Electricity and Magnetism
Dielectrics and Capacitors
Lana Sheridan
De Anza College
Oct 14, 2015
Last time
• Parallel plate capacitors
• Circuits and circuit diagrams
• Capacitors in series and parallel
• Energy stored in a capacitor
Warm Up Question
pictorial
epresentation of two
apacitorsTwo
connected
in
capacitors
eries to a battery
of
circuit as shown:
C1
"Q
!
A circuit diagram
showing the two
connected
values capacitors
4.0 nF and
6.0 nF
in series to a battery
C2
#V 2
!Q "Q
C1
C2
#V1
#V2
are
A circuit diagram
showing the equivalent
capacitanceinofathe
connected
capacitors in series
1
C eq
$
1
!
1
C1 C2
"
! "
#V
#V
(A) 4.0 nF
b
! "
#V
c
(B) 6.0 nF
dividual
capacitances.
Statement (2) makes sense because we are essentially
(C)
10 nF
ning the areas of all the capacitor plates when they are connected with connF
g wire,(D)
and 2.4
capacitance
of parallel plates is proportional to area (Eq. 26.3).
Warm Up Question
pictorial
epresentation of two
apacitorsTwo
connected
in
capacitors
eries to a battery
of
circuit as shown:
C1
"Q
!
A circuit diagram
showing the two
connected
values capacitors
4.0 nF and
6.0 nF
in series to a battery
C2
#V 2
!Q "Q
C1
C2
#V1
#V2
are
A circuit diagram
showing the equivalent
capacitanceinofathe
connected
capacitors in series
1
C eq
$
1
!
1
C1 C2
"
! "
#V
#V
(A) 4.0 nF
b
! "
#V
c
(B) 6.0 nF
dividual
capacitances.
Statement (2) makes sense because we are essentially
(C)
10 nF
ning the areas of all the capacitor plates when they are connected with connF ← of parallel plates is proportional to area (Eq. 26.3).
g wire,(D)
and 2.4
capacitance
Overview
• Dielectrics
• molecular view of dielectrics
• Guass’s law with dielectrics
• electric displacement
• some uses of dielectrics
Dielectrics
dielectric
an insulating material that can affects the strength of an electric
field passing through it
Different materials have different dielectric constants, κ.
Dielectrics
dielectric
an insulating material that can affects the strength of an electric
field passing through it
Different materials have different dielectric constants, κ.
κ tells us how the capacitance of a capacitor changes if the
material between the plates is changed.
For air κ ≈ 1. (It is 1 for a perfect vacuum.)
κ is never less than 1. It can be very large > 100.
Dielectrics and Capacitance
dielectric
an insulating material that can affects the strength of an electric
field passing through it
The effect of sandwiching a dielectric in a capacitor is to change
the capacitance:
C → κC
κ is the dielectric constant.
670 670 CHAPTER
CHAPTER
25 CAPACI
25 CAPACITANC
Dielectric in a Capacitor
Capacitance C
Capacitance κC
+ + + ++ + + +
+
– – – –– – – –
–
B
++ + + + + + ++++ + + + + + +
κ
κ
–– – – – – – –––– – – – – – –
B
Adding a dielectric increases the capacitance.
V = a constant
V = a constant
(a)
(a)
Effect of a Dielectric
The most straightforward way of tracking quantities that will
change when a dielectric is added is by replacing 0 in all
equations with using this relation:
= κ0
(Or just think of the effect of the dielectric being 0 → κ0 .)
The electrical permittivity increases.
Dielectric in a Capacitor
For a parallel plate capacitor with a dielectric, the capacitance is
now:
C=
κ0 A
d
Dielectric in a Capacitor
670
If we add
a dielectric
while the capacitor
is connected to a battery:
CHAPTER
25 CAPACITANCE
+ + + +
+
– – – –
–
B
++++++++
κ
––––––––
+
–
+ +
B
– –
V = a constant
(a)
(a) If the potential difference
between the plates of a capacitor is maintained, as by battery B, the effect of a dielectric is to increase the charge on the plates.
(b) If the charge on the capacitor plates is
Fig. 25-13
increase the cap
tanate, can incre
Another ef
difference that
the breakdown
Dielectric in a Capacitor
670
If we add
a dielectric
while the capacitor
is connected to a battery:
CHAPTER
25 CAPACITANCE
+ + + +
+
– – – –
–
B
++++++++
κ
––––––––
+
–
+ +
B
– –
V = a constant
(a)
• q Fig.
will increase.
25-13 (q
(a)=IfCV
the) potential difference
the plates
a capacitor
is main2)
• Ubetween
will increase.
(U =of1 CV
tained, as by battery 2B, the effect of a dielectric is to increase the charge on the plates.
(b) If the charge on the capacitor plates is
increase the cap
tanate, can incre
Another ef
difference that
the breakdown
Dielectric in a Capacitor
If we add a dielectric while the capacitor is isolated so charge
cannot leave the plates:
+
–
+ + + +
B
+ + + +
κ
– – – –
0
– – – –
0
VOLTS
VOLTS
q = a constant
(b)
increase the capacitance of a capacitor, and some materials, such as strontium
tanate, can increase the capacitance by more than two orders of magnitude.
Another effect of the introduction of a dielectric is to limit the poten
difference that can be applied between the plates to a certain value Vmax, ca
the breakdown potential. If this value is substantially exceeded, the dielec
Dielectric in a Capacitor
If we add a dielectric while the capacitor is isolated so charge
cannot leave the plates:
+
–
+ + + +
B
+ + + +
κ
– – – –
0
– – – –
0
VOLTS
VOLTS
q = a constant
(b)
q
will decrease.
= C)
increase •theV capacitance
of (V
a capacitor,
and some materials, such as strontium
tanate, can increase the capacitance
by more than two orders of magnitude.
q2
• U will
decrease.
= 2C
)
Another
effect
of the (U
introduction
of a dielectric is to limit the poten
difference that can be applied between the plates to a certain value Vmax, ca
the breakdown potential. If this value is substantially exceeded, the dielec
Effect of a Dielectric on Field
E
Imagine again the isolated conductor: charge density σ is constant.
+
–
+ + + +
B
+ + + +
κ
– – – –
0
– – – –
0
VOLTS
VOLTS
q = a constant
(b)
The electric field between the plates is E = σ0 originally.
increase the capacitance of a capacitor, and somematerials,
such as strontium t
tanate, can increase the capacitance by more than two orders of magnitude.
σ
With dielectric
E → κ
. of a dielectric is to limit the potenti
Another
effect added:
of the introduction
0
difference that can be applied between the plates to a certain value Vmax, calle
E substantially exceeded, the dielectr
the breakdown
potential.
If this value
The field strength
decreases!
E → is
κ
material will break down and form a conducting path between the plates. Ever
dielectric material has a characteristic dielectric strength, which is the maximu
value of the electric field that it can tolerate without breakdown. A few suc
Effect of a Dielectric on Field
E
Imagine again the isolated conductor: charge density σ is constant.
+
–
+ + + +
B
+ + + +
κ
– – – –
0
– – – –
0
VOLTS
VOLTS
q = a constant
(b)
The electric field between the plates is E = σ0 originally.
increase the capacitance of a capacitor, and somematerials,
such as strontium t
tanate, can increase the capacitance by more than two orders of magnitude.
σ
With dielectric
E → κ
. of a dielectric is to limit the potenti
Another
effect added:
of the introduction
0
difference that can be applied between the plates to a certain value Vmax, calle
E substantially exceeded, the dielectr
the breakdown
potential.
If this value
The field strength
decreases!
E → is
κ
material will break down and form a conducting path between the plates. Ever
dielectric
a characteristic
dielectric
strength, which is the maximu
Whatmaterial
happenshas
to the
energy density
u?
value of the electric field that it can tolerate without breakdown. A few suc
Effect of a Dielectric on Field
What happens to the energy density? Was: u0 = 12 0 E02 .
u =
1
(κ0 ) (E )2
2
Effect of a Dielectric on Field
What happens to the energy density? Was: u0 = 12 0 E02 .
u =
=
=
=
u =
Energy density decreases.
1
(κ0 ) (E )2
2
1
σ 2
(κ0 )
2
κ0
1
1
0 κ
E02
2
κ2
1 1
2
0 E0
κ 2
u0
κ
Dielectrics and Electric Field
Dielectrics effect the field around a charge
E→
E
κ
For example, for a point charge q in free space:
E0 =
kq
1 q
=
r2
4π0 r 2
But in a dielectric, constant κ:
E=
1
q
E0
=
2
4π(κ0 ) r
κ
Dielectrics and Electric Field
Dielectrics effect the field around a charge
E→
E
κ
For example, for a point charge q in free space:
E0 =
kq
1 q
=
r2
4π0 r 2
But in a dielectric, constant κ:
E=
But how does this happen?
1
q
E0
=
2
4π(κ0 ) r
κ
Dielectrics and Electric Field
Dielectrics become polarized by the presence of an electric field.
There are two types of dielectrics, the process is a little different in
each:
• polar dielectrics
• nonpolar dielectrics
tinuously jostling each other as a result of their random thermal
ment
is not complete,
but it becomes more complete as the magPolar
Dielectrics
plied field is increased (or as the temperature, and thus the
ased). The
of the electric
dipoles aligns
produces
electric of the
Thealignment
external electric
field partially
the an
molecules
d opposite
the
applied
field
and
is
smaller
in
magnitude.
dielectric with the field.
–
–
–
–
+
–
+
–
(a)
–
–
+
+
–
–
–
+
+
+
+
+
+
–
–
+
+
+
+
+
p
+
–
–
+
–
–
–
–
+
–
+
+
–
+
+
–
–
cules
ric dipole
random
ce of an
b) An
producthe
on prent.
+
(b)
Since the dielectric is an insulator, there are no free charges to
move through the substance, but molecules can align.
eg. distilled water
1
Figures from Halliday, Resnick, Walker, 9th ed.
Nonpolar Dielectrics
Nonpolar dielectrics are composed of molecules which are not
polar.
However, under the influence of a field, the distribution of the
electrons in the molecules, or the shape of the molecule, is altered.
Each molecule becomes slightly polarized.
"
!
"
a
S
E
"
b
!
"
Nonpolar Dielectrics
CHAPTER 25 CAPACITANCE
(a) are not
Nonpolar dielectrics are composed of molecules which
polar.
2. Nonpolar dielectrics.
The initial
electric
field of a field, the distribution of the
However,
under
the influence
Thedipole
applied moments,
field
mo
inside this
electrons
in thenonpolar
molecules, or the shape of the molecule, is altered.
aligns
the
atomic
placed in an externa
Each
moleculeslab
becomes
slightly polarized.
dielectric
is zero.
eg. nitrogen, (a)
benzene
slightly separating the
–+
–+
–+
–+
–
–
–
–
–
–
–
–
E0 = 0
+ + + + + + + +
dipole
thatmoments.
this occurs beca
Figure 25-15a shows
applied. In Fig. 25-15b, a
–+ –+ –+ –+
are charged as shown. T
E negative charge
tive and
–+ –+ 0–+ –+
on one face of the slab
charge(b)on the opposite f
as a whole remains elect
charge in any volume ele
Electric field inside
Thethe
fielddielectric
of the aligned
atoms is opposite the
applied field.
–
–
–
+
E
E'
E0
(c)
+
+
–
–
–
–
–
–
–
–
+ + + + + + + +
The polarized dielectric contributes its own field, E 0 .
dielectric m
shows a pa
dielectric. W
tions. Note
dielectric by
For the
:
field E0 betw
top plate wi
the magnitu
This reduces the electric field from the charged plates aloneor
E0 .
Fig. 25-15 (a) A nonpolar dielectric
slab. The circles represent the electrically
In Fig.
The resulting reduced field is E = Eκ0
neutral atoms within the slab. (b) An elecbetween the
tric field is applied via charged capacitor
face. Howev
!
:
:
(25-30)
#0 E
! dA "
#0EA " q,
Guass’s Law
with
dielectrics
E0 "
q κ Φ = q
free
. 0 E
#0 A
(25-31)
or:
25-16b, with the dielectric in place,
we can find the electric field
I
qfree
e plates (and within the dielectric)Eby
using
· dA
= the same Gaussian surver, now the surface encloses two
charge:
It still encloses
0
A types of κ
ate
withith a dinserted.
on the
med to
n both
+q
Gaussian surface
Gaussian surface
+ + + + + + + + + +
+ + + + + + + + + +
–
–
–
–
–
E0
+q
–q
– – – – – – – – – –
E
κ
+ + + + +
– – – – – – – – – –
–q'
+q'
–q
(a)
(b)
The charge qfree = q in the diagram. It is just the charge on the
plates, the charge that is free to move.
Capacitor
with
a Metal
Ex 26.7
opposite
in sign
to that
on theslab,
plates,
A
parallel-plate
capacitor
has
a
plate
separation
d and plate area
conductor (see Fig. 24.16).
A. An uncharged metallic slab of thickness a is inserted midway
between the plates. Find the capacitance of the device.
c Slab
tion d and plate
ness a is inserted
llic slab between
appears on one
of equal magnihe slab as shown
"
"
"
"
"
s
(d ! a)/2
d a
"
"
"
"
(d ! a)/2
!
!
!
!
! !s
!
!
!
!
"
"
"
"
"
"
"
"
"
!
!
s
(d ! a)/2
!
!
(d ! a)/2
!
!
! !s
!
!
a Metal slab, Ex 26.7
gnCapacitor
to that onwith
the plates,
A
parallel-plate
capacitor
has a plate separation d and plate area
ee Fig. 24.16).
A. An uncharged metallic slab of thickness a is inserted midway
between the plates. Find the capacitance of the device.
te
ed
en
ne
niwn
"
"
"
"
"
s
(d ! a)/2
d a
"
"
"
"
"
(d ! a)/2
!
!
!
!
! !s
!
!
!
!
!
"
"
"
"
"
"
"
"
"
"
!
!
!
s
(d ! a)/2
!
!
(d ! a)/2
!
!
! !s
!
!
gnitude but opposite in sign to that on the plates,
of zero in the conductor (see Fig. 24.16).
Capacitor with a Metal slab, Ex 26.7
of a Metallic Slab
a plate separation d and plate
c slab of thickness a is inserted
"
"
"
"
"
s
(d ! a)/2
e device.
d a
"
"
"
"
!
!
!
!
! !s
!
!
!
!
!
"
"
"
"
"
"
"
"
"
"
!
!
!
s
(d ! a)/2
hows the metallic slab between
ny charge that appears on one
duce a charge of equal magninear side of the slab as shown
ly, the net charge on the slab
ield inside the slab is zero.
"
(d ! a)/2
!
!
(d ! a)/2
!
a
!
! !s
!
!
b
This is just 2 capacitors
in series!
Figure 26.23 (Example 26.7) (a) A parallel-plate capacitor of plate separation
d partiallyfilled with a metallic slab
ge on the metallic slab’s upper
of thickness a. (b) The equivalent circuit
to the distribution of charges
1
1 −1 of the device in
(a) consists of two capacitors in series, each having a plate
=
+
The metal between the slab’s Ceq
separation (d 2 a)/2.
C1 C2
electrical connection between
model the edges of the slab as
lk of the slab as a wire. As a result, the capacitor in Figure 26.23a is equivalent to two
g a plate separation (d 2 a)/2 as shown in Figure 26.23b.
nd the rule for adding two
0) to find the equivalent
1
1
1
5
1
5
C
C1
C2
C5
P0 A
1
1
1
P0 A
P0 A
1 d 2 a2 /2
1 d 2 a2 /2
gnitude but opposite in sign to that on the plates,
of zero in the conductor (see Fig. 24.16).
Capacitor with a Metal slab, Ex 26.7
of a Metallic Slab
a plate separation d and plate
c slab of thickness a is inserted
"
"
"
"
"
s
(d ! a)/2
e device.
d a
"
"
"
"
!
!
!
!
! !s
!
!
!
!
!
"
"
"
"
"
"
"
"
"
"
!
!
!
s
(d ! a)/2
hows the metallic slab between
ny charge that appears on one
duce a charge of equal magninear side of the slab as shown
ly, the net charge on the slab
ield inside the slab is zero.
"
(d ! a)/2
!
a
!
(d ! a)/2
!
!
! !s
!
!
b
This is just 2 capacitors
in series!
Figure 26.23 (Example 26.7) (a) A parallel-plate capacitor of plate separation
d partiallyfilled with a metallic slab
ge on the metallic slab’s upper
of thickness a. (b) The equivalent circuit
to the distribution of charges
1
1 −1 of the device in
(a) consists of two capacitors in series, each having a plate
=
+
The metal between the slab’s Ceq
separation (d 2 a)/2.
C1 C2
electrical connection between
model the edges of the slab as
−1
lk of the slab as a wire. As a result, the capacitor(d
in Figure
26.23a is (d
equivalent
to two
− a)/2
− a)/2
=Figure 26.23b.
+
g a plate separation (d 2 a)/2 as shown in
0 A
0 A
1
1
1
1
1
1
5
1
5 A
nd the rule for adding two
PA
0P A
C
C1
C
0) to find the equivalent
= 2 1 d 20 a2 /2 1 d 20 a2 /2
(d − a)
C5
P0 A
trics
Partially-Filled Capacitor, Ex 26.8
A parallel-plate capacitor with a plate separation d has a
citor capacitance C0 in the absence of a dielectric. What is the
capacitance when a slab of dielectric material of dielectric constant
κ and thickness fd is inserted between the plates, where f is a
n d has
a between 0 and 1?
fraction
hat is the
k
fd
dielectric
the plates
1?
dielectrics
filled the
part of the
material.
fd
(1 ! f )d
k
d
(1 ! f )d
Partially-Filled Capacitor, Ex 26.8
What is the capacitance when a slab of dielectric material of
dielectric constant κ and thickness fd is inserted between the
plates, where f is a fraction between 0 and 1?
fd
fd
(1 ! f )d
k
C1
k
d
(1 ! f )d
C2
ance and Dielectrics
Partially-Filled Capacitor, Ex 26.8
Filled Capacitor
late separation d has a
a dielectric. What is the
ic material of dielectric
rted between the plates
etween 0 and 1?
fd
fd
C1
k
d
(1 ! f )d
scussions of dielectrics
the dielectric filled the
xample, only part of the
s the dielectric material.
k
(1 ! f )d
C2
a
b
ound that an infinitesiAgain, 2 capacitors in series!
between the plates of a
Figure 26.24 (Example 26.8) (a) A parallel-plate capacitor
−1 of thickcitance. Imagine sliding
of plate separation d partially filled with a dielectric
1 capacitor
ness fd. (b) The equivalent1
circuit of the
consists of
b along the bottom face
Ceq connected
=
+
two capacitors
in series.
.24a. We can model this
C
C2
1
wo capacitors as shown
s a plate separation fd and is filled with a dielectric; the other has a plate separation
ates.
nces in Figure 26.24b
rom Equation 26.10
C1 5
1
5
kP0 A
fd
and C 2 5
1
1
1
5
fd
P0 A
11 2 f 2d
1
11 2 f 2d
ance and Dielectrics
Partially-Filled Capacitor, Ex 26.8
Filled Capacitor
late separation d has a
a dielectric. What is the
ic material of dielectric
rted between the plates
etween 0 and 1?
fd
fd
C1
k
d
(1 ! f )d
scussions of dielectrics
the dielectric filled the
xample, only part of the
s the dielectric material.
k
(1 ! f )d
C2
a
b
ound that an infinitesiAgain, 2 capacitors in series!
between the plates of a
Figure 26.24 (Example 26.8) (a) A parallel-plate capacitor
−1 of thickcitance. Imagine sliding
of plate separation d partially filled with a dielectric
1 capacitor
ness fd. (b) The equivalent1
circuit of the
consists of
b along the bottom face
Ceq connected
=
+
two capacitors
in series.
.24a. We can model this
C
C2
1
wo capacitors as shown
other has a plate separation
s a plate separation fd and is filled with a dielectric; the
df
(1 − f )d −1
ates.
=
+
nces in Figure 26.24b
rom Equation 26.10
C1 5
1
5
κ0 A
0 A
P0 A
κ
11 2 f 2d
=
C0
f
+
κ(1
1 1 2 f 2−
fd
d f)
kP0 A
fd
and C 2 5
1
1
1
5
1
.00 mm, the dielectric is glass (k 5 4.50), and
apacitor
was charged
Partially-Filled
Capacitor to 2.00 3 103 V before
ielectric was inserted. Suggestion: The system can
onsidered
as two capacitors connected in parallel
What about this case?
!
!
!
!
x
k
"
"
! !Q
!
d
"
"
Figure P26.78
" "Q
Electric Displacement
It is sometimes convenient to package the effect of the electric
field together with the effect of the dielectric.
For this, people use a quantity, Electric Displacement.
D = κ0 E
Gauss’s law is very often written in terms of the electric
displacement, rather than the electric field, if the field being
studied is in a polarizable material.
Uses of
plates of the capacitor are
the wallboard
Dielectric
Effectsand air.
Capacitor
plates
Stud
finder
Stud
Wallboard
1
When
capacitor
Figures from Serway
& the
Jewett,
9th ed.
moves across
commonly
oil (Fig. 26.
Often, a
low voltage
tact with an
ions contai
electrolyte,
layer serves
an electroly
plate separ
Electroly
have a pola
device. Wh
rect. If the
oxide layer
Uses of Dielectric Effects
Computer keyboard:
Key
B
Movable plate
Insulator
Fixed plate
Figure 26.3
(Quick Quiz 26.2)
One type of computer keyboard
25.6) is sm
voltage of
the batter
more cha
plates. Wh
battery, th
the charg
result, the
Q uick Qu
as show
the mo
what ha
in a wa
Summary
• dielectrics
• molecular view of dielectrics
• Guass’s law with dielectrics
• electric displacement
• some uses of dielectrics
Homework
Serway & Jewett:
• PREVIOUS: Ch 26, onward from page 799. Problems: 13, 17,
21, 25, 31, 33, 35
• NEW: Ch 26. Problems: 43, 47, 49, 53, 63