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doPhysics · Electromagnetism 1

Capacitance and Dielectrics

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Why this chapter
Why this chapter exists

A charged conductor holds charge in proportion to its potential. Two conductors side by side, with equal and opposite charges, hold far more for the same voltage. Such a pair is a capacitor, and it stores energy in the field between its plates. Fans, flash guns and phones all contain capacitors. This chapter finds the capacitance of plates, spheres and cylinders, and the energy and force they carry. It joins capacitors together and follows the energy lost when they share charge. It ends with the insulator between the plates, which makes a capacitor larger, safer and able to store far more.

You will learn
By the end of this chapter you will
  1. Define capacitance as \(C = Q/V\), and find it for a sphere, parallel plates, concentric spheres and coaxial cylinders.
  2. Find the energy stored, \(\tfrac12CV^2\), and show that it lives in the field with density \(\tfrac12\varepsilon_0E^2\).
  3. Find the force between the plates, and follow \(Q\), \(V\), \(E\) and \(U\) when the plates move with the battery removed or connected.
  4. Combine capacitors in series and in parallel, solve networks and balanced bridges, and find the safe voltage of a series pair.
  5. Share charge between capacitors or conductors, and find the heat lost with either polarity.
  6. Explain how a dielectric reduces the field, and follow \(C\), \(Q\), \(V\), \(E\) and \(U\) when a slab goes in with the battery on or off.
  7. Find the capacitance with partial, side-by-side, stacked, sliding and varying dielectrics.
Before you begin
Before you begin

From the chapter Electric Potential, Energy and the Dipole, you should know the potential \(kQ/r\), the work \(qV\) and the torque on a dipole. From Electric Flux and Gauss's Law you should know the fields of a large sheet, a long wire and a shell. The chapter Conductors in Electrostatics helps: charge on a conductor sits on its surface, and the field inside the metal is zero. From Potential Energy and Conservation of Energy you should know \(F = -dU/dx\).

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Lessons

1
Capacitance and Its Geometries
The charge a conductor holds grows in step with its potential, and the ratio is its capacitance, fixed by shape and size. The isolated sphere, the parallel-plate capacitor, and the spherical and cylindrical capacitors.
Capacitance definedCapacitance of an isolated sphereThe parallel plate capacitorSpherical and cylindrical capacitors
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2
Energy and the Force Between the Plates
Charging a capacitor stores \(\tfrac12 CV^2\), and the battery gives twice that. The energy lives in the field, \(\tfrac12 \varepsilon_0 E^2\) in every cubic metre. The plates pull each other with \(\tfrac12 QE\), and pulling them apart changes \(Q\), \(V\), \(E\) and \(U\) in two different ways.
Energy stored in a capacitorEnergy density of the fieldForce between the platesPlates pulled apart at fixed q and at fixed v
15 min0%
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3
Capacitors in Series and in Parallel
In series every capacitor holds the same charge and the voltages add, so the reciprocals of C add. In parallel every capacitor has the same voltage and the charges add, so the capacitances add. Networks, safe voltages and the balanced bridge.
Capacitors in seriesCapacitors in parallelMixed networks of capacitorsThe balanced capacitor bridge
15 min0%
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4
Sharing Charge and the Heat Lost
Join a charged capacitor to another and charge flows until both have one voltage. The charge is conserved, but some stored energy always becomes heat, \(C_1 C_2(V_1 - V_2)^2/2(C_1 + C_2)\), whatever the wire. The same holds for two charged conductors.
Joining a charged to an uncharged capacitorEnergy lost as heatJoining plus to plus and plus to minusEnergy lost when conductors are joined
13 min0%
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5
Dielectrics: the Slab In, the Battery On or Off
An insulator between the plates is polarised, and its induced surface charges cut the field to \(E_0/K\), so \(C = KC_0\). With the battery removed \(Q\) stays and \(V\), \(E\) and \(U\) fall; with it connected \(V\) stays and \(Q\) and \(U\) rise. Breakdown, and why real capacitors are filled.
What a dielectric doesSlab in with the battery disconnectedSlab in with the battery connectedBreakdown and why real capacitors use dielectrics
15 min0%
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6
Partial, Stacked and Varying Dielectrics
A slab thinner than the gap acts as capacitors in series; dielectrics side by side act in parallel. A slab sliding in raises C in a straight line and is pulled in either way. A dielectric constant that varies is handled by adding thin slices.
A slab that fills part of the gapDielectrics side by side and stackedA slab sliding inA dielectric constant that varies
15 min0%
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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

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The Workbook
Questions to solve on paper, marked by you.
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Glossary

The key words of this chapter, with the meaning this chapter gives them.

capacitance
The charge a conductor, or a pair of conductors, holds for each volt of potential difference, \(C = Q/V\). It depends only on the sizes, shapes and spacing of the conductors and on the material between them.
capacitor
Two conductors, called plates, that carry equal and opposite charges \(+Q\) and \(-Q\). It stores charge, and energy in the field between the plates.
common potential
The single voltage that two capacitors, or two conductors, reach after they are joined and charge stops flowing between them.
dielectric
An insulating material, with no free charges, placed in an electric field; its molecules are polarised by the field.
dielectric constant
The factor \(K\) by which a material reduces the field of fixed charges that it fills, \(E = E_0/K\). It raises the capacitance of a filled capacitor to \(KC_0\).
dielectric strength
The largest field an insulator can bear before it breaks down and a spark passes; about \(3 \times 10^6\,\text{V/m}\) for dry air.
energy density
The energy stored per cubic metre of an electric field, \(u = \tfrac12\varepsilon_0E^2\) in vacuum or air, in J/m³.
farad
The unit of capacitance, one coulomb per volt, symbol F. It is very large, so μF, nF and pF are used.
fringing
The bowing outward of the field lines at the edges of a parallel-plate capacitor. It may be neglected when the gap is much smaller than the plates.
induced charge
The layers of bound charge left on the faces of a polarised dielectric, \(\sigma_i = \sigma(1 - 1/K)\) in a filled capacitor. They cannot leave the material.
parallel combination
Capacitors joined side by side between the same two points, so that each has the same voltage and their charges add; \(C = C_1 + C_2 + \cdots\).
polarisation
The lining up of molecular dipoles with an applied field, by stretching molecules or by turning molecules that are dipoles already.
series combination
Capacitors joined end to end, so that the same charge sits on each and their voltages add; \(1/C = 1/C_1 + 1/C_2 + \cdots\).