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Lecture 9-1
Capacitors
Q
Q
+Q
-Q
Q
• A capacitor is a device that is capable of
storing electric charges and thus electric
potential energy.
=> charging
• Its purpose is to release them later in a
controlled way.
=> discharging
• Capacitors are used in vast majority of
electrical and electronic devices.
Typically made of two
conductors and, when
charged, each holds equal
and opposite charges.
Lecture 9-2
DOCCAM 2 DEMO 5A-28
PARALLEL PLATE CAPACITOR
Lecture 9-3
Polar and Nonpolar Dielectrics
 Polar dielectrics: dielectric
material whose molecules have
permanent electric dipole
moments, such as water.
• Eext = 0: the orientations of the permanent electric dipoles are
distributed randomly, so the net dipole moment of the material is zero.
• Eext > 0 : the molecular dipoles try to align themselves with the field
against random thermal motion, resulting in a net dipole moment.
 Nonpolar dielectrics: molecules with
no permanent electric dipole moments.
• Eext > 0 : ± charges separate,
induced dipole moment emerges.
Lecture 9-4
READING QUIZ 1
A capacitor C0 with air dielectric κ =1 is charged to a potential V and then
is disconnected from the battery. A uniform dielectric with dielectric
constant κ > 1 is inserted into the capacitor and occupies the full volume
of the capacitor. If the original electric field is E0 and the final electric field
is Eκ , which of the following statements is correct.?
A|
B|
C|
D|
E 0 = Eκ
E 0 < Eκ
E 0 > Eκ
E 0 = Eκ / κ
Lecture 9-5
+Q
Dielectrics between Capacitor Plates
free
charges
-Q










• Electric field E between plates can
be calculated from Q – q.
E
(Q  q) / A
0
neutral
-q
+q
Polarization
Charges ± q
, V  Ed
Q
Q
C 
V (Q  q)d /  0 A

0 A
d

1
q
1
Q
A very polarizable substance can have q
nearly as large as Q, and this multiplies the
capacitance C by a large factor.
Lecture 9-6
DOCCAM 2
DEMO 5A-30
EFFECT OF DIELECTRICS ON CAPACITANCE
Lecture 9-7
Inserting Dielectrics (= Insulator)
• Inserting a dielectric between the plates of
a capacitor increases capacitance
Q=CV
holds more charges at fixed V
• Dielectric constant κ of a dielectric is the ratio
of the capacitance when filled with it to that
without it:
C
 
C0
κ > 1 always
(dimensionless)
•Under the same Q,
•V = V0/k
•E = E0/k
This is the answer to
your reading quiz
 Breakdown potential determined by
dielectric strength (Emax)
κ
Material
air (1 atm)
1.00059
paper
3.7
pyrex
5.6
water (20 o C)
80.4
strontium
titanate
310
Lecture 9-8
Capacitors in Parallel
V is common
q1 q2 q3
V 


C1 C2 C3
Equivalent Capacitor:
C
q
V
where
q  q1  q2  q3
q1  q2  q3
 Ceq 
 C1  C2  C3
V
Lecture 9-9
Capacitors in Series
q is common
 q  C1V1  C2V2  C3V3
Equivalent Capacitor:
C
q
V
where
V  V1  V2  V3
1 V1  V2  V3 1
1
1


 

Ceq
q
C1 C2 C3
Lecture 9-10
Conductor inserted between plates
• A parallel-plate capacitor with
conductor inserted in the middle
• Two capacitors of area A
in series
+q
-q
a
-Q
b
+q
-q
E=0 outside capacitors
q
E
 0 A between plates
+q
-q

-Q
+q
-q
E=0 in conductor
E
q
between plates
0 A
1
1
1
a
b
ab
0 A





C Ca Cb  0 A  0 A  0 A  C  a  b
Lecture 9-11
Warm-up quiz 2
All the capacitors have the same capacitance C,
What is the total capacitance between A and B?
A
A).
B).
C).
D).
E).
B
4/3 C
3/4 C
3/2 C
1/2 C
2/3
Lecture 9-12
Non-uniform Parallel-plate Capacitor 1
Equivalent to 2 capacitors in parallel
  A / 2  2 0 A / 2
C  C1  C2  1 0

d
d
 A    2 
 0  1

d  2 
V
Q
2d  Q

C  0 A( 1   2 )
Potential drop V in each is the same.
Q1  C1V 
Note
1
Q,
1   2
Q2  C2V 
Q1  Q2 if 1   2
even though V1  V2 , E1  E2
2
Q
1   2
Lecture 9-13
Non-uniform Parallel-plate Capacitor 2
Equivalent to 2 capacitors in series
1
1
1 d / 4 3d / 4




C C1 C2  0 A  0 A
d 1 3 1

 
 0 A  4 4  
Q d Q  1 3 
V 
 

C  0 A  4 4 
(Free) charge in each Q is the same.
V1  Q / C1 
Q
( 0 A / d ) 1
 V0 ,
( 0 A / d ) C1
4
Q
( 0 A / d )
3
V2  Q / C2 

V0 ,
( 0 A / d ) C2
4
Note V1 / V2  1/ 3
if
 1
Lecture 9-14
Inserting Dielectric Material with Battery Disconnected
1.
Charge a parallel plate capacitor filled with air
(or vacuum) to potential difference V0.
Deposits charge Q  C0V0
Q
2. Disconnect the battery
Q remains fixed
-Q
3. Insert a dielectric of dielectric constant κ
Q
C   C0  
V0
So, V and E decreases from V0 , E0 to
V 
-Q
E
V
 0
d

1
2

E
u0 2 0 0 1
  0  E 2
and u  


2
Q V0

and
C

Q2 U 0
U

2C 
Q
E
Lecture 9-15
Inserting Dielectric Material with Battery Disconnected
1.
Charge a parallel plate capacitor filled with air
(or vacuum) to potential difference V0.
Deposits charge Q  C0V0
Q
2. Disconnect the battery
Q remains fixed
-Q
3.
Insert a dielectric of dielectric constant κ
4.
So, V and E decreases from V0 , E0 to
2
Q
C   C0  
V01
Q
 0 E02
-Q
U0
Q
u
1
U

  0  E 2
and u  0  2
2C 


2
We know that the dielectric is pulled into the gap: an external agent
does negative work – hence U is smaller than Uo. Physically, the
induced polarization charges are attracted to the plates
Lecture 9-16
Inserting Dielectric Material with Battery Connected
1.
Charge a parallel plate capacitor filled with
vacuum (air) to potential difference V.
Deposits charge Q0  C0V
2. Keep the battery connected
V remains fixed
3. Insert a dielectric of dielectric constant κ
Q0
Q0
Q0
C   C0  
V
So, Q increases from Q0 and E remains fixed
Q  CV   Q0
and
Q
V
E
 E0Q
d
1
1
2
2
and
u


u


E


 0     0
U  CV  U 0
0
0
2
2
U>Uo. But the plate is still pulled INTO the gap!
Where does all the positive energy come from??? (the battery)

Lecture 9-17
Energy stored in a capacitor revisited
+Q
+Q
-Q
Separation ~ 0
-Q

Q
E 
0 0 A
Battery disconnected
Change in
separation Δ d
External work required to separate the
plates from zero to d m apart is
Q2
W  F d 
d
2 A
0
What if a battery
remains connected?
Q2
F
2 0 A
The factor of ½ is
spurious. Force is just
QE, Work is just QEd.
E = Q/Aεo.
Do the algebra!
Lecture 9-18
Force and energy with battery connected, a mystery?
 Force between plates?
attractive
+Q
V
Work to change separation from 0 to d = +
Note: infinite Q and infinite Force at the
unphysical d=0 location
-Q

Q
E 
0 0 A
+Q
Battery connected
Separation d
Separation
d+Δd
-Q
Separation ~ 0 Q changes
Separation Δ d
However, the energy stored DECREASES
Since V is fixed and C decreases as d increases
Why? Battery must be included in energy conservation !
Lecture 9-19
Lecture 9:30 quiz 3 February 8, 2011
Five identical capacitors have the same capacitance C.
Then capacitor (e) is filled with a material with dielectric
constant k. What is the capacitance from A to B?
a
b
e
A
c
A).
B).
C).
D).
E).
B
d
2C
1/(1+k) C
(1+k)C
5 C
[4/(4+k)] C
Lecture 9-20
Lecture 10:30 quiz 3 February 8, 2011
Five identical capacitors have the same capacitance C.
Then capacitor (e) is filled with a material with dielectric
constant k. What is the total capacitance from A to B?
a
b
e
A
c
A). 2C
B). [k/(1+k)] C
C). (1+k)C
D). [1/(1+k)] C C
E). (4+k) C
d
B
Lecture 9-21
Lecture 11:30 quiz 3, February 8, 2011
Four identical capacitors have the same capacitance C.
Then capacitor (a) is filled with a material with dielectric
constant k. What is the total capacitance from A to B?
a
b
A
c
A).
B).
C).
D).
E).
(3/2 + k) C
(3+k) C
5/2 C
(2/3 +k) C
(3+k) C
B
d