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Chapter 22
Capacitance, Dielectrics,
Electric Energy Storage
Capacitors & capacitance
2
Capacitor / condenser: device that stores charge
For a given capacitor: Q ∝ V
Q
R
Q CV
C: Capacitance of capacitor
or Q
CV
+Q - QUnit: farad (F), μF & pF
Symbol in diagram:
Capacitors
3
Size, shape, relative position, insulating material
Capacitance depends on
Determine capacitance
4
For a parallel-plate capacitor:
+Q - Q
Sd
0 0
QE
S
0 0
d QdV Ed
S
(ignoring edge effect)
0 SQ
CV d
Cylindrical capacitor
5
Example1: A capacitor consists of two coaxial cylindrical shells (R1, R2), both with length
L>>R2. Calculate the capacitance.
L
R1R21 2
0
, 2
E R r Rr
2
1
212
0 1
ln2
R
R
RV Edr
R
0
12 2 1
2
ln( )
LQC
V R R
Spherical capacitor
6
Question: A capacitor consists of a sphere surrounded by a concentric spherical shell (R1, R2).
Calculate the capacitance.
Q2 Q1
R1
R2
- Q1 Q1+Q2
11 22
0
, 4
QE R r R
r
2
1
112
0 1 2
1 1( )
4
R
R
QV Edr
R R
0 1 2
12 2 1
4
R RQC
V R R
Capacitors in series and parallel
7
1) In series
1 2 ,Q Q Q
1C 2C
C1 2V V V
1 2
1 1 1 C C C
2) In parallel
1 2 ,V V V 1 2Q Q Q
1 2 C C C
1C
2C
C
Capacitor combination
8
Example2: Are the capacitors in series or parallel?
1)1C
2C
2)
Example3: How does C, Q, V change if d → 2d ?
1C 2C
V
d
parallel parallel
1C C Q
2V 1V
Electric energy storage
9
Charged capacitor stores electric energy
dW Vdq
+q - q
dq
21
2
q QW dq
C C
qdq
C
221 1 1
2 2 2
QU CV QV
C
Work required to store charges:
Energy stored:
Energy in the field
10
How does the electric energy distribute?
21
2U CV
Energy per unit volume / energy density:
201( )
2
SEd
d
20
1
2E Sd
20
1
2u E
Electric energy is stored in
the electric field!
d
S
Energy in capacitor
11
Example4: A parallel-plate capacitor is charged at 220V. (a) What is the capacitance; (b) how much electric energy can be stored?
d = 1mm
S =10cm2Solution: (a) 0 S
Cd
128.85 10 F
(b) Energy stored:
21
2U CV 72.14 10 J Maximum U ?
Different methods
12
Example5: Determine the electric energy stored in a spherical capacitor.
0 1 2
2 1
4 R RC
R R
Q R1
R2
- Q
Solution: 2 methods to solve it.
a) By using the capacitance:
Energy stored:21
2
QU
C
22 1
0 1 2
( )
8
Q R R
R R
13
20
1
2u E Q R1
R2
- Q
b) By using the energy density:
Total energy:
U udV2
0 1 2
1 1( )
8
Q
R R
20 2
0
1( )
2 4
Q
r
2
1
2 20 2
0
1( ) 4
2 4
R
R
Qr dr
r
22 1
0 1 2
( )
8
Q R R
R R
Discussion: Isolated conductor sphere? Q
Dielectrics
14
Insulating material in capacitors → dielectric
0 ,C KC
①Harder to break down → V ↗
②Distance between plates ↘
③ Increases the capacitance:
where K is the dielectric constant
Also noted as : relative permittivity r
Permittivity of material
15
If a dielectric is completely filled the space:
0C KC 0K S
d
S
d
: the permittivity of material
In a specific region filled with dielectric:
0K 0All in electric expressions →
00 0
QE
S
Q
ES
Electrostatics in dielectric
16
In a specific region filled with dielectric:
0 E
EK
field is reduced but not zero
The energy density in a dielectric: 21
2u E
Example6: How does C, Q, U change?
C Q U
CV battery disconnected?
Half filled dielectric
17
Question: How does the capacitance change if the capacitor is half filled with dielectric? (S, d, K)
1) 2)
Molecular description (1)
18
1) Polar dielectrics, such as H2O and CO
Have permanent electric dipole moments.
- +
p Ql
2) Nonpolar dielectrics, such as O2 and N2
No permanent electric dipole moments.
There are two different types of dielectrics:
Molecular description (2)
19
Polar Nonpolar
E
----
++++
Polarization
Bound chargeInduced field Eind
0 E
EK
Q -Q
indQ 1(1 )Q
K
Thinking & question
20
Thinking: Why charged objects can attract small paper scraps or water flow?
Question: Move a dielectric plate out of a charged capacitor, the electric energy increases, where does the extra energy come from?
CV
Brief review of Chapter 19-22
21
Coulomb’s law: 1 22
0
1
4
Q QF r
r
Gauss’s law: 0
= inE
QE dS
Electric potential: E V
aV E dl
Capacitance; Electric energy:
21,
2U CV 2
0
1
2u E
Dielectrics:
0
Chapter 23 & 24
22
Some useful concepts and equations:
These chapters should be studied by yourself
Ij
S
Ohm’s law: V = I R Current & resistance
Current density: I j dS
Microscopic statement of Ohm’s law:
/j E E
conductivity & resistivity
(ch23, pr.24)