Download Lecture 7.1 : Current and Resistance

Lecture 7.1 :
Current and Resistance
Lecture Outline:
Current and Current Density
Conductivity and Resistivity
Resistance and Ohm’s Law
Textbook Reading:
Ch. 30.3 - 30.5
Feb. 25, 2014
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Announcements
•Homework #7 is now posted.
Due next Tue., March 4, at 9am.
•Optional review session for Exam #2 is scheduled for Tue., March
4, from 5-7pm in Stolkin. Bring your clickers.
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Last Lecture...
Conduction electrons are like free particles
moving through a metal lattice.
No Electric Field
With Electric Field
Average Velocity is Zero
Average Velocity is NonZero
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Last Lecture...
Electrons are the charge carriers in metals!
electron current (ie) = number of electrons per second that pass
through a conductor in a given time.
ie = ne A
d
ne e⇥ A
=
E
m
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Clicker Question #1
A wire carries a current. If both the wire diameter
and the electron drift speed are doubled, the
electron current increases by a factor of
A.
B.
C.
D.
E.
2.
4.
6.
8.
Some other value.
ie ∝ Aνd
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Current and Current Density
Current was known long before electrons were discovered, so
conventional definition of current is in terms of Charge:
I
dQ
dt
1 ampere = 1 A ≡ 1 coulomb per second = 1 C/s
Conventional current and electron current are related:
Q
eNe
I=
=
= eie
t
t
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Current and Current Density
By convention, current is defined to
flow in the direction in which
positive charges would move! (i.e. in the direction of the Electric Field)
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Current and Current Density
We define current density (J) to arrive at a property intrinsic to a
material, and not specific to the cross-sectional Area of a conductor.
J
I
= ne e
A
d
The current in a 100 Watt lightbulb is 0.85 A. The filament inside the bulb
is 0.25 mm in diameter. What is the current density in the filament?
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Current and Current Density
Current is a conserved quantity!
The current is the same at all points in a current-carrying wire.
Current is not “used up” by the paddle...but energy is required to
turn the paddle.
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Clicker Question #2
A and B are identical lightbulbs
connected to a battery as
shown. Which is brighter?
A. Bulb A.
B. Bulb B.
C. The bulbs are equally bright.
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Current and Current Density
Kirchoff’s Junction Law:
Iin =
Iout
Current = ?
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Conductivity and Resistivity
J = ne e
Define a new
property of materials
d
= ne e
e⇥ E
m
⇥
2
ne e ⇥
=
E
m
2
ne e ⇥
= conductivity =
m
J= E
Conductivity describes how easily electrons can move
through a material. (Also depends on temperature)
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Clicker Question #3
The current density in this wire is
A.
B.
C.
D.
E.
4 × 106 A/m2.
2 × 106 A/m2.
4 × 103 A/m2.
2 × 103 A/m2.
Some other value.
Current Density: J = Current/Area = (8A)/(0.002m x 0.002m) = 2x106 A/m2
Which way does the current density vector point?
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Clicker Question #4
Both segments of the wire are made of the same
metal. Current I1 flows into segment 1 from the left.
How does current I1 in segment 1 compare to
current I2 in segment 2?
I 1 > I 2.
B. I1 = I2.
C. I1 < I2.
D. There’s not enough information to
compare them.
A.
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Clicker #5
Both segments of the wire are made of
the same metal. Current I1 flows into
segment 1 from the left. How does
current density J1 in segment 1 compare
to current density J2 in segment 2?
J1 > J2. Smaller cross-section area
B. J1 = J2.
C. J1 < J2.
D. There’s not enough information to
compare them.
A.
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Conductivity and Resistivity
Inverse of conductivity (resistivity) tells how reluctant
electrons are to move through a material.
1
m
= resistivity = =
⇥
ne e 2 ⇤
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Resistance and Ohm’s Law
Potential Difference across conductor creates Electric field.
E=
V
V
=
s
L
I = JA = A⇥E =
A
Ohm’s Law: I =
V
L
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V
R
A
E
R = resistance
Resistance and Ohm’s Law
Resistance (R) characterizes ability of current to flow through
specific piece of conductor with a specific geometry.
L
R=
A
1 ohm = 1 Ω ≡ 1 Volt/Ampere = 1 V/A
Wire 2 is twice the length and twice
the diameter of wire 1. What is the
ratio R2/R1 of their resistances?
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Conductivity and Resistivity
Superconductivity!
Some materials exhibit zero resistance to current if they are
cooled to low enough temperatures.
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Conductivity and Resistivity
A 3.0 mm diameter wire carries a 12 A current when the
electric field is 0.085 V/m. What is the wire’s resistivity?
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Resistance and Ohm’s Law
Ohm’s Law: Establishing a potential difference ΔV across the ends of a
conductor with resistance R causes a current I through the current.
V
I=
R
Example: What resistor would have a 15 mA current if
connected across the terminals of a 9.0 V battery?
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Resistance and Ohm’s Law
In this course, we will assume wires are ideal (R = 0 Ω ), so
they don’t have any potential drop across them.
Real wires have some resistance, and use up
some energy in a circuit.
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Resistance and Ohm’s Law
Resistors are placed in electrical circuits to control the
voltage at different points in the circuit.
Resistors are made out of materials, such as
carbon, that are poor conductors.
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Reminders
•Homework #7 is now posted.
•Come see me if you have any questions or concerns.
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