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Ch 26
Capacitors
•Conductors are commonly used as places to store charge
•You can’t just “create” some positive charge somewhere, you have to
have corresponding negative charge somewhere else
Definition of a capacitor:
•Two conductors, one of which stores charge +Q, and the other of
which stores charge –Q. Can we relate the charge Q that develops to
the voltage difference V?
–Q
•Gauss’s Law tells us the electric field
keQ
E  2 rˆ
+Q
between the conductors
r
b a
•Integration tells us the potential difference
V
a
a
ke Q
ke Q
V    2 dr 
r
r b
b
1 1
V  keQ   
a b
1 1
V  keQ   
a b
Capacitance
•The relationship between voltage difference and charge is always linear
•This allows us to define capacitance
Q  C V
•Capacitance has units of Coulomb/Volt
•Also known as a Farad, abbreviated F
C
F
•A Farad is a very large amount of capacitance
V
•Let’s work it out for concentric conducting spheres:
What’s the capacitance of the Earth, if we put
Q
ab
C

the “other part” of the charge at infinity?
V
ke  b  a 
a  6370 km
b
ab
ab
6.37 106 m
C


ke  b  a  ke b 8.988 109 N  m 2 / C2
7.09 104

2  709  F
Nm / C
Parallel Plate Capacitors
A
•A “more typical” geometry is two large,
closely spaced, parallel conducting plates
•Area A, separation d.
Let’s find the capacitance:
•Charge will all accumulate on the inner surface
d
Q  A
•Let + and – be the charges on each surface

•As we already showed using Gauss’s law, this means
E
nˆ
0
there will be an electric field given by
•If you integrate the electric field over the distance d,
you get the potential difference
Circuit symbol

 d Qd
for a capacitor:
dx 

V   E  ds 


Q
C
V
C
0
0 A
d
0
0 A
To get a large capacitance, make the
area large and the spacing small
Warmup 08
Ex A 20.0 F spherical capacitor is composed of two metallic speheres, one having twice the
radius of the other. If the region between the spheres is a vacuum, determine the volume of this
region.
Solve on board
Capacitors in Parallel
2V
3 F
Q  C V
6 F
When we close the switch, how much
charge flows from the battery?
A) 36 C
B) 4 C
C) 18 C
D) 8 C
E) 10 C
•The voltage difference across each capacitor will be 2 V.
Q1  C1V   3  F 2 V   6 C
Q2  C2 V   6  F 2 V   12 C
Qtot  Q1  Q2  18  C
•When capacitors are connected like this at both ends, we say they are
connected in parallel
•The combined capacitors act like a single capacitor with capacitance:
Ctot  C1  C2
Capacitors in Series
V1
V2
+Q -Q +Q -Q
V
C1
C2
Q  C1V1  C2 V2
When we close the switch, which
capacitor gets more charge Q on it?
A) The one with the bigger capacitance
B) The one with the smaller capacitance
C) They get the same amount of charge
D) Insufficient information
1
•Charge flows to the left side of the first capacitor
 1
1 
C 

•Since capacitors have balanced charge, the right

 C1 C2 
side must have the equal and opposite charge
•The only place it can get this charge is the left side of the right capacitor
•The charges on the second capacitor are also balanced
•The voltage difference on each capacitor is not the same, since only one
end is connected. But the charge is the same.
Q
C
•The two voltages differences can be added together
 1
1 
V  V1  V2  Q C1  Q C2  Q 


 C1 C2 
V
Warmup 08
Series and Parallel
•When two circuit elements are connected at one end, and nothing else is
connected there, they are said to be in series
C1
C2
1
1
1


C C1 C2
•When two circuit elements are connected at both ends, they are said to
be in parallel
C1
C2
C  C1  C2
•These formulas work for more than two circuit elements as well.
C4
C2
C3
1
1
1
1
1
1
C1
C5
C

C1

C2

C3

C4

C5
Ct1- Two capacitors C1 and C2 are connected in a series connection. Suppose that their
capacitances are in the ratio C2/C1 = 2/1. When a potential difference, V, is applied across the
capacitors, what is the ratio of the charges Q2 and Q1 on the capacitors? Q2/Q1 =
A. 2
B. 1
C. 1/2
D. none of the above
E. Need more information
C2
C1
V
CT2 - For the capacitors above the ratio of the voltage drops across each one is V2/V1 =
A. 2
B. 1
C. 1/2
D. none of the above
E. Need more information
CT3 - If the capacitors were connected in parallel, how would you answer the above two
questions?
Sample Problems
60 F
60 F
60 F
What is the capacitance of the structure
shown at left?
A) 180 F
B) 120 F
C) 30 F
D) 20F
E) none of the above
•They are connected end-to-end, so they are in series
1
1
1
1
1
1
1
3
1








C C1 C2 C3 60 60 60 60 20
•Now add a wire across the middle. The two ends of the
middle capacitor are at the same potential; therefore,
there is no charge on this capacitor
•You might as well ignore it
1
1
1
1
1
1





60 60 30
C C1 C3
Complicated Capacitor Circuits
•For complex combinations of capacitors, you can replace small
structures by equivalent capacitors, eventually simplifying everything
The capacitance of the capacitors in pF below is marked.
What is the effective capacitance of all the capacitors shown?
4
3
2
2
10 V
6 5
1
•Capacitors 1 and 5 are connected at both
ends- therefore they are parallel
C  1 5  6
•Capacitors 3 and 6 are connected at just
one end – therefore they are series
1 1 1 1
  
C 3 6 2
C2
•All three capacitors are now connected at both ends – they are all in
parallel
C  242 8
Exs – Serway 26 - 13 and 14. Two capacitors C1 = 5.00 F and C2 = 12.0 F are connected in
parallel, and the resulting combination is connected to a 9.00 V battery. (a) What is the value of
the equivalent capacitance of the combination, (b) what are the potential differences across each
capacitor?, and (c) what are the charges on each capacitor? Repeat if they are in series.
Solve on
board
Warmup09
Energy in a capacitor
•Suppose you have a capacitor with charge q already on it,
and you try to add a small additional charge dq to it, where
dq is small. How much energy would this take?
q  C V
•The side with +q has a higher potential
•Moving the charge there takes energy
•The small change in energy is:
–q
+q
dU   dq  V
dq  q


C
•Now, imagine we start with zero charge and build it up
dq
gradually to q = Q
•It makes sense to say an uncharged capacitor has U = 0
U  
q Q
q 0
Q
dU  0
Q2
U 
2C
q  dq 
q
Q2


2C 0 2C
C
Q  C V
2 Q
C  V 
Q
U

2C
2
2
2
Energy in a capacitor (2)
20 V
1 F
2 F
•Capacitors in parallel
have the same voltage
difference V
•The larger capacitor
has more energy
20 V
1 F 2 F
•Capacitors in series
have the same charge Q
•The smaller capacitor
has more energy
C  V 
Q
U

2C
2
2
2
For each of the two circuits, which
capacitor gets more energy in it?
A) The 1 F capacitor in each circuit
B) The 2 F capacitor in each circuit
C) The 1 F capacitor in the top circuit,
the 2 F capacitor in the bottom circuit
D) The 2 F capacitor in the top circuit,
the 1 F capacitor in the bottom circuit
E) They are equal in each circuit
CT4- Consider 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 until they are
separated by a distance D > d. The electrostatic energy stored in the capacitor is
A. greater than
B. the same as
C. . smaller than
before the plates were pulled apart.
Energy density in a capacitor
Suppose you have a parallel plate capacitor with
area A, separation d, and charged to voltage V.
(1) What’s the energy divided by the volume between the plates?
(2) Write this in terms of the electric field magnitude
U  C  V  
1
2
2
0 A
2d
 V 
2
•Energy density is energy over volume
 0 A  V  1  V 
U
U
 0 
u



2
2  d 
V
Ad
2 Ad
2
V
E 
d
u  0 E
1
2
2
A
2
•We can associate the energy with the electric field itself
•This formula can be shown to be completely generalizable
•It has nothing in particular to do with capacitors
d
Ex- A uniform electric field E = 3000 V/m exists within a certain region of space. What volume
of space contains an energy equal to 1.00 x 10-7 J? Express your answer in m3 and liters.
Solve
on
Board
Warmup09
Dielectrics in Capacitors
•What should I put between the metal  A
C 0
plates of a capacitor?
d
•Goal – make the capacitance large
•The closer you put the plates together, the
bigger the capacitance
•It’s hard to put things close together –
unless you put something between them
•When they get charged, they are also very
attracted to each other
•Placing an insulating material – a dielectric –
allows you to place them very close together
•The charges in the dielectric will also shift
•This partly cancels the electric field
•Small field means smaller potential difference
•C = Q/V, so C gets bigger too
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
C
–
–
–
–
–
–
–
–
–
–
–
–
 0 A
 1
d
Choosing a dielectric
 A
What makes a good dielectric?
C 0
d
•Have a high dielectric constant 
•The combination 0 is also called , the permittivity
•Must be a good insulator
•Otherwise charge will slowly bleed away
•Have a high dielectric strength
•The maximum electric field at which the insulator suddenly
(catastrophically) becomes a conductor
•There is a corresponding breakdown voltage where the capacitor fails
CT -5-Consider a capacitor made of two parallel metallic plates separated by a distance d. The
top plate has a surface charge density +s, the bottom plate –s. A slab of metal of thickness l < d is
inserted between the plates, not connected to either one. Upon insertion of the metal slab, the
potential difference between the plates
A. increases.
Consider isolated capacitor
B. decreases.
C. remains the same.
CT - 6 - A dielectric is inserted between the plates of a capacitor. The system is then charged and
the dielectric is removed. The electrostatic energy stored in the capacitor is
A. greater than
Consider isolated capacitor
B. . the same as
after charging
C. . smaller than
it would have been if the dielectric were left in place.
CT - 7- A 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 in-serted so
as to just fill the space between the plates. The stored energy
A. increases.
B. . decreases.
C. . remains the same.
What are capacitors good for?
•They store energy
•The energy stored is not extremely large, and it tends to leak
away over time
•Gasoline or fuel cells are better for this purpose
•They can release their energy very quickly
•Camera flashes, defibrillators, research uses
•They resist changes in voltage
•Power supplies for electronic devices, etc.
•They can be used for timing, frequency filtering, etc.
•In conjunction with other parts