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Chapter 24
Capacitance, Dielectrics,Electric Energy Storage
Physics for Scientists & Engineers, 3rd EditionDouglas C. Giancoli
© Prentice Hall
P07 -
Capacitors and Capacitance
P07 -
Capacitors: Store Electric EnergyCapacitor: two isolated conductors with equal and opposite charges Q and potential difference ∆V between them.
QCV
=∆
Units: Coulombs/Volt or Farads
20P07 -
Parallel Plate Capacitor
0=E
0=E
Q Aσ+ =
Q Aσ− = −
d?E =
21P07 -
Parallel Plate CapacitorWhen you put opposite charges on plates, charges move to the inner surfaces of the plates to get as close as possible to charges of the opposite sign
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/35-capacitor/35-capacitor320.html
22P07 -
Calculating E (Gauss’s Law)
0S
inqdε
⋅ =∫∫ E A00 εε
σAQE ==( )
0
GaussGauss
AE A σε
=
Note: We only “consider” a single sheet! Doesn’t the other sheet matter?
23P07 -
Alternate Calculation Method
+ + + + + + + + + + + + + +Top Sheet:02
E σε
= −
02E σ
ε=
- - - - - - - - - - - - - -Bottom Sheet: 02E σ
ε=
02E σ
ε= −
0 0 0 02 2QE
Aσ σ σε ε ε ε
= + = =
24P07 -
Parallel Plate Capacitor
top
bottom
V d∆ = − ⋅∫ E S0
Q dAε
=Ed=dA
VQC 0ε=∆
=
C depends only on geometric factors A and d
Figure 24-5 (b)
Figure 24-6
26P07 -
Spherical CapacitorTwo concentric spherical shells of radii a and b
What is E?
Gauss’s Law E ≠ 0 only for a < r < b, where it looks like a point charge:
rE ˆ4 2
0rQπε
=
27P07 -
Spherical Capacitor
For an isolated spherical conductor of radius a:
20
ˆ ˆ4
b
a
Q drrπε
= − ⋅∫r r
( )1104−− −
=∆
=baV
QC πε
outside
inside
V d∆ = − ⋅∫ E S0
1 14Q
b aπε⎛ ⎞= −⎜ ⎟⎝ ⎠
Is this positive or negative? Why?
aC 04πε=
28P07 -
Capacitance of EarthFor an isolated spherical conductor of radius a:
aC 04πε=
mF1085.8 120
−×=ε m104.6 6×=a
mF7.0F107 4 =×= −C
A Farad is REALLY BIG! We usually use pF (10-12) or nF (10-9)
29P07 -
1 Farad Capacitor
How much charge?
( )( )1F 12 V12C
Q C V= ∆==
33P07 -
Energy To Charge Capacitor
1. Capacitor starts uncharged.2. Carry +dq from bottom to top.
Now top has charge q = +dq, bottom -dq3. Repeat4. Finish when top has charge q = +Q, bottom -Q
+q
-q
34P07 -
Work Done Charging CapacitorAt some point top plate has +q, bottom has –q
Potential difference is ∆V = q / CWork done lifting another dq is dW = dq ∆V
+q
-q
35P07 -
Work Done Charging CapacitorSo work done to move dq is:
dW dq V= ∆1qdq q dq
C C= =
Total energy to charge to q = Q:
0
1 Q
W dW q dqC
= =∫ ∫ +q
-q212
QC
=
36P07 -
Energy Stored in CapacitorQCV
=∆
Since
22
21
21
2VCVQ
CQU ∆=∆==
Where is the energy stored???
37P07 -
Energy Stored in Capacitor
Energy stored in the E field!
ando AC V Eddε
= =Parallel-plate capacitor:
212
U CV= ( )2
21 ( )2 2
o oA EEd Addε ε
= = × ( )Eu volume= ×
2
field energy density2
oE
Eu E ε= =
38P07 -
1 Farad Capacitor - EnergyHow much energy?
( )( )
2
2
121 1F 12 V272 J
U C V= ∆
=
=
Compare to capacitor charged to 3kV:
( )( )
( )( )
22
24 3
1 1 100µF 3kV2 21 1 10 F 3 10 V 450 J2
U C V
−
= ∆ =
= × × =
P8-
Capacitors: Store Electric Energy
QCV
=∆
To calculate:1) Put on arbitrary ±Q 2) Calculate E3) Calculate ∆V
Parallel Plate Capacitor:
0 ACdε
=
18P8-
Capacitors in Parallel
19P8-
Capacitors in Parallel
Same potential!
1 21 2,Q QC C
V V= =∆ ∆
20P8-
Equivalent Capacitance
?
( )1 2 1 2
1 2
Q Q Q C V C VC C V
= + = ∆ + ∆
= + ∆
1 2eqQC C CV
= = +∆
21P8-
Capacitors in Series
Different Voltages NowWhat about Q?
22P8-
Capacitors in Series
23P8-
Equivalent Capacitance
1 21 2
Q QV , VC C
∆ = ∆ =
1 2V V V∆ = ∆ + ∆(voltage adds in series)
1 2eq
Q Q QVC C C
∆ = = +
1 2
1 1 1
eqC C C= +
25P8-
Dielectrics
27P8-
DielectricsA dielectric is a non-conductor or insulatorExamples: rubber, glass, waxed paper
When placed in a charged capacitor, the dielectric reduces the potential difference between the two plates
HOW???
28P8-
Molecular View of Dielectrics
Polar Dielectrics : Dielectrics with permanent electric dipole moments Example: Water
29P8-
Molecular View of Dielectrics
Non-Polar DielectricsDielectrics with induced electric dipole momentsExample: CH4
30P8-
Dielectric in Capacitor
Potential difference decreases because dielectric polarization decreases Electric Field!
31P8-
Gauss’s Law for Dielectrics
Upon inserting dielectric, a charge density σ’ is induced at its surface
0S
insideqd EAε
⋅ = =∫∫ E A0
'εσσ −
=E( )0
' Aσ σε−
=
What is σ’?
32P8-
Dielectric Constant κDielectric weakens original field by a factor κ
0
0 0
' EE σ σ σε κ κε−
= ≡ = 1' 1σ σκ
⎛ ⎞= −⎜ ⎟⎝ ⎠
⇒
Gauss’s Law with dielectrics:
0S
freeinsideqdκε
⋅ =∫∫ E A
Dielectric constantsVacuum 1.0Paper 3.7Pyrex Glass 5.6Water 80
33P8-
Dielectric in a CapacitorQ0= constant after battery is disconnected
0VVκ
=Upon inserting a dielectric:
0 00
0 0/Q QQC C
V V Vκ κ
κ= = = =
34P8-
Dielectric in a CapacitorV0 = constant when battery remains connected
Upon inserting a dielectric: 0Q Qκ=
00
0
QQC CV V
κ κ= = =
36P8-
Group: Partially Filled Capacitor
What is the capacitance of this capacitor?