Short on time? This page is the recap. For explanations, Figures 1 and 2 and the worked examples, use the full study guide.
Recap
- A parallel-plate capacitor is two conducting plates holding +Q and −Q. Q means the charge on one plate; the total is zero.
- Capacitance C = Q/ΔV is the charge stored per volt. Unit: farad (F = C/V).
- C depends only on the capacitor’s geometry and the material between the plates, not on Q or ΔV.
- Between the plates the field is uniform, from + to −, except near the edges (ignored).
- A charge between the plates has constant acceleration qE/m: it moves like a projectile, along a parabola.
- The stored energy equals the work needed to separate the charge; it is the area under a Q–ΔV graph.
- A dielectric sets up an induced field opposite to the plates’ field. The net field drops and C rises by κ.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| C = Q / ΔV | C in F, Q in C, ΔV in V | define or measure capacitance |
| C = κε₀A / d | ε₀ = 8.85 × 10⁻¹² C²/(N·m²); A of one plate in m²; d in m | find C from the geometry |
| E = ΔV / d = Q / (ε₀A) | V/m = N/C (second form: vacuum or air) | find the uniform field between the plates |
| U_C = ½QΔV = ½C(ΔV)² = Q²/(2C) | J | find the energy stored |
| a = qE / m | m/s² | treat a particle between the plates as a projectile |
| Situation | Fixed | Example: dielectric κ added |
|---|---|---|
| Isolated (disconnected) | Q | ΔV, E and U_C fall by κ |
| Connected to a battery | ΔV | Q and U_C rise by κ; E between the plates is the same |
Assumptions behind the numbers
- Plates are flat and parallel, with d much smaller than the plate size, so edge effects are ignored.
- A dielectric, when present, fills the gap.
- Gravity is negligible for electrons and protons between the plates.
Mistakes to avoid
- Saying C changes because Q or ΔV changed. Only A, d and κ set C.
- Using the charge on both plates. Q is on one plate.
- Dropping the ½ in U_C.
- Not checking what is fixed before a “what happens if” question.
- Deflecting a negative charge toward the negative plate. It goes toward the positive plate.
- Forgetting unit prefixes. pF = 10⁻¹² F, nF = 10⁻⁹ F, µF = 10⁻⁶ F; mm must become m.
Quick self-check
- Plates of area 0.025 m² are 1.0 mm apart in air. What is C? (2.21 × 10⁻¹⁰ F, 221 pF)
- An isolated charged capacitor has a dielectric with κ = 2 slid between its plates. What happens to ΔV? (It halves)
- How much energy does a 10 µF capacitor store at 20 V? (2.0 × 10⁻³ J)
Next: practice questions.