Capacitance and Dielectrics
Young and Freedman
Chapter 24
Two conductors separated by an insulator (or a
vacuum) form a capacitor
If the two objects are charged, there will be an
electric field and a potential difference between them.
Definition of Capacitance
r
dl
#V = "
b
a
r r
E ! dl
1) a and b are conductor so they are
equipotential surface and it doesn’t matter
where I start from.
2) The electric field is conservative so it
doesn’t matter what path I take.
There is a single voltage difference between two charged conductors.
If I double the charge Q, I will double the voltage difference.
Define C = Q/V as the “capacitance”
Capacitance depends only on the shape, size, and position of conductors
Calculating Capacitance in Vacuum
C = Q/V Capacitance has units of “farads” 1 farad = 1 Coulomb/Volt
Capacitance between two parallel plates
Va " Vb = #
b
a
r r
E ! dl = Ed
Integral is easy because field is uniform
"
Q
E=
=
!0 !0 A
Recall result from infinite flat plate
1 Qd
Vab = Ed =
!0 A
Q
A
C=
= "0
Vab
d
How much is a Farad?
A parallel plate capacitor has a capacitance of 1.0 F.
If the plates are 1 mm apart, what is the area of the
plates?
A
Cd
C = "0 # A =
d
"0
%3
(1.0F)(1.0 $10 m)
A=
%12
8.85 $10 F / m
8
2
A = 1.1$10 m
Imagine a square ~ 6miles on each side (about 1/3
larger than Manhattan Island)
!
Spherical Capacitor
Example 24.3
From Gauss’s Law we know field between a and b
Q
E (r ) =
4)* 0 r 2
+V = (
b
a
Q
dr
2
4)* 0 r
Q
V=
4)* 0
(
b
a
1
dr
2
r
Q
V=
4)* 0
&1 1#
$$ ' !!
% rb ra "
V
1 &1 1#
$$ ' !!
=
Q 4)* 0 % rb ra "
'1
&1 1#
& ra rb #
Q
$
!
!!
= 4)* 0 $ ' ! = 4)* 0 $$
V
% rb ra "
% ra ' rb "
& ra rb #
!!
C = 4() 0 $$
% ra ' rb "
Capacitors in Parallel
Q1 = C1V
Q2 = C2V
Q = Q1 + Q2 = (C1 + C2 )V
Q
= C1 + C2
V
Cequiv = C1 + C2
Cequiv = C1 + C2 + C3 + ....
Capacitors in series
Q
Vac = V1 =
C1
Vcb = V2 =
& 1
1 #
Vac = V = V1 + V2 = Q$$ + !!
% C1 C2 "
V
1
1
=
+
Q C1 C2
1
Cequiv
1
Cequiv
=
1
1
+
C1 C2
1
1
1
=
+
+
+ ....
C1 C2 C3
Q
C2
Example 24.6 – What is equivalent capacitance for network “a”
C’
C’’
Ceq
1
1
1
=
+
C' 12 µF 6 µF
1
1
=
C' 4 µF
C' = 4 µF
C' ' = 3 µF + 11µF + 4 µF
C' ' = 18 µF
1
1
1
=
+
Ceq 18 µF 9 µF
1
1
=
Ceq 6 µF
Energy stored in a capacitor
+Q
Q
V=
C
-Q
To add more charge, I must do Work
To add a small amount of charge dq I must do work dW -
dW = Vdq =
q
dq
C
To add a finite charge Q, I must integrate Q final
W=
Q final
Q final
' dW = ' Vdq = '
Qinitial
Qinitial
Qinitial
Q final
q
1
1 &1 2
1 2 #
dq =
qdq = $ Q final = Qinitial !
'
C
C Qinitial
C%2
2
"
If I start from NO net charge I get -
1 Q2 1
1
W=
= QV = CV 2 Stored Energy in a Charged Capacitor
2 C 2
2
A few words on dielectrics
A few words on dielectrics
• Dielectrics are insulators and are often used in
capacitors, giving three advantages:
– mechanical: maintaining a small gap, ability to roll
– increases maximum potential difference: chose dielectric
material which can tolerate larger electric fields than air
without dielectric breakdown (conduction through ionization).
– Increases the capacitance for given dimensions
Chapter 24 Summary
Chapter 24 Summary cont.
End of Chapter 24
You are responsible for the material covered in T&F Sections 24.1-24.3
You are expected to:
•
Understand the following terms:
Capacitor, Capacitance, Equivalent Capacitors, Series, Parallel, Stored
Energy
•
Be able to calculate capacitance in simple geometries (e.g. Parallel planes,
spheres,…)
•
Be able to determine the equivalent capacitance of (relatively) simple networks
of capacitance using series and parallel rules (see example 24.6 for a typical
network)
•
Calculate the energy stored in a capacitor using integration.
Recommended F&Y Exercises chapter 24:
•
1,3,4,9,14,15,20