Capacitance
Capacitance is a measure of a system's ability to store electric charge per unit of applied potential difference, C = Q/V, determined entirely by the geometry of the conductors involved and the material between them.
Curious
The simplest capacitor is just two parallel metal plates facing each other, separated by a small gap. Connect them to a battery, and charge builds up — positive on one plate, negative on the other — creating an electric field between them and storing energy, ready to be released later. Capacitance measures how effectively a particular arrangement of conductors stores this charge for a given applied voltage.
Crucially, capacitance is a fixed property of the physical setup — plate size, spacing, and the material in between — not of how much charge is currently stored. A capacitor with C = 10 microfarads has that same capacitance whether it's fully charged, half-charged, or empty; C relates charge and voltage to each other, but doesn't itself change as charge flows in or out.
Exploring
| Symbol | Name | SI Unit |
|---|---|---|
| C | Capacitance | farad (F) = C/V |
| Q | Stored charge (on either plate) | coulomb (C) |
| A | Plate area (parallel-plate case) | m² |
| d | Plate separation | m |
The energy stored in a charged capacitor is U = ½CV² — this is why capacitors, unlike batteries, can release their stored energy almost instantaneously (limited mainly by circuit resistance), making them ideal for applications needing a very fast, powerful pulse of energy.
Worked example: A parallel-plate capacitor has plates of area 0.02 m² separated by 1 mm of vacuum. C = ε₀A/d = (8.854×10⁻¹²)(0.02)/(0.001) = 1.771 × 10⁻¹⁰ F = 177.1 pF. If charged to 12 V, it stores Q = CV = 1.771×10⁻¹⁰ × 12 ≈ 2.13 × 10⁻⁹ C, and energy U = ½CV² = 0.5 × 1.771×10⁻¹⁰ × 144 ≈ 1.28 × 10⁻⁸ J.
Deep Dive
The parallel-plate formula C = ε₀A/d follows directly from combining Gauss's law (which gives the field between the plates for a given charge density) with the definition V = ∫E·dl (which gives the potential difference from that field). Filling the gap with a dielectric material (an insulator that polarises in response to the field) increases capacitance by a factor equal to the material's relative permittivity (dielectric constant), because the dielectric's induced polarisation partially cancels the field for a given stored charge, allowing more charge to be stored for the same voltage.
Historical origin: The earliest known capacitor, the Leyden jar, was independently discovered around 1745–1746 by Ewald Georg von Kleist and Pieter van Musschenbroek, decades before the underlying physics (charge, field, and potential) was properly understood — an early instance of a useful electrical device preceding, rather than following from, its theoretical explanation.
In circuits, capacitors in parallel combine by simple addition (C_total = C₁ + C₂ + ...), while capacitors in series combine reciprocally (1/C_total = 1/C₁ + 1/C₂ + ...) — the opposite pattern from how resistors combine, a frequent point of confusion when first learning circuit analysis.
Concept Relationship Map
Requires: electric potential, Gauss's law
Enables: RC circuits [PENDING FULL ENTRY — simulation available], dielectrics [PENDING FULL ENTRY]
Commonly confused with: confusing capacitance (fixed, geometric) with stored charge (variable, depends on applied voltage)
Sources
- Griffiths, D.J. (2017). Introduction to Electrodynamics, 4th ed. Cambridge University Press, §2.5.