Capacitance and Dielectrics Electric devices function through storing and manipulating the flow of charges. This is achieved through several types discrete circuit components. Capacitors can store electric charge (and therefore electric potential energy) for a certain period of time.
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Capacitance and DielectricsElectric devices function through storing and manipulating the flow of charges.This is achieved through several types discrete circuit components.Capacitors can store electric charge (and therefore electric potential energy) for a certain period of time.
Capacitance
Two conductors with equal but opposite charges
Electric field
Potential difference between the charges
Now turn the problem around …Given a potential difference (a voltage) between two conductors, what is the charge on them?
VQCΔ
≡ (C/V=F)
Calculating Capacitance
Assume a charge q on the plates
Calculate the e-field using Gauss’ Law
Calculate the potential
Find the capacitance
qAdE =⋅∫vv
0ε
qEA =0ε
∫ ⋅−=−f
iif sdEVV vv
∫+
−
=Δ EdsV
VqCΔ
=
Parallel Plate Capacitor
Parallel Plate Capacitor
Increased area results in increased charge storage for a constant voltage.
Capacitance is proportional to area
Parallel Plate Capacitor
V E
+Q
-Q-
+
VEdV ==Δ
V E+Q
-Q-
+
VdEV <′=′Δ
The potential difference between the battery terminals and the plates will create a field and charges will flow from the battery to the plates until the potential difference is zeroed.
Capacitance is inversely proportional to the distance between the plates.
Parallel Plate Capacitor
AQ
=σ
AQE00 εε
σ==
AQdEdV
0ε==Δ
dA
VQC 0ε=Δ
=
Surface charge density on the plates:
The field (assuming d<<A1/2):
The potential difference:
The capacitance:
Cylindrical Capacitor
⎟⎠⎞
⎜⎝⎛
=
⎟⎠⎞
⎜⎝⎛
=
abkl
Cabk
lC
e
e
ln2
1
ln2lQ λ=rkE eλ2
=
⎟⎠⎞
⎜⎝⎛−=−=−=− ∫∫ a
bkrdrkEdrVV e
b
ae
b
aab ln22 λλ
⎟⎠⎞
⎜⎝⎛
=Δ
=
abk
lV
QCe ln2 λ
λ
Spherical Capacitor
)( abkabC
e −=
2rQkE e=
⎟⎠⎞
⎜⎝⎛ −==−=− ∫∫ ba
QkrdrEdrVV e
b
a
b
aab
112
⎟⎠⎞
⎜⎝⎛ −
=Δ
=
baQk
QV
QCe
11
Circuits with CapacitorsA battery drives charges to accumulate on the capacitor plates.A switch controls the flow of charge.
ΔV
C,Q
+
Parallel Combination
VCVCQVCVCQΔ=Δ=Δ=Δ=
2222
1111
21
21
VVVQQQ
Δ=Δ=Δ+=
VCQ eqΔ=
21 CCCeq +=
For parallel capacitors
∑=
=n
jjeq CC
1
Parallel Connections
All are parallel connections
This is not
Series Combination
21
21
QQQVVV
==Δ+Δ=Δ
22
22
11
11
CQ
CQV
CQ
CQV
==Δ
==Δ
eqCQV =Δ
21
111CCCeq
+=
∑=
=n
j jeq CC 1
11For series capacitors
Mixed Combination of Capacitors
Metallic Slab Inside a Capacitor
adAC−
= 0ε
Energy Stored in a CapacitorSince capacitors can carry charge after a battery is disconnected, they can store energy.To calculate this energy, consider the work done to move a small amount of charge from one plate of the capacitor to the other.As more and more charges are moved, the work required increases.
dqCqVdqdW =Δ=
CQdqq
Cdq
CqW
21 2
00
=== ∫∫
22
)(21
21
2VCVQ
CQU Δ=Δ==
Concept QuestionConsider a simple parallel-plate capacitor whose plates are given equal and opposite charges and are separated by a distance d. Suppose the plates are pulled apart untilthey are separated by a distance D > d. The electrostatic energy stored in the capacitor is
1. greater than2. the same as3. smaller than
before the plates were pulled apart.
Energy DensityDefined as stored energy per unit volume.Consider the parallel plate capacitor.
2)(21 VCU Δ=
dA0ε 2)(Ed
20 )(
21 EAdU ε=
20
202
1
21
)(
Eu
AdEAd
VolUu
E
E
ε
ε
=
==
DielectricsDielectrics are insulators like glass, rubber or wood.They are used to increase the charge storage capacity and/or maximum voltage in capacitors while providing mechanical support.They are characterized by a dielectric constant, κ, and are limited by their dielectric strength.
An Atomic View
As we apply a voltage to the plates, the dielectric in between is polarized.The dielectric constant is a measure of the polarizability of the material.This creates a field in the opposite direction to the applied field and therefore reduces the effective voltage or under constant voltage allows a larger charge to be collected on the plates. The dielectric strength is the maximum electric field the material can withstand before breaking down (becoming a conductor).
Capacitors With Dielectrics
κ0VV Δ
=Δ
0
00
VQ
VQC
Δ=
Δ= κ
0CC κ=
Concept QuestionA parallel-plate capacitor is attached to a battery that maintains a constant potential difference V between the plates. While the battery is still connected, a glass slab is inserted so as to just fill the space between the plates. The stored energy
1. increases.2. decreases.3. remains the same.
A Partially Filled Capacitor
( ) 00
323131131C
dAC
κκε
κκ
+=
−+=
SummaryCapacitors are geometries that can hold charges.Use Gauss’ Law symmetries to calculate capacitance.Series and parallel connections.Dielectrics increase capacitance.
For Next ClassReading Assignment
Chapter 27 – Current and Resistance
WebAssign: Assignment 4
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