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Lecture 10-1
©2008 by W.H. Freeman and Company
Lecture 10-2
C
Capacitor Examples
C
2C
C  C1  C2 
C
C
C
C/2
 1 0 A / 2
d
 A    2 
 0  1

d 
2 

 2 0 A / 2
d
C
C
?C
?=2/3
1
1
1
d / 4 3d / 4




C
C1
C2
0 A
 0 A

d 1
3 1


0 A 
4 
4
Lecture 10-3
READING QUIZ 1
An electric toaster (resister) uses 2000 watts when the it
is connected to a 100 volt potential difference. What is the
resistance R (Ω) of the toaster?
A| 10 Ω
B| 5 Ω
C| 15 Ω
D| 20 Ω
E| 12 Ω
Lecture 10-4
DOCCAM 2 DEMO 5B-01
OHMS LAW BOARD
Lecture 10-5
DOCCAM 2 DEMO 5B-O3
VOLTAGE PARADOX
Lecture 10-6
Electric Current
Current = charges in motion
q dq
Magnitude
I  lim

x 0 t
dt
rate at which net positive charges
move across a cross sectional surface
I  J dA
Units:
[I] = C/s = A (ampere)
J = current density
(vector) in A/m²
Current is a scalar, signed quantity, whose
sign corresponds to the direction of motion of
net positive charges by convention
A
Lecture 10-7
Microscopic View of Electric Current in Conductor
All charges move with some velocity ve
A
random motion with high speeds
(O(106)m/s) but with a drift in a certain
direction on average if E is present
Why random
motion?
• thermal energy
• scattering off each
other, defects, ions,
…
Drift velocity vd is orders of magnitudes less
than the actual velocity of charges.
Lecture 10-8

Current and Drift Velocity in Conductor
Drift velocity vd is orders of
magnitudes less than the actual velocity
of charges.
I  nAvd q
I
 vd 
nAq
where n =carrier density
or
J  nqv d
( E )
if ohmic
Lecture 10-9
Ohm’s Law Summary
Current-Potential (I-V) characteristic of a
device may or may not obey Ohm’s Law:
I V
(J  E)
or V = IR with R constant
V V

Resistance  R    I   A   (ohms)
tungsten wire
gas in fluorescent tube
diode
Lecture 10-10
Resistance and Resitivity for Ohmic Material
J E
E  J
(= I/A if current
density is uniform)
resistivity
 I
 L
 V  EL     L  I   
 A
 A
A
L
R (in) Ohms Ω
resistance
Lecture 10-11
Resistance
Resistance
(definition)
V
R
I
R
I
V
constant R
L
R
A
Ohm’s Law
Lecture 10-12
Warm up quiz 2
There are 2x1014 electrons entering a resistor
in10 seconds. What is the current through the
resistor?
a) 2.0 μA
b) 1.6 mA
c) 3.2 nA
d) 1.6 A
e) 3.2 μA
Note: e = 1.6x10-19 C
R
I
V
Lecture 10-13
Temperature Dependence of Resistivity
  0 1   (T  T0 )
• Usually T0 is 293K (room temp.)
• Usually α > 0 (ρ increases as T )
Material
ρ0 (Ωm)
α (K-1)
Ag
1.6x10-8
3.8x10-3
Cu
1.7x10-8
3.9x10-3
Si
6.4x102
-7.5x10-2
glass
1010 ~ 1014
sulfur
1015
Copper
Lecture 10-14
Electric Current and Joule Heating
electron gas
• Free electrons in a conductor gains
kinetic energy due to an externally
applied E.
• Scattering from the atomic ions of the
metal and other electrons quickly leads to
a steady state with a constant current I.
Transfers energy to the atoms of the solid
(to vibrate), i.e., Joule heating.
Mean drift of electrons, i.e., current
Lecture 10-15
Energy in Electric Circuits
• Steady current means a
constant amount of charge ΔQ
flows past any given cross
section during time Δt, where
I= ΔQ / Δt.
Energy lost by ΔQ is
V
U  Q  (Va  Vb )  I t V
=> heat
So, Power dissipation = rate of decrease of U =
dU
P
 IV  I 2 R  V 2 / R
dt
Lecture 10-16
EMF – Electromotive Force
• An EMF device is a charge pump that can maintain a potential
difference across two terminals by doing work on the charges
when necessary.
Examples: battery, fuel
cell, electric generator,
solar cell, fuel cell,
thermopile, …
• Converts energy (chemical, mechanical, solar, thermal, …)
into electrical energy.
 Within the EMF device, positive charges
are lifted from lower to higher potential.
 If work dW is required to lift charge dq,

dW

dq
   Volt
EMF
Lecture 10-17
Energy Conservation
A circuit consists of an ideal battery
(B) with emf ε, a resistor R, and two
connecting wires of negligible
resistance.
Energy
conservation
• Ideal battery: no internal
energy dissipation
• Real battery: internal
energy dissipation exists
Work done by battery is equal
to energy dissipated in resistor
dW  i 2 Rdt or  i dt  i 2 R dt
   iR
EMF ε = terminal voltage V
dW > i2Rdt then εi > iR=V
Lecture 10-18
DOCCAM 2 DEMO 5B-02
TERMINAL VOLTAGE ON A BATTERY
Lecture 10-19
Resistors in Series
 The current through devices
in series is always the same.
i
R1
R2
i
i
Req

ε
ε
  iR1  iR2  0
  iReq  0
Req  ( R1  R2 )
For multiple
resistors in series:
Req  R1  R2  R3  ...
Lecture 10-20
Internal Resistance of a Battery
Life story (ups and downs) of a charge
load
 i r iR  0
internal
resistance
 i

Rr
terminal
voltage
, Vb  Va    ir

R
Rr
Lecture 10-21
Lecture 10:30 quiz 3 September 22, 20ll
There are 1014 electrons entering in 10 seconds a
resistor which has a potential drop of 3.2μV.
What is the resistance of the resistor?
a) 4.0 Ω
b) 1.0 mΩ
c) 2.0 Ω
d) 3.0 μΩ
e) 2.0 kΩ
Note: e = 1.6x10-19 C
R
I
V
Lecture 10-22
Lecture 11:30 quiz 3 September 22, 2011
There are 1014 electrons entering a resistor of
resistance 1.0 Ω in 10 seconds. What is the
potential drop across the resistor?
a) 3.2 mV
b) 8.0 V
c) 2.5 V
d) 1.6 μV
e) 1.6 mV
Note: e = 1.6x10-19 C
R
I
V
Lecture 10-23
Lecture 11:30 quiz 3 February 10, 2011
The potential drop is 6.4mV across a resistor of
resistance 1.0Ω. How many electrons enter the
wire in 5 seconds?
a)3.2×1014
b)8.0×1015
c)2.5×1012
d)2.0×1017
e)1.6×1019
Note: e = 1.6x10-19 C
R
I
V