5 電容 Capacitance
National Ignition Facility
Explosions in Airborne Dust
Flour explosion sugar
5-1電容與電容器
•電容
（Capacitance）•平行板電容器
• C = ( F 法拉 ) K: 介電常數，真空 = 1
0: 真空電容率 A: 板面積 d : 兩板距離
電位差電荷
V C q
d K0
The Uses of Capacitors
As storehouses of electric potential energy
As vital elements in tuned circuits (radio and TV) and RAM chips
Charging a capacitor
5-2 Calculating the Capacitance
1)
Put a charge q on the plates
2)
Find E using Gauss’s Law
3)
Find V between the plates
4)
Calculate C by its definition
Calculating the electric field
EA q
q A
d
E
00
Calculating the potential difference
) (
V V V
Eds V
s d E
V V
i f
f i i
f
A parallel-plate capacitor
d A V
C q
Ed ds
E Eds
V
EA q
d
0
0 0
A cylindrical capacitor
) /
ln(
or 2 )
/ 2 ln(
) 2 ln(
2 2
) 2
(
0 0
0
0 0
0 0
a b
L C a
b C L
a b L
q
r dr L
Eds q V
Lr E q
rL E
EA q
b a
A spherical capacitor
4
) 4 1 ( 1
4
4 4
1
) 4
(
0
0 0
2 0
2 0
2 0
0
a b
ab V
C q
ab a b
q b
a q
r dr Eds q
V
r E q
r E
EA q
b a
An Isolated Sphere – the
missing
plate is a conducting sphere of infinite radius
) (b
/ 4
4
01
0
R
b a
C a
Ex.1 1.0 F 電容器
• d = 1.0mm
2 2
8
12
3
0
100
~ 10
1 . 1
/ 10
85 . 8
10 0
. 1 0
. 1
km m
m F
m F
A Cd
Ex.2 a hyperbaric chamber
•Spark：> 2000V
•fire：> 0.20 mJ
•V
f = 600V→
•V
f：> 6000V•U
f：> 0.45 mJ= , 1/ 2 2
i i f f
q CV C V U CV
Ex.3 the threshold value Ut ( 150 mJ) required to ignite airborne grains.
2 0
3
0 0
4
4 , 1/ 2
2 2(150 10 )
4 4 (1.8m)
3.9 10 V
t
C R U CV
V U
R
Conducting floor
5-3 Capacitors in parallel
等效電容
n
j
j eq
eq
C C
C C
V C C q
V C
C C
q q
q q
V C
q V
C q
V C
q
1
3 2
1
3 2
1 3
2 1
3 3
2 2
1 1
) (
,
,
5-4 Capacitors in series
等效電容
3 2
1
3 2
1
3 2
1 3
2 1
3 3
2 2
1 1
1 1
1 1
/ 1 /
1 /
1
1
1 ) 1
( 1 ,
,
C C
C C
C C
C V
C q
C C
q C V
V V
V
C V q
C V q
C V q
eq eq
Ex.4 Finding eq. capacitor
q1 = ?
12 1 2
123 12 3
12
123 123 12 12 123
12
1 1 1 1 12
1 1 1
,
= ( )
( ) 31.0
C C C
C C C
q C V V q q q
C
q C V V V F
5-5 Storing Energy in an Electric Field
2 2
2 0
2 1 2
2 1
C CV U q
C q q
d C q
dW W
q C d
q q d V dW
q
Ex.5 A RAM Chip
electrons 10
8 . 10 1
60 . 1
3 . 5 10
55 6
19
15
C V F
e CV e
n q
C = 55fF（10-15） V = 5.3 volt
Ex.6
電容式鍵盤Ex.6
(cont)•
A = 9.50 10-5m2 ; k = 3.50•
按鍵後d由5.00mm減為0.150mm•
按鍵前 C = 0.589PFPF C 19.0
F
d PF
k 0 12
10 6
.
19
Touch Screen
Indium Tin Oxide；ITO 氧化銦錫
電 容 式 麥 克風
Ex.7 defibrillator (除纖顫機)
kW s 100
10 0
. 2
J 200
J 875 )
V 5000 )(
F 10
70 2 (
1 2
1
3
2 6
2
t P U
CV U
Z machine
Z pinch/Θ pinch 2 10 K/2.9 10 W 9 14
5-6 Capacitor with a dielectric
• The capacitance is increased by κ times κ: the dielectric constant
(Continued from the last slide)
9 7
0.07V, 7 10 m , =10 V/m V d E
0
0 0
0 0
2
0 0
, 2 , 4
ln( / ) ,
1 , 4
air air
A L ab
C C C
d b a b a
C L C C L
E q E
r
q increased V reduced
The effect of dielectrics
q
0A C V d
Ex 5-8 A porcelain slab as the dielectric
pJ U
U W
U pJ C
U q
C pJ q
CV U
f i
i f
i
893 2 162
1055 2 2
1
2
2 2
5-7 Dielectrics: An Atomic View
Polar and Nonpolar Dielectrics
The resultant electric field is
weakened
5-8 Dielectrics and Gauss’ Law
Without and with Dielectrics
q q A q
q E E
A q E q
q q
EA A
d E
A E q
q A
E A
d E
0 0
0 0
0
0 0
0 0
0
Gauss’ Law revised
nt disp laceme
electric the
or
0
0 0