Parallel-Plate Capacitance⚠ unverified
Physics / Electromagnetics · Capacitance of a parallel-plate capacitor (vacuum permittivity)
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| A | A | m^2 | 0.01 | Plate area |
| d | d | m | 0.001 | Plate separation |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| C | C | F | Capacitance |
The science & history
Understanding the Parameters
- $A$ — larger area stores more charge at the same voltage.
- $d$ — smaller gap → larger $C$ (and larger field $E = V/d$ for fixed $V$).
- $\varepsilon$ — vacuum $\varepsilon_0$; dielectrics multiply by $\varepsilon_r > 1$.
Derivation (Approaching a Proof)
Uniform field between plates (Gauss): $E = \sigma/\varepsilon = Q/(\varepsilon A)$. Potential $V = E d = Q d/(\varepsilon A)$. By definition $C = Q/V$:
$$C = \frac{\varepsilon A}{d}.$$
Energy stored: $U = \tfrac12 C V^{2}$ (see electrical Capacitor Energy).
History
Parallel-plate capacitors are the canonical electrostatic storage element; the $C=\varepsilon A/d$ formula is standard from introductory EM through RF design (with fringing corrections when needed).
Related Concepts: Electric Field point charge, Electric Potential, Capacitor Energy, Energy Electric Field
Notes: Registry calculator capacitance-parallel-plate (unverified). Ideal infinite-plate model.
Possible registry gap: $\varepsilon$ not a user input — vacuum assumed.