Chapter 26

Capacitance

and

Dielectrics

Capacitors

Capacitors are devices that store

electric charge

Examples where capacitors are used:

radio receivers

filters in power supplies

energy-storing devices in electronic flashes

Definition of Capacitance

The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors

Capacitance is a positive quantity.

It is a measure of the ability to store charge

The SI unit of capacitance is the farad (F) large, e.g. microfarads (µF) or picofarads (pF)

QC

V

Makeup of a Capacitor

A capacitor consists of

two conductors

These are called plates

When the plates are

charged, they have equal

and opposite charges

A potential difference

exists between the plates

due to this charge

Quick Quiz 26.1

A capacitor stores charge Q at a potential difference ΔV. If

the voltage applied by a battery to the capacitor is doubled to

2ΔV:

(a) the capacitance falls to half its initial value and the

charge remains the same

(b) the capacitance and the charge both fall to half their

initial values

(c) the capacitance and the charge both double

(d) the capacitance remains the same and the charge doubles

Answer: (d). The capacitance is a property of the physical

system and does not vary with applied voltage. According to

C=Q/V, if the voltage is doubled, the charge is doubled.

Quick Quiz 26.1

Demo Ea9: Parallel plate

capacitor and plate separation

Voltage proportional

to distance between

plates

V=Ed

Voltage decreases on

inserting dielectric

between plates

V = Q/κC0

Parallel Plate Capacitor

Each plate is connected

to a terminal of the

battery

Suppose initially

uncharged

Battery then establishes

an electric field in the

connecting wires

Parallel Plate Capacitor

Consider, first, the negative terminal: Field applies a force on electrons in the wire

Force causes the electrons to move onto the negative plate

Continues until equilibrium is achieved i.e. the plate, wire and terminal are all at the same

potential

There is now no field present in the wire

Hence the movement of electrons ceases

The plate is now negatively charged

Parallel Plate Capacitor

Now consider the positive terminal

A similar process occurs at the other plate,

with electrons moving away from the plate

and leaving it positively charged

In its final configuration, the potential

difference across the capacitor plates is

the same as that between the terminals

of the battery

Capacitance – Parallel Plates

Charge density σ = Q/A

Electric field E = /0 (for conductor) Uniform between plates, zero elsewhere

i.e. proportional to the area of plates and inversely proportional to the distance between them

C Q

VQ

Ed

Q

Q

0Ad

0A

d

Parallel Plate Assumptions

Assumption that electric field is uniform is valid in the

central region, but not at the ends of the plates

If separation between plates is small compared with

their length, effect of non-uniform field can be ignored

Quick Quiz 26.2

Many computer keyboard buttons are constructed of

capacitors, as shown in the figure below. When a key is

pushed down, the soft insulator between the movable plate

and the fixed plate is compressed. When the key is pressed,

the capacitance

(a) increases

(b) decreases

(c) changes in a way that we

cannot determine because the

complicated electric circuit

connected to the keyboard

button may cause a change

in ΔV.

Answer: (a). When the key is pressed, the plate separation is

decreased and the capacitance increases (since C=0A/d).

Capacitance depends only on how a capacitor is constructed

and not on the external circuit.

Quick Quiz 26.2

Capacitance of Isolated Sphere

Assume a spherical charged conductor

Assume V = 0 at infinity. Then

independent of charge and potential

difference

C Q

V

Q

keQ / R

R

ke

4 0R

Spherical Capacitor (ex 26.3)

V Erdr

a

b

keQ

dr

r2

a

b

keQ

1

r

a

b

keQ

1

b

1

a

C Q

V

1

ke

1

a

1

b

Cylindrical Capacitor (ex. 26.2)

Q= l

E = 2ke / r

From Gauss’s Law

(exercise…)

V Erdr

a

b

2 ke

dr

r

a

b

2 ke ln b

a

C Q

V

l

2 ke

ln ba

Geometry of Some Capacitors

Circuit Symbols

A circuit diagram is a

simplified representation

of an actual circuit

Circuit symbols are used

to represent the various

elements

Lines are used to

represent wires

The battery’s positive

terminal is indicated by the

longer line

ECM05ANA: Charging &

Discharging a Capacitor

Capacitors in Parallel

When capacitors are first

connected in the circuit,

electrons are transferred

from the left plates

through the battery to the

right plate, leaving the left

plate positively charged

and the right plate

negatively charged

Capacitors in Parallel, 2

The flow of charges ceases when the voltage across the capacitors equals that of the battery Maximum charge

Total charge is sum of the charges

Qtotal = Q1 + Q2

Potential difference across the capacitors is the same, equal to voltage of battery (V=V1=V2)

Hence Qtotal/V = Q1/V1 + Q2/V2, so

Ceq = C1 + C2

Capacitors in Parallel, 3

Capacitors can be replaced

with one capacitor with a

capacitance of Ceq

The equivalent capacitor

must have exactly the same

external effect on the circuit

as the original capacitors

Capacitors in Series

When a battery is

connected to the

circuit, electrons are

transferred from the

left plate of C1 to the

right plate of C2

through the battery

Capacitors in Series, 2

As negative charge accumulates on right

plate of C2, an equivalent amount of

negative charge is removed from left plate

of C2, leaving it with excess positive

charge

All the right plates gain charges of –Q, all

left plates have charges of +Q

Capacitors

in Series, 3

The potential differences

add up to the battery

voltage

Q Q1 Q

2

V V1 V

2

V

Q

V1

Q1

V

2

Q2

1

C

1

C1

1

C2

Capacitors in Combination

When two or more capacitors are connected in parallel, the potential differences across them are the same Charge on each capacitor proportional to its capacitance

Capacitors add directly to give the equivalent capacitance

When two or more capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same Capacitances add as reciprocals

Equivalent capacitance always less than smallest individual capacitor

Equivalent Capacitance

(exercise)

The 1.0-µF and 3.0-µF capacitors are in parallel as are the 6.0-µF and 2.0-µF capacitors

These parallel combinations are in series with the capacitors next to them

The series combinations are in parallel and the final equivalent capacitance can be found

Quick Quiz 26.3

Two capacitors are identical. They can be connected in

series or in parallel. If you want the smallest equivalent

capacitance for the combination, you should connect them in

(a) series

(b) parallel

(c) Either combination has the same capacitance.

Answer: (a). When connecting capacitors in series, the

inverses of the capacitances add (1/C = 1/C1 + 1/C2),

resulting in a smaller overall equivalent capacitance.

Quick Quiz 26.3

Quick Quiz 26.4

Consider two identical capacitors. Each capacitor is charged

to a voltage of 10 V. If you want the largest combined

potential difference across the combination, you should

connect them in

(a) series

(b) parallel

(c) Either combination has the same potential difference.

Answer: (a). When capacitors are connected in series, the

voltages add, for a total of 20 V in this case. If they are

combined in parallel, the voltage across the combination is

still 10 V.

Quick Quiz 26.4

Exercise

Connect plates of capacitor to battery. What

happens to the charge when the connecting

wires to the battery are removed?

Nothing! Charge remains on the plates.

What happens if the wires are now connected

to one another?

Charges move along wires and plates until the

entire conductor is at a single potential and the

capacitor is discharged.

Demo Eb14

Energy Storage in Capacitor

Charge up capacitor

using DC power

supply.

Disconnect and

attach leads to

electric motor.

Rotation of motor

enables work to be

done.

ECA05AN2:

Energy storage in a capacitor

Energy in a Capacitor –

Overview

Before switch is closed,

energy is stored as

chemical energy in

battery

When switch is closed,

energy is then

transformed from

chemical to electric

potential energy

Energy in a Capacitor –

Overview, cont

Electric potential energy related to

separation of positive and negative

charges on plates

Thus, a capacitor is a device that stores

energy as well as charge

Energy Stored in a Capacitor

Assume capacitor is being charged and, at some point, has a charge q on it and a potential difference V

The work then needed to transfer a charge dq from one plate to the other is

The total work required is

qd W V d q d q

C

2

0 2

Q q QW d q

C C

Energy, cont

The work done in charging the capacitor appears as electric potential energy U:

(remember C = Q/V)

Applies in any geometry

Energy stored increases as charge increases and as potential difference increases

In practice, there is a maximum voltage before discharge occurs between the plates

2

21 1( )

2 2 2

QU Q V C V

C

Energy, final

Energy is stored in the electric field

For parallel-plate capacitor, the energy

can be expressed in terms of the field:

U = ½ CV2 =½ (εoA/d)(Ed)2= ½ (εoAd)E2

Can also be expressed as the energy

density (energy per unit volume [Ad])

uE = U/[Ad] = ½ εoE2

Quick Quiz 26.5

You have three capacitors and a battery. In which of the

following combinations of the three capacitors will the

maximum possible energy be stored when the combination

is attached to the battery?

(a) series

(b) parallel

(c) Both combinations will store the same amount of energy.

Answer: (b). For a given voltage, the energy stored in a

capacitor is proportional to C: U = C(ΔV)2/2. Thus, you want

to maximize the equivalent capacitance and the potential

difference cross it. You do this by connecting the three

capacitors in parallel, so that the capacitances add, and each

capacitor has the same potential difference, ΔV, across it.

Quick Quiz 26.5

Quick Quiz 26.6

You charge a parallel-plate capacitor, remove it from the

battery, and prevent the wires connected to the plates from

touching each other. When you pull the plates apart to a

larger separation, do the following quantities increase,

decrease, or stay the same?

(a) C;

(b) Q;

(c) E between the plates;

(d) ΔV ;

(e) energy stored in the capacitor.

Answers:

(a) C decreases (C=0A/d).

(b) Q stays the same because there is no place for the charge

to flow.

(c) E remains constant (E=/20).

(d) ΔV increases because ΔV = Q/C, Q is constant (part b),

and C decreases (part a).

(e) The energy stored in the capacitor is proportional to both

Q and ΔV2 and thus increases. The additional energy comes

from the work you do in pulling the two plates apart.

Quick Quiz 26.6

Quick Quiz 26.7

Repeat Quick Quiz 26.6, but this time answer the questions

for the situation in which the battery remains connected to

the capacitor while you pull the plates apart.

Do this in your own time.

Answers:

(a) C decreases (C=0A/d).

(b) Q decreases. The battery supplies a constant potential

difference ΔV; thus, charge must flow out of the capacitor if

C = Q /ΔV is to decrease.

(c) E decreases because the charge density on the plates

decreases.

(d) ΔV remains constant because of the presence of the

battery.

(e) The energy stored in the capacitor decreases (U =

C(ΔV)2/2).

Quick Quiz 26.7

Some Uses of Capacitors

Defibrillators When fibrillation occurs, the heart produces a

rapid, irregular pattern of beats

A fast discharge of electrical energy through the heart can return the organ to its normal beat pattern

In general, capacitors act as energy reservoirs that can be slowly charged and then discharged quickly to provide large amounts of energy in a short pulse

Demo Eb4

Energy storage in a capacitor

Light bulb placed in

series with a capacitor.

Currents generated in

charging and

discharging the

capacitor.

Note that light bulb does

not have constant

resistance as the

temperature changes.

Capacitors with Dielectrics A dielectric is a nonconducting material that,

when placed between the plates of a

capacitor, increases the capacitance

Dielectrics include rubber, plastic, and waxed

paper

For a parallel-plate capacitor

C = κCo = κεo(A/d)

The capacitance is multiplied by the factor κ when

the dielectric completely fills the region between

the plates

Dielectrics, cont

In theory, d could be made very small to create a very large capacitance

In practice, there is a limit to how small: d is limited by the electric discharge that could

occur though the dielectric medium separating the plates

For a given d, the maximum voltage that can be applied to a capacitor without causing a discharge depends on the dielectric strength of the material

Dielectrics, final

Dielectrics provide the following advantages:

Increase in capacitance

Increase the maximum operating voltage

Possible mechanical support between the plates Allows plates to be close together without

touching

This decreases d and increases C

Dielectrics – An Atomic View

(not examinable)

The molecules that

make up the

dielectric are

modeled as dipoles

The molecules are

randomly oriented in

the absence of an

electric field

Dielectrics – Atomic View, 2

An external electric

field is applied

This produces a

torque on the

molecules

The molecules

partially align with

the electric field

Induced Charge and Field

The electric field due to the

plates is directed to the right

and it polarizes the dielectric

The net effect on the

dielectric is an induced

surface charge that results

in an induced electric field

If the dielectric were

replaced with a conductor,

the net field between the

plates would be zero

Quick Quiz 26.9

A fully charged parallel-plate capacitor remains connected to

a battery while you slide a dielectric between the plates. Do

the following quantities increase, decrease, or stay the same?

(a) C;

(b) Q;

(c) E between the plates;

(d) ΔV.

Answers:

(a) C increases (C= C0).

(b) Q increases. Because the battery maintains a constant

ΔV, Q must increase if C increases (Q=CV).

(c) E between the plates remains constant because ΔV = Ed

and neither ΔV nor d changes. The electric field due to the

charges on the plates increases because more charge has

flowed onto the plates. However, the induced surface

charges on the dielectric create a field that opposes the

increase in the field caused by the greater number of charges

on the plates.

(d) The battery maintains a constant ΔV.

Quick Quiz 26.9

End of Chapter

Dielectrics – Atomic View, 3

Degree of alignment of the molecules

with the field depends on temperature

and the magnitude of the field

In general,

the alignment increases with decreasing

temperature

the alignment increases with increasing

field strength

Dielectrics – Atomic View, 4

If the molecules of the dielectric are

nonpolar molecules, the electric field

produces some charge separation

This produces an induced dipole

moment

The effect is then the same as if the

molecules were polar

Dielectrics – Atomic View,

final

An external field can

polarize the dielectric

whether the molecules

are polar or nonpolar

The charged edges of the

dielectric act as a second

pair of plates producing

an induced electric field

in the direction opposite

the original electric field

Quick Quiz 26.8

If you have ever tried to hang a picture or a mirror, you

know it can be difficult to locate a wooden stud in which to

anchor your nail or screw. A carpenter’s stud-finder is

basically a capacitor with its plates arranged side by side

instead of facing one another, as shown in the figure below.

When the device is moved over a stud, the capacitance will:

(a) increase

(b) decrease

Answer: (a). The dielectric constant of wood (and of all

other insulating materials, for that matter) is greater than 1;

therefore, the capacitance increases (C=C0). This increase

is sensed by the stud-finder's special circuitry, which causes

an indicator on the device to light up.

Quick Quiz 26.8