Physics 122B
Electricity and Magnetism
Lecture 13 (Knight: 30.6 and 30.7)
Capacitance and Capacitors
April 25, 2007
Martin Savage
Lecture 13 Announcements
 Lecture HW #4 has been posted on the
Tycho system, it is due at 10 PM tonight
7/19/2016
Physics 122B - Lecture 13
2
Combining Capacitors
Parallel: Same DV, but different Qs.
Cparallel 
Q  Q2  Q3 
Q
 1
DVC
DVC
 C1  C2  C3 
Series: Same Q, but different DVs.
Cseries 
Q
Q

DVC DV1  DV2  DV3 

1
 DV1 / Q    DV2 / Q    DV3 / Q  

1
1/ C1  1/ C2  1/ C3 
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 C1 || C2 || C3 ||
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Reminder: Combining Resistors
Conducting material that carries current
across its length can form a resistor,
a circuit element characterized by an
electrical resistance R:
R ≡ rL/A
where L is the length of the conductor and A is
its cross sectional area. R has units of ohms.
Multiple resistors may be combined in
series, where resistances add, or in parallel,
where inverse resistances add.
I
Rnet
Rnet
Parallel Connection [(1/A)]:
Series Connection [L]:
Rnet  R1  R2  R3
1
1
1
1
 

Rnet R1 R2 R3
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Example: A Capacitor Circuit
Find the charge and
potential difference
across each capacitor
shown in the figure.
Q
Cparallel  C1  C2
DVC
CC
1
Cseries 
 1 2
1/ C1  1/ C2 C1  C2
C
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Energy Stored in a Capacitor
1
DU  dqDV  qdq
C
Q
2
1
1 Q
U C   qdq  2
C0
C
2
Q
U C  12
 12 C DVC 2
C
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Example:
Storing Energy in Capacitor
How much energy is stored in a 2.0 mF
capacitor that has been charged to 5000 V?
What is the average power dissipation if
the capacitor is discharged in 10 ms?
2.0 mF
5 kV
UC  12 CDV 2  12 (2.0 10-6 F)(5000 V)2  25 J
DU
(25 J)
6
P


2.5

10
W  2.5 MW
5
Dt (1.0 10 s)
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Energy in the Electric Field
Volume of E-field
U C  C DV 
1
2
2
1
2
0 A
d
 Ed 
2

0
2
 Ad  E 2
energy stored U C  0 2
uE 

 E
storage volume Ad 2
Example: d=1.0 mm, DVC=500 V
E
DVC
500 V
5


5.0

10
V/m
d
1.0 10-3 m
uE 
0
2
E 
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2
1
2
 5.0 10
5
V/m  /  4  9.0 109 Vm/C   1.1 J/m 3
2
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Dielectric Materials*
There is a class of polarizable dielectric
materials that have an important application in
the construction of capacitors. In an electric
field their dipoles line up, reducing the E field
and potential difference and therefore
increasing the capacitance:
E off
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 0 A
Q
C

DVC
d
E on
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Electric Fields and Dielectrics
In an external field EO, neutral molecules can polarize. The induced
electric field E’ produced by the dipoles will be in the opposite direction
from the external field EO. Therefore, in the interior of the slab the
resulting field is E = EO-E’.
The polarization of the material has the net effect of producing a sheet
of positive charge on the right surface and a sheet of negative charge on
the left surface, with E’ being the field made by these sheets of charge.
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Capacitors and Dielectrics*
If a capacitor is connected to a
battery, so that it has a charge q,
and then a dielectric material of
dielectric constant e is placed in
the gap, the potential is unchanged
but the charge becomes eq.
If a capacitor is given a charge
q, and then a dielectric material of
dielectric constant e is placed in
the gap, the charge q is unchanged,
but the potential drops to V/e.
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Conductivity and Resistivity
 e E  ne 
J  nevd  ne 
E

m
 m 
2
ne2
  conductivity 
m
AC
1
1
 Units:


2
Nm
ohm m  m
r  resistivity 
1


m
ne 2
J E 
r Units: ohm m   m
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E
r
12
Example:
The Electric Field in a Wire
A 2.0 mm diameter aluminum wire carries a current of 800 mA.
What is the electric field strength inside the wire?
J
I
I
(0.800 A)
3
E 



7.2

10
N/C
2
7
-1 -1
2
  A  r
(3.5 10  m ) (0.001 m)
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Example:
Mean Time Between Collisions
What is the mean time between collisions for electrons in copper, for
which the electron density is 8.5 x 1028 electrons per cubic meter?
m (9.1110-31 kg)(6.0 107 -1m-1 )
-14
 2

2.5

10
s
28
-3
-19
2
ne
(8.5 10 m )(1.60 10 C)
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Potential and Current (1)
DVwire  DVbat
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Potential and Current (2)
L
DVwire    Es ds; E s  E wire
0
L
DVwire  E wire  ds
0
 E wire L
E wire
Assuming uniform J across A
I  AJ  A E wire 
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A
r
E wire
 A 
I 
 DVwire
 rL 
R
Physics 122B - Lecture 13
rL
A
DVwire

L
; (Units: =V/A)
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Resistors and Resistance
Conducting material that carries current
across its length can form a resistor,
a circuit element characterized by an
electrical resistance R defined by:
R ≡ rL/A
where L is the length of the conductor and A is
its cross sectional area. R has units of ohms
( = V/A).
Multiple resistors may be combined in
series, where resistances add, or in parallel,
where inverse resistances add.
I
Rnet
Rnet
Parallel Connection [(1/A)]:
Series Connection [L]:
Rnet  R1  R2  R3
1
1
1
1
 

Rnet R1 R2 R3
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Question
What is the relation of the currents at the points shown?
(a) Ia=Ib=Ic=Id;
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(b) Ia=Ib>Ic=Id;
(c) Ia>Ib>Ic>Id;
Physics 122B - Lecture 13
(d) Ia>Ib>Ic=Id;
18
Resistors and Ohm’s Law
J E 
I  JA 
R
rL
A
E
r
E
r
A
DV / L
r
I
A
DV
;
R
DV
 r L / A
DV  IR
Ohm’s Law
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Example:
The Current in a Wire
What is the current in a 1.0 mm diameter 10.0 cm long copper wire
that is attached to the terminals of a 1.5 V battery.
R  r L / A  r L /( r 2 )  (1.7 10-8  m)(0.10 m) /  (0.0005 m) 2
 2.2 10-3 
I  DV / R  (1.5 V) /(2.2 10-3 )  680 A
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Ohmic and Non-ohmic Materials
Despite its name, Ohm’s Law is not a law of
Nature (in the sense of Newton’s Laws). It is
a rule about the approximately linear
potential-current behavior of some materials
under some circumstances.
Important non-ohmic devices:
1. Batteries, where DV=E is determined by
chemical reactions independent of I;
2. Semiconductors, where I vs. DV can be very
nonlinear;
3. Light bulbs, where heating changes R;
4. Capacitors, where the relation between I and
DV differs from that of a resistor.
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The Ideal Wire Model
In considering electric circuits, we will make
the following assumptions:
1. Wires have very small resistance, so that
we can take Rwire=0 and DVwire=0 in circuits.
Any wire connections are ideal.
2. Resistors are poor conductors with constant
resistance values from 10 to 108 .
3. Insulators are ideal non-conductors, with
R=∞ and I=0 through the insulator.
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Circuit Elements & Diagrams
These are some of the symbols we will use to represent objects in
circuit diagrams.
Other symbols: inductance, transformer, diode, transistor, etc.
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Circuit Diagram
Actual Circuit
Circuit Diagram
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Circuit Diagrams
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Anatomy of a Light Bulb
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Question
(u)
(v)
(w)
(x)
Which of these diagrams show the same circuit?
(a) All show different circuits;
(b) (u) and (v);
(c) (u), (v), and (w);
(d) (u), (v), and (x);
(e) All show the same circuit.
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End of Lecture 13
 Before the next lecture, read Knight,
sections 30.5 and 31.4.
 Lecture HW #4 has been posted on the
Tycho system. and is due at 10 PM tonight.
7/19/2016
Physics 122B - Lecture 13
28