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Physics 2

Capacitors: capacitance, energy and dielectrics

Capacitance and the parallel-plate capacitor with proof, series and parallel, stored energy, dielectrics and polarisation with a table of materials, real capacitors.

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The capacitor is the component that stores charge and energy in an electric field. An electronic board holds dozens or hundreds of them: for filtering supplies, coupling stages, timing, tuning. In this lesson we cover capacitance, the parallel-plate capacitor, series and parallel connections, stored energy and the role of the dielectric.

Capacitance

Two conductors insulated from each other, with charges +Q and -Q, have a potential difference V proportional to Q. The ratio is the capacitance:

    \[ C = \frac{Q}{V}\qquad[\text{F} = \text{C/V}] \]

It depends only on the geometry and on the material between the conductors, not on the charge. The farad is a huge unit: common capacitors range from picofarads (pF) to millifarads, and only supercapacitors reach thousands of farads.

The parallel-plate capacitor

Parallel-plate capacitor with charged plates and a uniform field between them
Two plates of area A a distance d apart with charge densities +σ and −σ: the field between them is uniform and equals σ/ε₀; outside it is almost zero, apart from edge effects (dashed lines).

Proof of the formula

From Gauss’s law (lesson on the electric field) the field between two planes with densities \pm\sigma is E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A), uniform. The potential difference is V = E\,d = \frac{Q\,d}{\varepsilon_0 A}. So

    \[ C = \frac{Q}{V} = \frac{\varepsilon_0 A}{d} \]

Capacitance grows with area and falls with distance. Example: two 10 cm × 10 cm plates 1 mm apart in air have C = 8.85\times 10^{-12}\cdot 0.01/0.001 \approx 88 pF. To reach 1 µF you would need plates 10 m on a side: that is why real capacitors use very thin dielectrics with a high permittivity, wound or stacked.

Other geometries: the cylindrical capacitor (a coaxial cable of length L) has C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)}; the spherical capacitor C = 4\pi\varepsilon_0\frac{ab}{b - a}. An RG-58 coaxial cable has about 100 pF per metre.

Series and parallel connections

  • Parallel: same voltage, the charges add: C_{eq} = C_1 + C_2 + \dots It is like increasing the plate area.
  • Series: same charge on each capacitor, the voltages add: \frac{1}{C_{eq}} = \frac1{C_1} + \frac1{C_2} + \dots It is like increasing the distance. Two equal capacitors in series have half the capacitance but withstand twice the voltage.

They are the opposite of the rules for resistors.

Stored energy

Charging a capacitor means carrying charge from one plate to the other against a rising voltage. When the charge is q the voltage is q/C, and the work to add dq is \frac qC\,dq. Integrating from 0 to Q:

    \[ U = \int_0^Q\frac qC\,dq = \frac{Q^2}{2C} = \frac12\,C\,V^2 \]

For the parallel-plate capacitor, \frac12 CV^2 = \frac12\frac{\varepsilon_0 A}{d}(Ed)^2 = \frac12\varepsilon_0 E^2\cdot(Ad): the energy density \frac12\varepsilon_0E^2 times the volume between the plates. The energy really is in the field.

Example. A camera flash capacitor, 470 µF charged to 330 V, holds \frac12\cdot 470\times 10^{-6}\cdot 330^2 \approx 26 J, dumped into the tube in a millisecond: a peak power of tens of kilowatts. It is also why power supply capacitors must be discharged before you put your hands in.

Dielectrics

Inserting an insulator between the plates increases the capacitance by a factor \varepsilon_r, the relative permittivity:

    \[ C = \varepsilon_r\,\frac{\varepsilon_0 A}{d} \]

The reason is polarisation: the field aligns the dipoles of the molecules (or creates them, by distorting the atoms), and the bound charges appearing on the faces of the dielectric reduce the internal field by a factor \varepsilon_r. For the same charge the voltage drops, so the capacitance rises.

Material\varepsilon_rTypical use
air1.0006variable capacitors
polypropylene2.2film capacitors, audio and power
FR-4 fibreglassabout 4.4printed circuit boards
aluminium oxideabout 9electrolytic capacitors
tantalum oxideabout 26tantalum capacitors
barium titanate ceramicsfrom 1000 to over 10,000multilayer ceramic capacitors

Real capacitors

A multilayer ceramic capacitor (MLCC) stacks hundreds of dielectric layers a few micrometres thick, alternating with electrodes, and so packs microfarads into a one-millimetre cube. An electrolytic uses as dielectric a layer of aluminium oxide nanometres thick, grown on a foil etched to multiply its area; that is why it has a polarity to respect.

A real capacitor is not ideal: it has a series resistance (ESR) and a series inductance (ESL), and high-\varepsilon_r ceramics lose much of their capacitance when a DC voltage is applied. These details are read in datasheets and they matter when designing a power supply.

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