far-separated spheres, use V₁=V₂ together with Q₁+Q₂=Q_total. Since V≈kQ/R for each isolated sphere, Q₁/Q₂=R₁/R₂. The larger sphere carries more charge, but the final potentials are equal. State the separation approximation explicitly.
Dielectric insertion: isolated versus battery-connected
Start with the same capacitor and insert a dielectric. In the isolated case Q is fixed; in the battery-connected case ΔV is fixed. Build two side-by-side ledgers for C, Q, ΔV, E, and U. The direction of the energy change differs because the battery can exchange energy with the capacitor system.
Coaxial capacitor from field integration
Use a cylindrical Gaussian surface to obtain E(r), integrate the radial field from a to b for |ΔV|, then divide Q by the result to derive C=2πκε₀L/ln(b/a). This links Unit 8 Gauss reasoning directly to Unit 10 capacitance.
Laboratory and data-analysis tasks
Equipotential conductor mapping
Map potential around a conducting shape. Verify constant surface potential, field-perpendicular geometry, and enhanced field near tighter curvature using equipotential spacing.
Capacitance versus plate separation
Measure or simulate C for several d values at fixed A. Test C∝1/d and use a transformed plot to estimate ε from slope.
Capacitance versus area
Vary overlap area at fixed separation and dielectric. Test linearity of C versus A and quantify fringing deviations at small overlap.
Dielectric constraint experiment
Compare voltage and charge responses when dielectric is inserted with the capacitor isolated versus maintained at fixed source voltage. Use energy changes to identify the role of the source.
Unit 10 traps that cost points
Potential inside conductor = 0
Electrostatic potential is constant throughout a connected conductor, but the constant depends on reference and surroundings.
C depends on Q
For a linear capacitor, capacitance is set by geometry and material. Q and ΔV change together according to Q=CΔV.
Mixing fixed-Q and fixed-V dielectric results
Always identify whether the capacitor is isolated or connected to a source before predicting field, charge, voltage, or energy changes.
Grounding as “zero charge”
Ground fixes potential relative to the reference and provides a charge reservoir; it does not mean the grounded conductor must have zero charge.
Using isolated-sphere formulas for nearby conductors
Mutual influence can change surface-charge distributions and invalidate simple Q/R sharing.
Ignoring capacitor geometry limits
Parallel-plate, coaxial, and spherical formulas assume ideal geometry and neglect fringing/end effects unless otherwise modeled.
Where Unit 10 goes next
Unit 11 · Electric circuits
Capacitors become circuit elements. Current, Kirchhoff rules, resistance, and RC transients determine how stored charge and voltage evolve with time.
Unit 9 · Potential
Capacitance is defined through potential difference; conductor equipotential behavior and capacitor energy depend directly on Unit 9 ideas.
Unit 8 · Gauss’s law
Symmetry and field integrals provide the most efficient derivations of spherical and coaxial capacitance.
Unit 13 · LC circuits
Capacitor energy later exchanges with inductor magnetic energy, producing an ideal electromagnetic oscillator.
Unit 10 mastery checklist
Unit 10 lesson sequence
This dedicated unit hub owns the chapter intent. Every lesson keeps its own canonical topic URL and returns here for unit context.
Topic 10.1 · Electrostatics with Conductors
An ideal conductor in electrostatic equilibrium is an equipotential: the electric field inside the conducting material is zero, excess charge resides on its surface, and the electric field immediately outside the surface is perpendicular to that surface. External fields can polarize the conductor, sharp curvature concentrates surface charge, and a closed conducting shell can shield an interior region from external electrostatic fields.
Open Topic 10.1 →Topic 10.2 · Redistribution of Charge Between Conductors
When conductors are connected, mobile charge flows until the connected surfaces reach the same electric potential—not necessarily until the conductors carry equal charge. Ground is modeled as a zero-potential charge reservoir, and induction with grounding can leave a conductor with net charge even without direct contact with the external inducing object.
Open Topic 10.2 →Topic 10.3 · Capacitors
A capacitor stores separated equal-and-opposite charge on two conductors. Capacitance C=Q/|ΔV| is a geometric/material property, not the amount of charge currently stored. For an ideal parallel-plate capacitor C=κε₀A/d; the field is nearly uniform away from edges, energy can be written as U=Q²/(2C)=½C(ΔV)²=½QΔV, and the current course also expects quantitative concentric-spherical and coaxial-cylindrical capacitor models.
Open Topic 10.3 →Topic 10.4 · Dielectrics
A dielectric is an insulating material whose bound charges polarize in an external electric field. The polarization field opposes the applied field. For a fully filled ideal capacitor, the dielectric constant κ=ε/ε₀ increases the capacitance to C=κC₀. What happens to charge, voltage, field, and stored energy then depends on the constraint: an isolated capacitor keeps Q fixed, while a battery-connected capacitor keeps ΔV fixed.
Open Topic 10.4 →Unit 10 resource network
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