Capacitance

1
Capacitance

• Chapter 26

2
Electric Potential of Conductors
V2
V1
E0
E0

• The electric field E is zero within a conductor
  at equilibrium.

3
Electric Potential of Conductors
V2
V1
E0
E0

• The electric field E is zero within a conductor
  at equilibrium.
• Consequently the conductor has the same electric
  potential V everywhere inside.

4
Electric Potential of Conductors
V2
V1
E0
E0

• The electric field E is zero within a conductor
  at equilibrium.
• Consequently the conductor has the same electric
  potential V everywhere inside.

5
Electric Potential of Conductors
V2
V1
E0
E0

• The electric field E is zero within a conductor
  at equilibrium.
• Consequently the conductor has the same electric
  potential V everywhere inside.
• The electric potential depends on how much excess
  charge has been added to the metal. The
  relationship between these quantities defines the
  capacitance.

6
Capacitance
A capacitor consists of two pieces of metal
carrying equal and opposite charges Q and Q.
7
Capacitance
A capacitor consists of two pieces of metal
carrying equal and opposite charges Q and
Q. An electric potential difference V develops
between the two pieces of metal.
8
Capacitance
A capacitor consists of two pieces of metal
carrying equal and opposite charges Q and
Q. An electric potential difference V develops
between the two pieces of metal. It turns out
that Q and V are proportional. The constant of
proportionality is called the capacitance C.
9
Capacitance
A capacitor consists of two pieces of metal
carrying equal and opposite charges Q and
Q. An electric potential difference V develops
between the two pieces of metal. It turns out
that Q and V are proportional. The constant of
proportionality is called the capacitance C.
Q CV Units Coulomb /Volt
Farad
10
Capacitance
A capacitor consists of two pieces of metal
carrying equal and opposite charges Q and
Q. An electric potential difference V develops
between the two pieces of metal. It turns out
that Q and V are proportional. The constant of
proportionality is called the capacitance C.
Q CV Units Coulomb /Volt
Farad
The capacitance C depends on the size and shape
of the capacitor, and the material (if any)
between them.
11
Parallel Plate Capacitor
A
d
-Q
Q
12
Parallel Plate Capacitor
A
d
-Q
Q
The charge on the top and bottom plates attract
so that the it all ends up on the inner facing
surfaces. These have surface charge density ?s
with sQ/A.
13
Parallel Plate Capacitor
A
d
-Q
E
Q
The charge on the top and bottom plates attract
so that the it all ends up on the inner facing
surfaces. These have surface charge density ?s
with sQ/A.
14
Parallel Plate Capacitor
A
d
-Q
E
Q
The charge on the top and bottom plates attract
so that the it all ends up on the inner facing
surfaces. These have surface charge density ?s
with sQ/A. Gausss law
15
Parallel Plate Capacitor
A
d
-Q
E
Q
The charge on the top and bottom plates attract
so that the it all ends up on the inner facing
surfaces. These have surface charge density ?s
with sQ/A. Gausss law
16
Parallel Plate Capacitor
A
d
-Q
E
Q
The charge on the top and bottom plates attract
so that the it all ends up on the inner facing
surfaces. These have surface charge density ?s
with sQ/A. Gausss law gives EAQ/e0 so
EQ/e0A.
17
Parallel Plate Capacitor
A
d
-Q
E
Q
The charge on the top and bottom plates attract
so that the it all ends up on the inner facing
surfaces. These have surface charge density ?s
with sQ/A. Gausss law gives EAQ/e0 so
EQ/e0A. The potential difference between the
two plates is ?V?Edl Ed or ?V Qd/e0A.
18
Parallel Plate Capacitor
A
d
-Q
E
Q
The charge on the top and bottom plates attract
so that the it all ends up on the inner facing
surfaces. These have surface charge density ?s
with sQ/A. Gausss law gives EAQ/e0 so
EQ/e0A. The potential difference between the
two plates is ?V?Edl Ed or ?V
Qd/e0A. Turning this about gives Q (e0A/d)?V
i.e. QC?V with Ce0A/d.
19
Parallel Plate Capacitor
A
d
-Q
E
Q
QC?V with C e0A/d The capacitance is like the
capacity of the object to hold charge. Bigger C
means the object can hold more charge (at a given
?V). The capacitance is proportional to the area
A (more capacity) and inversely proportional to
the separation d (smaller ?VEd for given Q).
20
How Do You Get Q and -Q?
Q
-Q
21
How Do You Get Q and -Q?
Q
-Q
Move Q from one plate to the other. How?
22
How Do You Get Q and -Q?
Q
-Q
-

Move Q from one plate to the other. How? With a
gadget that pushes charge for instance, a
battery.
23
How Do You Get Q and -Q?
Q
-Q
-

Move Q from one plate to the other. How? With a
gadget that pushes charge for instance, a
battery. Hooked to metal plates, a 1.5 Volt
battery moves charge until the potential
difference between the plates is also 1.5 V.
24
How Do You Get Q and -Q?
Q
-Q
A battery is like a charge escalator. With just
a wire between the plates, the charges making up
Q would repel each other and run through the wire
to neutralize -Q. But the battery pushes the
charges uphill.
25
Thinking About Capacitors
The black lines are metal wires attached to metal
rods. Suppose the battery has been hooked up
for a long time so that it has finished pushing
charge and the system has come to equilibrium.
2
1

6V
-
3
4
26
Thinking About Capacitors
The black lines are metal wires attached to metal
rods. Suppose the battery has been hooked up
for a long time so that it has finished pushing
charge and the system has come to equilibrium.
2
1

6V
-
3
4
What is ?V12? What is ?V34? What is ?V23?
27
Thinking About Capacitors
The black lines are metal wires attached to metal
rods. Suppose the battery has been hooked up
for a long time so that it has finished pushing
charge and the system has come to equilibrium.
2
1

6V
-
3
4
What is ?V12? Zero What is ?V34?
Zero What is ?V23? 6V
28
More About Capacitors
2
1
Now hook the battery to a parallel plate
capacitor.
3

5
3 mm
6V
-
4
What is ?V12? What is ?V23? What is ?V34?
29
More About Capacitors
2
1
Now hook the battery to a parallel plate
capacitor.
3

5
3 mm
6V
-
4
What is ?V12? Zero What is ?V23? Zero
What is ?V34? 6V
30
More About Capacitors
2
1
Now hook the battery to a parallel plate
capacitor.
3

5
3 mm
6V
-
4
What is ?V12? Zero What is ?V23? Zero
What is ?V34? 6V What is E at point 5?
31
More About Capacitors
2
1
Now hook the battery to a parallel plate
capacitor.
3

5
3 mm
6V
-
4
What is ?V12? Zero What is ?V23? Zero
What is ?V34? 6V What is E at point 5? E
6V/(3x10-3m) 2000 V/m
32
More About Capacitors
2
1
3
Now suppose you pull the plates apart slightly
while keeping the battery attached. What happens
to ?V, Q, and E?

5
3 mm
6V
-
4
33
More About Capacitors
2
1
3
Now suppose you pull the plates apart slightly
while keeping the battery attached. What happens
to ?V, Q, and E?

5
3 mm
6V
-
4
?V depends on the battery it stays at 6V.
34
More About Capacitors
2
1
3
Now suppose you pull the plates apart slightly
while keeping the battery attached. What happens
to ?V, Q, and E?

5
3 mm
6V
-
4
?V depends on the battery it stays at 6V. Use
QC?V. Here Ce0A/d decreases so Q decreases.
35
More About Capacitors
2
1
3
Now suppose you pull the plates apart slightly
while keeping the battery attached. What happens
to ?V, Q, and E?

5
3 mm
6V
-
4
?V depends on the battery it stays at 6V. Use
QC?V. Here Ce0A/d decreases so Q
decreases. E ?V / (new length) gets smaller.
36
What Does a Capacitor Do?

• Stores electrical charge.
• Stores electrical energy.

Capacitors are used when a sudden release of
energy is needed (such as in a photographic
flash). Capacitors are basic elements of
electrical circuits both macroscopic (as discrete
elements) and microscopic (as parts of integrated
circuits).
37
What Does a Capacitor Do?

• Stores electrical charge.
• Stores electrical energy.

The charge is easy to see. If a certain
potential, ?V, is applied to a capacitor C, it
must store a charge QC?V
(Symbol for a capacitor)
?V
38
Energy Stored in a Capacitor

• Build the charge up a little at a time, letting
  the charge q on the plate grow from 0 to Q.

39
Energy Stored in a Capacitor

• Build the charge up a little at a time, letting
  the charge q on the plate grow from 0 to Q.
• When the charge is q the potential is Vq/C.

40
Energy Stored in a Capacitor

• Build the charge up a little at a time, letting
  the charge q on the plate grow from 0 to Q.
• When the charge is q the potential is Vq/C.
• Now transfer a little more charge dq.

41
Energy Stored in a Capacitor

• Build the charge up a little at a time, letting
  the charge q on the plate grow from 0 to Q.
• When the charge is q the potential is Vq/C.
• Now transfer a little more charge dq.
• This requires a work dW Vdq (1/C) q dq.

42
Energy Stored in a Capacitor

• Build the charge up a little at a time, letting
  the charge q on the plate grow from 0 to Q.
• When the charge is q the potential is Vq/C.
• Now transfer a little more charge dq.
• This requires a work dW Vdq (1/C) q dq.
• Integrating q from 0 to Q gives the total stored
  (potential) electric energy

43
Energy Stored in a Capacitor

• Build the charge up a little at a time, letting
  the charge q on the plate grow from 0 to Q.
• When the charge is q the potential is Vq/C.
• Now transfer a little more charge dq.
• This requires a work dW Vdq (1/C) q dq.
• Integrating q from 0 to Q gives the total stored
  (potential) electric energy

44
Energy Density
Q -Q
?V

• Look at an energy density, i.e., energy per unit
  volume.
• For the parallel plate capacitor the volume is
  Ad, so uE U/(Ad) (1/2 C?V2)/Ad
• Now also use C e0A/d. Then

uE (e0/2)(?V/d)2 (e0/2)E2
45
Energy Density
Q -Q
V
uE (e0/2)(?V/d)2 (e0/2)E2

• This leads to another way to understand the
  energy.
• We can think of the energy as stored in the
  FIELD, rather than in the plates.
• If an electric field exists, then you can
  associate anelectric potential energy density of
  (e0/2)E2.