# The Capacitor

Laplace’s Mini World · Plate 71 · Matter, Light & Sound · Matter

> Aluminium foil with an oxide film on it, rolled up with paper and a second foil in a can. Connect a battery and no charge crosses the film: it moves round the wires outside, fast at first, then slower, over a time set by RC.

- Interactive model: https://world.lapla.si/en/capacitor/
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- 中文: https://world.lapla.si/zh/capacitor/
- Updated: 2026-10-31

## 01 An Insulator in Between

A capacitor is two conductors with an insulator between them. Connect a battery and one takes a positive charge, the other an equal negative charge; the capacitor as a whole stays neutral.

The one on the bench is a screw-terminal aluminium electrolytic modelled on Cornell Dubilier’s 550C113T400FG2D: 11,000 µF and 400 V, 88.9 mm across and 219 mm tall, about 1.66 kg. A quarter of it has been sawn away to show what is rolled up inside.

Inside are two strips of aluminium foil with paper between them. The insulator is a film of aluminium oxide grown on the anode foil. The paper is soaked in electrolyte, which conducts: the electrolyte is the other “plate”, and the cathode foil only carries current to it.

*Two conductors, one insulator · +Q on one, −Q on the other · 11,000 µF, 400 V · 88.9 × 219 mm*

## 02 A Bottle in 1745

In the autumn of 1745 von Kleist, dean of a cathedral in Pomerania, put a nail into a medicine bottle with water in it, touched the nail to a static electricity machine, then touched the nail and got a nasty shock. He wrote to others about it but didn’t say the bottle had to be held in the hand, and nobody could repeat it.

Soon after, Musschenbroek in Leiden did it with a glass bottle half full of water; holding the bottle was first found to matter by Cunaeus, a friend who came to his laboratory. On 20 January 1746 he wrote to Réaumur in Paris that he would not do it again for all the kingdom of France. The jar took its name from the Leiden experiment.

It was soon found that the water was not needed: foil on the inside and outside worked better, with a brass rod reaching the inner foil. In 1747 Franklin wrote that the top of the bottle was charged plus and the bottom minus, in exact proportion; in 1749 he joined eleven panes of window glass faced with lead on both sides and called it an electrical battery. At the back left of the bench is a Leyden jar.

*Autumn 1745: von Kleist · January 1746: Musschenbroek’s letter · 1749: Franklin’s eleven panes*

## 03 Charge Goes the Long Way Round

With the switch to the left, the battery is joined to the capacitor. The battery pulls electrons off the positive side and pushes them, through the switch, onto the negative side: one side is short of electrons and positive, the other has extra and is negative. No electron crosses the insulator.

The wires are full of free electrons from the start. When the switch closes, the electrons all round the circuit move together; once the capacitor’s voltage matches the battery’s, they stop.

The charge separated is Q = CV: this 11,000 µF capacitor on a 6 V battery has +0.066 C on one side and −0.066 C on the other (computed here). A galvanic cell drives a current by a chemical reaction; a capacitor has no such reaction, and only keeps apart the charge the battery moved.

*Q = CV · 11,000 µF × 6 V = 0.066 C (computed) · no electron crosses the insulator*

## 04 Wide Plates, a Narrow Gap

For two parallel metal plates the capacitance is C = εᵣε₀A/d: the larger the area A, the smaller the gap d and the higher the insulator’s relative permittivity εᵣ, the more charge the same voltage separates. ε₀, the vacuum permittivity, is about 8.854 × 10⁻¹² F/m.

Two metal plates 10 cm square, 1 mm apart in air, make only about 88.5 pF. To reach this capacitor’s 11,000 µF with plates 1 mm apart in air, each would need about 1.24 million square metres, about 1.1 km on a side (computed here).

A farad is a very large capacitance. Ordinary capacitors run from a few picofarads to a few millifarads; supercapacitors reach thousands of farads.

*C = εᵣε₀A/d · 10 cm square, 1 mm apart: about 88.5 pF · 11,000 µF: about 1.1 km square (computed)*

## 05 Square Metres in a Can

An aluminium electrolytic makes the gap tiny. Its insulator is aluminium oxide grown electrochemically on the anode foil, about 1.2 nm thick per volt, with a relative permittivity of about 9.5. At 400 V that is a film only about 0.48 µm thick (an estimate, computed here).

The anode foil is first etched full of microscopic tunnels to enlarge its surface: about 50⁠–⁠100 times for low-voltage foil, about 20 times for high-voltage foil. On those figures this capacitor needs about 63 m² of effective area, about 3 m² of foil (estimates, computed here).

To fit that much foil in an 88.9 mm can it has to be rolled: anode foil, paper, cathode foil and paper, wound turn on turn into the winding, whose cross-section shows on the step. Aluminium tabs join each foil to one of the terminals on top.

*Oxide about 1.2 nm per volt · at 400 V about 0.48 µm · high-voltage foil about ×20 · about 63 m² effective (estimates)*

## 06 A Capacitor with a Plus Side

An aluminium electrolytic has a plus and a minus: the oxide grows only on the anode foil, and the supply’s positive side must go there. The wrong way round, electrolysis grows a film on the cathode foil, with strong heating and gas that destroy the capacitor; makers allow at most 1.5 V reversed, and only briefly.

So the cover disc is marked + and −, and the sleeve carries a column of minus signs on the negative side. The can is joined to the negative side through the electrolyte.

Its seal is deliberately not airtight: the can’s edge is rolled onto a rubber-faced phenolic disc, and the disc carries a pressure vent. If the pressure inside builds up, the vent swells, then bursts and lets out the electrolyte’s vapour.

*Plus to the anode foil · at most 1.5 V reversed, briefly · a pressure vent in the cover*

## 07 Fast at First, Then Slower

The moment the switch closes the capacitor has no voltage yet, so the battery’s whole 6 V lies across the resistor and the current is at its largest: 6 V ÷ 220 Ω, about 27 mA. As charge builds up, the capacitor’s voltage rises, less is left for the resistor, and the current falls.

So the voltage rises as 1 − e^(−t/RC) and the current falls as e^(−t/RC). RC is the time constant τ: 220 Ω × 11,000 µF = 2.42 s (computed here).

After one τ the voltage is at 63 % of the battery’s; after two about 86 %, after five about 99.3 % (computed here). It keeps closing in; there is no moment when it is exactly full.

*τ = RC = 220 Ω × 11,000 µF = 2.42 s · 1τ: 63 % · 5τ: 99.3 % (computed)*

## 08 Down the Same Curve

Throw the switch to the right and the battery is cut out; the capacitor’s two sides are joined through the same resistor. The extra electrons leave the negative side and go round through the resistor to the positive side: the current runs the other way.

The voltage, the charge and the current all fall as e^(−t/RC), with the same 2.42 s time constant: 37 % is left after one τ, 0.7 % after five (computed here).

The stored energy turns to heat in the resistor. A larger resistor means a smaller current and a slower discharge, but the same heat.

*Discharging: e^(−t/RC) · 37 % left after 1τ · 0.7 % after 5τ (computed)*

## 09 The Energy Is in the Field

What a capacitor stores is energy: U = ½CV². It sits in the electric field between the two sides, ½εE² in each unit of volume. Double the voltage and the energy is four times as much.

On the 6 V battery this capacitor holds only 0.198 J; charged to its rated 400 V it holds 880 J (computed here).

While charging, the battery gives CV²; only half goes into the capacitor, and the other half turns to heat in the resistor, whatever its resistance (computed here).

*U = ½CV² · 6 V: 0.198 J · 400 V: 880 J · half turns to heat while charging (computed)*

## 10 All at Once

A capacitor doesn’t hold much energy, but it can let it all go in a very short time. A camera flash uses an aluminium electrolytic of a few hundred microfarads at over 300 V: one of 300 µF at 330 V holds about 16 J (computed here); a flashgun at full output lasts about 1/980 s.

Defibrillators use metallized polypropylene film capacitors of tens to two hundred microfarads at thousands of volts. The European Resuscitation Council’s 2021 guidelines give a first biphasic shock of at least 150 J. One automated external defibrillator charges to 200 J in under 9 s.

The National Ignition Facility stores about 400 MJ in 3,840 capacitors over 60 s, then releases it in 400 µs into 7,680 flashlamps, which pump the laser.

*Flash: 300 µF at 330 V, about 16 J (computed) · defibrillation: at least 150 J · NIF: about 400 MJ in 400 µs*

## 11 Filling the Dips

The commonest job of an aluminium electrolytic is smoothing in power supplies and inverters. Rectified mains comes as one hump after another; the capacitor charges near each peak and discharges in the dips, holding the voltage up.

Rectified 50 Hz mains has 100 peaks a second. If this 11,000 µF capacitor alone supplied 1 A between two peaks, its voltage would drop only about 0.91 V (an estimate from ΔV = I/(fC), computed here).

That is why makers mostly rate capacitance at 100 Hz or 120 Hz: the ripple of 50 Hz and 60 Hz mains after rectifying. The type on the bench is a high-ripple capacitor made for inverters.

*Rectified 50 Hz: 100 peaks a second · 1 A from 11,000 µF: about 0.91 V of droop (estimate)*

## 12 A Gap Under a Nanometre

At the back right is a supercapacitor modelled on Maxwell’s BCAP3000 P270 K04: 3,000 F and 2.7 V, 60.3 mm across, 138 mm long and about 475 g.

It has no solid insulating film. Its electrodes are activated carbon with over 1,500 m² of surface per gram; on charging, ions from the electrolyte gather at the carbon’s surface, with less than a nanometre between the positive and negative charges: the double layer. It is electrostatic, with no chemical reaction, and lasts about a million cycles.

It holds about 10.9 kJ, over twelve times what the electrolytic holds at 400 V, but at only 2.7 V (computed here). It is not a battery either: as it discharges, its voltage falls in proportion to the charge left.

*3,000 F, 2.7 V · about 10.9 kJ (computed) · carbon over 1,500 m² per gram · about a million cycles*

## 13 The Capacitor, Piece by Piece

First the cover disc comes off (terminals, vent and the gasket under it), then the winding lifts out with its aluminium tabs; the can and its sleeve stay. The voltmeter, switch, resistor and battery lift one by one as well.

Point at a piece and it is lifted out alone and named; left alone, the pieces lift in turn.

*Cover disc · winding · can · voltmeter · knife switch · resistor · battery*

## 14 Charge, Discharge

Pick the resistance, the capacitance and the number of cells, then throw the switch to charge or discharge. Each throw starts the clock again: the voltage and current follow the exponential curve set by RC, and the chart marks where one τ falls.

The readings are computed here from the formulas: τ = RC, V = V₀(1 − e^(−t/RC)) charging and V = V₀e^(−t/RC) discharging, energy ½CV². The battery’s internal resistance, the wires’ resistance and the capacitor’s leakage are left out.

*τ = RC · 63 % at 1τ · energy ½CV² (computed)*

**Try this**: Raise the resistance from 220 Ω to 470 Ω: how many more seconds does it take to reach 63 % of the battery’s voltage? Does the energy it ends up storing change?

## What the model leaves out

The bench is a model of the principle. The winding is drawn with 30 turns, every layer too thick: real anode foil is 80⁠–⁠130 µm, cathode foil 15⁠–⁠50 µm and paper 25⁠–⁠90 µm, so a winding like this one has well over a hundred turns (an estimate from those thicknesses). The sleeve, tabs and cover follow the usual pattern, not this type’s drawings. The voltmeter, switch, resistor and battery are sketches. The red dots in the wires show only which way the electrons go and how their pace changes, not their real speed. The lab leaves out the battery’s internal resistance, the wires’ resistance, leakage and the capacitor’s equivalent series resistance.

## Easily misread

A capacitor as a whole carries no charge: +Q on one side, −Q on the other; what it stores is energy, and “charging” is a custom of speech. Franklin wrote in 1749 that there is no more electricity in the bottle after its “charging” than before. Current does not pass through the insulator; it runs round the outside of the circuit, and charge crosses only if the insulator breaks down. A capacitor is not a battery: no chemical reaction stores its energy, and its voltage falls in proportion to its charge. One time constant brings it to 63 %, not full. In an electrolytic the electrolyte is a conductor; the insulator is the oxide film. Size and capacitance don’t go together: in the same case, a 200 V part holds 14,000 µF, a 400 V part only 5,300 µF.

## Common questions

### What is inside a capacitor?

A capacitor is two conductors with an insulator between them. Connect a battery and one takes a positive charge, the other an equal negative charge; the capacitor as a whole stays neutral.

See: [01 An Insulator in Between](https://world.lapla.si/en/capacitor/#gap)

### Does charge flow through the insulator when a capacitor charges?

The battery pulls electrons off the positive side and pushes them, through the switch, onto the negative side: one side is short of electrons and positive, the other has extra and is negative. No electron crosses the insulator.

See: [03 Charge Goes the Long Way Round](https://world.lapla.si/en/capacitor/#push)

### How long does a capacitor take to charge fully?

After one τ the voltage is at 63 % of the battery’s; after two about 86 %, after five about 99.3 % (computed here). It keeps closing in; there is no moment when it is exactly full.

See: [07 Fast at First, Then Slower](https://world.lapla.si/en/capacitor/#curve)

### How much energy does a capacitor store?

What a capacitor stores is energy: U = ½CV². It sits in the electric field between the two sides, ½εE² in each unit of volume. Double the voltage and the energy is four times as much.

See: [09 The Energy Is in the Field](https://world.lapla.si/en/capacitor/#energy)

### Why does a supercapacitor have such a large capacitance?

Its electrodes are activated carbon with over 1,500 m² of surface per gram; on charging, ions from the electrolyte gather at the carbon’s surface, with less than a nanometre between the positive and negative charges: the double layer.

See: [12 A Gap Under a Nanometre](https://world.lapla.si/en/capacitor/#super)

## Sources

1. [OpenStax, University Physics Volume 2, chapter 8: capacitors and capacitance (two conductors, +Q and −Q, C = Q/V, parallel plates C = ε₀A/d, dielectrics)](https://openstax.org/books/university-physics-volume-2/pages/8-1-capacitors-and-capacitance)
2. [OpenStax, University Physics Volume 2, §8.3: energy stored in a capacitor (U = ½CV², energy density ½ε₀E²)](https://openstax.org/books/university-physics-volume-2/pages/8-3-energy-stored-in-a-capacitor)
3. [OpenStax, University Physics Volume 2, §10.5: RC circuits (charging, discharging, the time constant τ = RC, 63 %)](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)
4. [NIST: CODATA 2022 recommended value of the vacuum electric permittivity ε₀](https://physics.nist.gov/cgi-bin/cuu/Value?ep0)
5. [Cornell Dubilier: Aluminum Electrolytic Capacitor Application Guide (construction, etching, the oxide, electrolyte, the seal, the vent, polarity, uses)](https://default.assets-knowles-prod.roccommercecloud.com/assets%2Ff6c30cb949d042ea95f7b1ab3c23b463/KNO_CD_AEappGuide_R2.pdf)
6. [TDK: Aluminum Electrolytic Capacitors — General technical information (August 2026: 1.2 nm/V, relative permittivity 9.5, the liquid cathode, reverse voltage)](https://www.tdk-electronics.tdk.com/download/530704/31a5416e653dd6e4e428b8208d65cc2e/pdf-generaltechnicalinformation.pdf)
7. [Cornell Dubilier: Type 550C screw-terminal aluminium electrolytic data sheet (550C113T400FG2D: 11,000 µF, 400 V, 88.90 × 219.08 mm, 1,658 g)](https://www.cde.com/resources/catalogs/550C.pdf)
8. [Maxwell (UCAP Power): BCAP3000 P270 K04 ultracapacitor data sheet (3,000 F, 2.7 V, 475 g, the double layer)](https://maxwell.com/wp-content/uploads/2022/11/3003279-EN.3_DS_2.7V-3000F-Cell-BCAP3000-P270.pdf)
9. [W. B. Ashworth Jr., Linda Hall Library: Ewald Jürgen von Kleist](https://www.lindahall.org/ewald-jurgen-von-kleist/)
10. [W. B. Ashworth Jr., Linda Hall Library: Pieter van Musschenbroek (the letter of 20 January 1746)](https://www.lindahall.org/about/news/scientist-of-the-day/pieter-van-musschenbroek/)
11. [Revista Brasileira de Ensino de Física 46 (2024): three accounts of the Leyden jar (Cunaeus)](https://www.scielo.br/j/rbef/a/x7jCNGdWHMqkCd6Mk9vYsMz/?lang=pt)
12. [Whipple Museum of the History of Science, Cambridge: Leyden jar, object 9598 (foil inside and out, the brass rod, the ball)](https://collections.whipplemuseum.cam.ac.uk/objects/9598)
13. [B. Franklin: Experiments and Observations on Electricity (London, 1751; Project Gutenberg edition)](https://www.gutenberg.org/cache/epub/45515/pg45515.txt)
14. [Cornell Dubilier: Type 7P photoflash capacitor data sheet (7P301V330J032: 300 µF, 330 V)](https://www.cde.com/resources/catalogs/7P.pdf)
15. [Nikon: SB-⁠5000 Speedlight specifications (flash about 1/980 s at full output)](https://imaging.nikon.com/imaging/lineup/speedlights/sb-5000/)
16. [European Resuscitation Council guidelines 2021, adult advanced life support (German edition, Notfall + Rettungsmedizin): a first biphasic shock of at least 150 J](https://pmc.ncbi.nlm.nih.gov/articles/PMC8185697/)
17. [Stryker: LIFEPAK CR Plus specifications (150⁠–⁠360 J, 200 J charged in under 9 s)](https://www.stryker.com/content/dam/stryker/ems/resources/data-sheets/lifepak_cr_plus_one-sheet.pdf)
18. [TDK: B32365 film capacitors for defibrillators (metallized polypropylene, 32⁠–⁠200 µF, up to 5,500 V)](https://en.tdk-electronics.tdk.com/inf/20/50/ds/DBL_B32365_201014_final.pdf)
19. [Lawrence Livermore National Laboratory: NIF’s power conditioning system (about 400 MJ, 3,840 capacitors, 60 s, 400 µs)](https://lmf.llnl.gov/about/how-nif-works/power-conditioning-system)

## Data

Charging, discharging, the energy and the parallel-plate areas are computed here from the textbook formulas (checked against OpenStax University Physics Volume 2, chapters 8 and 10.5) with ε₀ from CODATA 2022; the oxide, the area it needs and the foil are estimates from Cornell Dubilier’s and TDK’s figures; the ripple is the estimate ΔV = I/(fC). The capacitor on the bench is Cornell Dubilier’s 550C113T400FG2D (data sheet); its construction is from Cornell Dubilier’s Aluminum Electrolytic Capacitor Application Guide and TDK’s general technical information; the supercapacitor is Maxwell’s BCAP3000 P270 K04 (data sheet); the flash capacitor Cornell Dubilier’s Type 7P; the flash duration Nikon’s SB-⁠5000; the defibrillator from the European Resuscitation Council’s 2021 guidelines, Stryker’s LIFEPAK CR Plus and TDK’s B32365 series; NIF’s bank from Lawrence Livermore National Laboratory. The Leyden jar from W. B. Ashworth Jr. (Linda Hall Library), the Revista Brasileira de Ensino de Física (2024), the Whipple Museum and Franklin’s Experiments and Observations on Electricity (1751). The bench is a model of the principle, not any maker’s drawing. Material presets and the softbox studio of the site’s model kit are ported from AirsupHQ/machine-skill (MIT, © 2026 Air Supply Technologies Inc.).

## Licence CC BY-NC 4.0

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## Cite this plate

Laplace’s Mini World, Plate 71, “The Capacitor”, https://world.lapla.si/en/capacitor/, CC BY-NC 4.0, updated 2026-10-31.