introduction to capacitance
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25. Capacitance25. Capacitance
• Capacitor : storing charges, e.g. Photoflashconsists two conductors, which are charged oppositely.
Opposite Charges.
• Capacitance : Capacitor 의용량 , C
∆V
VqC
∆≡
VCF 11 =
C
Unit : farad (F)
∆V : 1 Coulomb per Voltq
VCq ∆⋅=
1F is too large. => Most capacitor units: µF (10-6 F) or pF (10-12 F)
: Typical Capacitor
• Charging a Capacitor :
25-3. Calculating the Capacitance• Spherical Conductor
+q Rq
RqkV e
041πε
==∆
RkR
VqC
e04πε==
∆=
Assume another conductor at ∞.
RC ∝
• Calculating the capacitance
(i) Assume stored charges +q and –q in the two conductors
(ii) Use Gauss’s Law to find E-field and ∆V between two conductorsqAdE =⋅∫rr
0ε
VqC
∆=(iii) Determine the capacitance
• Parallel Plate Capacitor Supposed stored charges +q and –q
Electric field
qAdE =⋅∫rr
0ε
AqE
AAE
00
0
εεσ
σε
==
=
Potential difference ∆V
dA
qV ⋅=∆0ε
Capacitance C
dA
VqC 0ε
=∆
= d,A 1∝
mpFmFmNC /85.8/1085.8/1085.8 1222120 =×=⋅×= −−εPermittivity :
For A = 0.012 m2 = 10-4 m2, d = 1 mm= 10-3 m C = 0.885 pF
• A Cylindrical Capacitor
λπεεε ⋅=⋅⋅=⋅=Φ ∫ LELrAdEc 2000
rr
rr
E ˆ2
1
0
λπε
=r
drr
drEV b
a
a
b
λπε
=⋅−=∆ ∫∫02
1
Lq
=λ ⎟⎠⎞
⎜⎝⎛
πελ
=abln
02⎟⎠⎞
⎜⎝⎛=
abLq ln
2 0πε
Linear charge density λ =q/L
L
Capacitance C( )ab
LVqC
/ln2 0πε
=∆
=
Capacitance per unit length
( ) ( )abkabLC
e /ln21
/ln2 0 ==πε
• A Spherocal Capacitor
rrqE ˆ
41
20πε
=r
drrqdrEV
b
a
a
b 202
1∫∫ =⋅−=∆
πε
ababq
baq −
=⎟⎠⎞
⎜⎝⎛ −=
00 411
4 πεπε
Capacitance C
abab
VqC
−=
∆= 04πε
25-4. Capacitors in Parallel and in Series• Symbols in Circuits
Capacitor
+− Battery
Switch
• Capacitors in Parallel (병렬)
33 qVC =11 qVC = 22 qVC =
( )VCCCqqqq 321321 ++=++=
321 CCCCeq ++=
⎟⎟⎠
⎞⎜⎜⎝
⎛=+++= ∑
jjeq CCCCC L321
• Capacitors in Series (직렬)
333
222
111
VCqVCqVCq
===
qqqq === 321
Charge Conserved
33 C
qV =1
1 CqV =
22 C
qV =
eqCQ
CCCqVVVV =⎟⎟
⎠
⎞⎜⎜⎝
⎛++=++=
321321
111
321
1111CCCCeq
++=
⎟⎟⎠
⎞⎜⎜⎝
⎛=+++= ∑
j jeq CCCCC11111
321
L
• Example. Capacitors in Parallel and Series
2112 CCC +=321312123
11111CCCCCC
++
=+=
( )( )
321
321123
321
321
123
1
CCCCCCC
CCCCCC
C
+++
=
+++
=
25-5. Energy Stored in an Electric Field+++++++
−−−−−−−
CVq = CdVdq =⇒
CVdVVdqdW ==
CQCVCVdVW22
1 22 === ∫
Work
Q V
dV dq
C
Electrostatic Energy : 221
21
2
2CVQV
CQU ===
Parallel Plate with Area AdAC 0ε
=
( )2020
21
21 Ed
dAV
dAU ⋅
ε=⋅
ε=
AdEU 202
1ε=
202
1 EudA
UVolU
ε==⋅
=Energy Density :(Energy per unit Volume)
202
1 Eu ε=* Energy density of any electrostatic field :
Example. Rewiring Two Charged Capacitors
Connect both switches
+ −
− +
Q1 = C1V1
Q2 = C2V2
21 QQQ −=
21 CCCeq +=
21
21
21 CCQQ
CCQV
+−
=+
=∆21
2211
CCVCVC
+−
=
021
21 VCCCCV ∆
+−
=∆⇒021 VVV ∆==If
2021
202
201 ))((
21)(
21)(
21 VCCVCVCUi ∆+=∆+∆=
2
21
21⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛⇒
CCCC
UU
i
f
20
21
2212
021 )()(21))((
21 V
CCCCVCCU f ∆
+−
=∆+=
25-6. Capacitor with a Dielectric• Dielectric Materials
0=Er 0≠E
r
1
00
>
=−=
κκQqQQeff
κ0EE =
AQE0
0
0 ε=
εσ
=r
dA
QV0
0 ε=∆
−−−−−−−
− Q0
+++++++
dAC 00 ε=
0
202
000 2)(
21
CQVCU =∆=
κκεσ
εκε0
00
0
0
1 EA
QA
QE eff
d ====+ Q0
Dielectric Materials
AdQVVd
0
00 1εκκ
==∆ +++++++
−−−−−−−
− + − + − + − +− + − + − + − +− + − + − + − +− + − + − + − +− + − + − + − +
+ Q0 − Q0
000 C
dA
VQC
dd κκε
==∆
=
Increase of Capacitance
κκ
κ
020
20
02
1)(21
21)(
21
UVC
VkCVCU dd
=∆=
⎟⎠⎞
⎜⎝⎛ ∆
=∆=
1
00
>
=−=
κκQqQQeff
Materials Dielectric Constant κ Dielectric Strength (V/m)
Vacuum Air (dry) Bakelite Fused quartz Pyrex glass Polystyrene Teflon Neoprene rubber Nylon Paper Strontium titanate Water Silicone Oil
1.00000 1.00059 4.9 3.78 5.6 2.56 2.1 6.7 3.4 3.7
233 80 2.5
---- 3 × 106 24 × 106 8 × 106 14 × 106 24 × 106 60 × 106 12 × 106 14 × 106 16 × 106
8 × 106 ----
15 × 106
• Dielectric Constant and Dielectric Strength
Example
++++++
−−−−−−
− +− +− +− +− +
+ Q0 − Q0
κ
ld
I II III
∆V0
∆Vd
x⎟⎠⎞
⎜⎝⎛ −−=
+−=∆
κεε
εκε11
1)(
00
00
AQld
AQ
lA
QldA
QVd
κ
εlld
AVQC
d +−=
∆= 0
AQE00 ε
=εσ
=
I ; II ;
AQ
AQ
E eff
00
1εκε
==
III ;
AQE00 ε
=εσ
=
25-8. Dielectrics and Gauss’s Law• Gauss Law in vacuum
qAEAdE ==⋅∫ 000 εεrr
000 ε
σε
==A
qE
κεε qqqEAAdE =′−==⋅∫ 00
rr
κεσ
ε 00
=′−
=AqqE
• Gauss Law in dielectric material
qAdE =⋅∴ ∫rr
κε0
qAdD =⋅∫rr
EEDrrr
εκε == 0
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