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
C 



Q
V
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)
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
Quick Quiz 26.1
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.
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
C 
Q
V

Q
Ed

Q
Q
0A



d
0A
d
i.e. proportional to the area of plates and inversely
proportional to the distance between them
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.
Quick Quiz 26.2
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.

Capacitance of Isolated Sphere


Assume a spherical charged conductor
Assume V = 0 at infinity. Then
C 

Q
V

Q
k eQ / R

R
ke
 4 0R
independent of charge and potential
difference
Spherical Capacitor (ex 26.3)
b
V  

b
E r dr   k e Q
a
a
 1
1 
 k e Q   
 b
a 
C 
Q
V


1
 1
1 
k e   
 a
b 
b
1 
dr
 k e Q  
2
r a
r
Cylindrical Capacitor (ex. 26.2)


Q= l
E = 2ke / r

From Gauss’s Law
(exercise…)
b
V  

a
C 
Q
V
b
E r dr   2 k e 



a
l
 a
2 k e ln b
dr
r
 a
  2 k e  ln b
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
1
C

 V1

Q1
1
C1


1
C
2
V2
Q2
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.
Quick Quiz 26.3
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 26.4
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.
Quick Quiz 26.4
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.
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
dW  Vdq 
q
dq
C

The total work required is
W 

Q
0
q
C
dq 
Q
2
2C
Energy, cont

The work done in charging the capacitor
appears as electric potential energy U:
(remember C = Q/V)
U 
Q
2
2C




1
2
Q V 
1
C ( V )
2
2
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
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 CV2 =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.
Quick Quiz 26.5
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 26.6
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.
Quick Quiz 26.6
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=/20).
(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 26.7
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.
Quick Quiz 26.7
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).
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.
Quick Quiz 26.9
Answers:
(a) C increases (C= C0).
(b) Q increases. Because the battery maintains a constant
ΔV, Q must increase if C increases (Q=CV).
(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.
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
Quick Quiz 26.8
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.