Lecture 10-1
Physics 241 – Question 1 – September 22, 2011
Below, all capacitors are identical. Which
combination has the smallest equivalent capacitance?
a
b
d
c
e
Lecture 10-2
Batteries
• We use batteries as devices that provide direct currents in
circuits
• The voltage written on it is the potential difference that it can
provide to a circuit
• You will also find its rating in units of mAh
– This rating provides information on the total charge that a single battery
can deliver over its lifetime
• The quantity mAh is another unit of charge:
3
1 mAh  (10 A)(3600 s)  3.6 As  3.6 C
Lecture 10-3
Circuits
In this circuit, electrons flow around
the circuit counterclockwise.
(The conventionally defined current
is clockwise; remember, electrons
are negative charges.)
The electrons can’t disappear so the
current requires a whole loop!
+
Chemical action pumps electrons from the positive
terminal (+) to the negative terminal (-) in the battery.
The emf (electromotive force, or electric field) pushes
electrons around the wire from (-) to (+).
Direct electric current flows in one direction
lightbulb
Lecture 10-4
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 [O(10-4 m/s)] is orders of
magnitudes less than the actual velocity of
charges [O(106 m/s)].
Lecture 10-5
Drift Velocity (1)
• Consider a conductor with cross
sectional area A and electric field E
• Suppose that there are
n electrons per unit volume
• The negatively charged electrons will drift in a direction opposite to
the electric field
• We assume that all the electrons have the same drift velocity vd and
that the current density J is uniform.
• In a time interval dt, each electron moves a distance vd dt
• The volume that will pass through area A is then Avd dt; the number of
electrons is dn = nAvd dt
Lecture 10-6
Drift Velocity (2)
The volume that will pass through area A is then Avddt;
the number of electrons is dn = nAvd dt
• Each electron has charge e so that the charge dq that
flows through the area A in time dt is
• So the current is
dq
I
 nevd A
dt
… and the current density is
I
J   nevd
A
• For a positive charge, the current density and the drift velocity are
parallel vectors, pointing in the same directions. As vectors:
J  nevd
• For electrons:
J  nevd
Lecture 10-7
Drift Velocity (3)
• Consider a wire carrying a current

• The physical current carriers are negatively charged electrons
• These electrons are moving to the left in this drawing
• However, the electric field, current density and current are directed to the
right
Comments
Electrons are negative charges!
On top of the coherent motion the electrons
have random (thermal) motion.
Lecture 10-8
The Ampere
• The unit of current is coulombs per second, which has been
given the unit ampere, named after French physicist AndréMarie Ampère, (1775-1836)
• The ampere is abbreviated as A and is defined by
1C
1A
1s
• Some typical currents are
–
–
–
–
Flashlight - 1 A
The starter motor in a car - 200 A
iPod - 50 mA
Lightning strike (for a very short time) – 100,000 A
Lecture 10-9
Example: Current Through a Wire
• The current density J in a cylindrical wire of radius R = 2.0 mm is
uniform across a cross section of the wire and has the value
J = 2.0 105 A/m2.
Question: What is the current i through the outer portion of the wire between
radial distances R/2 and R?
Answer:
• J = current per unit area = dI/dA
Area A’ (outer portion)
2
3R
2
A   R2   R / 2   
 9.424 10 6 m 2
4
Current through A’
I  JA   2.0 105 A/m2  9.424 106 m2   1.9 A
R
Lecture 10-10
Ohm’s Law
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 10-11
Resistance and Resistivity
Resistance
(definition)
V
R
I
R
I
V
Constant R
Ohm’s Law
R (in )
L
R
A
resistivity (in  m)
Material
 (m)
Ag
1.6 x10-8
Cu
1.7x10-8
Si
6.4x102
glass
1010 ~ 1014
wood
108 ~ 1011
Resistivity  at 20C
Lecture 10-12
Physics 241 – Question 2 – September 22, 20011
Which one(s) of the wires (made of the same
material) below have the largest resistance in the
vertical direction?
A
a)
b)
c)
d)
e)
1 and 3
2 only
3 and 4
4 only
2 and 4
L
1
2A
2
3
2L
A/2
L
4
Lecture 10-13
Example: Resistance of a Copper Wire (1)
• Standard wires that electricians put into residential
housing have fairly low resistance.
Question:
What is the resistance of a length of 100 m of standard 12gauge copper wire, typically used in household wiring for
electrical outlets?
Answer:
• The American Wire Gauge (AWG) size convention
specifies wire cross sectional area on a logarithmic scale.
• A lower gauge number corresponds to a thicker wire.
• Every reduction by 3 gauges doubles the cross-sectional
area.
(36 AWG ) / 39
d  0.127  92
mm
Lecture 10-14
Example: Resistance of a Copper Wire (2)
• The formula to convert from the AWG size to the wire
diameter is
(36 AWG ) / 39
d  0.127  92
mm
• So a 12-gauge copper wire has a diameter of 2.05 mm
• Its cross sectional area is then
A   d  3.3 mm
1
4
2
2
• Look up the resistivity of copper in the table …
L
100 m
-8
R    (1.72 10 m)
 0.52 
-6
2
A
3.3 10 m
Lecture 10-15
Physics 241 – Question 3 – September 22, 2011
What is the resistance of a copper wire that
has length L=70.0 m and diameter
-8


1.72

10
m
d=2.60 mm ?
a) 0.119 
b) 0.139 
c) 0.163 
d) 0.190 
e) 0.227 
Lecture 10-16
Demo - Resistance and Resistivity
Resistance
(definition)
V
R
I
R
I
V
Ohm’s Law
Constant R
R (in )
L
R
A
 I new  2 I
R
new
R

2
 (m)
Ag
1.6 x10-8
Cu
1.7x10-8
Si
6.4x102
glass
1010 ~ 1014
wood
108 ~ 1011
Resistivity  at 20C
resistivity (in  m)
L
L
2
Material
V
I 
R
I
new
V
V

2
( R / 2)
R
Lecture 10-17
•
•
•
•
•
Resistors
In many electronics applications one needs a range
of resistances in various parts of the circuits
For this purpose one can use commercially
available resistors
Resistors are commonly made from carbon,
inside a plastic cover with two wires sticking out at the
two ends for electrical connection
The value of the resistance is indicated by four colorbands on the plastic capsule
The first two bands are numbers for the mantissa, the
third is a power of ten, and the fourth is a tolerance for
the range of values
Lecture 10-18
Resistor Color Codes
Example:
Colors (left to right)
red, yellow, green, and gold
Using our table, we can see that
the resistance is
24×105  = 2.4 M
with a tolerance of 5%
Lecture 10-19
Temperature Dependence of Resistivity
  0 1   (T  T0 )
• Usually T0 is 293K (room temp.)
• Usually  > 0 (ρ increases as T )
Material
0 (m)
 (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 10-20
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 10-21
Energy in Electric Circuits
• Steady current means a
constant amount of charge Q
flows past any given cross
section during time t, where
I= Q / t.
Energy lost by Q 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