Lecture 6 Current and Resistance Chp. 27

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Lecture 6 Current and Resistance Chp. 27

• Cartoon -Invention of the battery and Voltaic
  Cell
• Opening Demo - Lemon Battery
• Warm-up problem
• Physlet
• Topics
• Demos
• Lemon Battery estimate internal resistance
• Ohms Law demo on overhead projector
• T dependence of resistance
• Three 100 Watt light bulbs
• Puzzles
• Resistor network figure out equivalent resistance

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Loop of copper wire
Nothing moving electrostatic equilibrium
Now battery forces charge through the conductor.
We have a field in the wire.
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What is Current?
It is the amount of positive charge that moves
past a certain point per unit time.
I
Copper wire with voltage across it


A
?L
Drift velocity of charge
?Q charge per unit volume x volume
nq x Av?t
?Q nqAv ?t
Density of electrons
1.6 x 10-19 C
Divide both sides by ?t.
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Question What causes charges to move in the
wire?
How many charges are available to move?
Example What is the drift velocity for
1 Amp of current flowing through
a 14 gauge copper wire of radius 0.815
mm?

• I 1 Amp
• q 1.6x10-19 C
• A ?(.0815 cm)2
• 8.9 grams/cm 3
• No 6x1023 atoms/mole
• M 63.5 grams/mole

Drift velocity
8.4x1022 atoms/cm3
The higher the density the smaller the drift
velocity
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Drift speed of electrons and current density
Directions of current i is defined as the
direction of positive charge.
(Note positive charge moves in direction of E)
electron flow is opposite E.
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Currents Steady motion of charge and
conservation of current
i i1 i2
(Kirchoffs 2nd rule)
Current is the same throughout all sections in
the diagram below it is continuous. Current
density J does vary.
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Question How does the drift speed compare to the
instantaneous speed?
Instantaneous speed ? 106 m/s
vd ? 3.5x10-11 vinstant
(This tiny ratio is why Ohms Law works so well
for metals.) At this drift speed 3.5x10-5 m/s, it
would take an electron 8 hours to go 1 meter.
Question So why does the light come on
immediately when you turn on the light switch?
Its like when the hose is full of water and you
turn the faucet on, it immediately comes out the
ends. The charge in the wire is like the water. A
wave of electric field travels very rapidly down
the wire, causing the free charges to begin
drifting.
Example Recall typical TV tube, CRT, or PC
monitor. The electron beam has a speed 5x107 m/s.
If the current is I 100 microamps, what is n?
Take A 1mm2 (10-3)2
10-6 m2
For CRT
n 1.2x1013 e/m3 1.2x107 e/cm3
The lower the density the higher the speed.
For Copper
n 8.5x1022 e/cm3
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What is Resistance?
The collisions between the electrons and the
atoms is the cause of resistance and a very slow
drift velocity of the electrons. The higher
density, the more collisions.
field off
field on
extra distance electron traveled
e-

• The dashed lines represent the straight line
  tracks of electrons in between collisions
• Electric field is off.
• Electric field is on. When the field is on, the
  electron traveled drifted further to BI.

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Ohms Law
Want to emphasize here that as long as we have
current (charge moving) due to an applied
potential, the electric field is no longer zero
inside the conductor.
I
Potential difference VB - VA E ?L


A
B
Constant E
?L
I current ? E ?L (Ohms Law)
True for many materials not all. Note that this
is an experimental observation and is not a true
law.
Best conductors Silver w/ sulpher Copper
oxidizes Gold pretty inert Non-ohmic
materials Diodes Superconductors
Constant of proportionality between V and I is
known as the resistance. The SI unit for
resistance is called the ohm.
V RI
R V/I
Ohm volt/amp
Demo Show Ohms Law
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A test of whether or not a material satisfies
Ohms Law
V IR
Here the slope depends on the potential
difference. Ohms Law is violated.
I V/R
Slope 1/R constant
Ohms Law is satisfied.
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Resistance What is it? Denote it by R

• Depends on shape, material, temperature.
• Most metals R increases with increasing T
• Semi-conductors R decreases with increasing T

Define a new constant which characterizes
materials.
Resistivity
A
L
Demo Show temperature dependence of resistance
For materials ? 10-8 to 1015 ohms-meters
Example What is the resistance of a 14 gauge Cu
wire? Find the resistance per unit length.
Build circuits with copper wire. We can neglect
the resistance of the wire. For short wires 1-2
m, this is a good approximation.
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Example Temperature variation of resistivity.
? ?20 1 ? (T-20)
can be positive or negative
Consider two examples of materials at T 20oC.
?(?-m) ?(k-1) L Area R (20oC)
Fe 10-7 .005 6x106 m 1mm2(10-6m2) 60,000 ?
Si 640 -.075 1 m 1 m2 640 ?
Fe conductor - a long 6x106 m wire. Si
insulator - a cube of Si 1 m on each
side
Question You might ask is there a temperature
where a conductor and insulator are one and the
same?
Condition RFe RSi at what temperature?
Set RFe RSi and solve for T
Use
?20 1 ? (T-20)
T 20 -196C (pretty low temperature)
RFe 10-7 ?-m 1 .005 (T-20)
T -176C
RSi 640 ?-m 1 .075 (T-20)
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Resistance at different Temperatures Resistance at different Temperatures Resistance at different Temperatures Resistance at different Temperatures
Cu .1194 ? .0152 ? conductor
Nb .0235 ? .0209 ? impure
C .0553 ? .069 ? semiconductor
T 300 K 77 K
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Power dissipation resistors
I
Potential energy decrease
?U ?Q (-V)
P IV
(drop the minus sign)

• Rate of potential energy decreases equals rate of
  thermal energy increases in resistor.
• Called Joule heating
• good for stove and electric oven
• nuisance in a PC need a fan to cool computer

Also since V IR,
P I2R or V2/R
All are equivalent.
Example How much power is dissipated when I
2A flows through the Fe resistor of R 10,000
?. P I2R 22x104 ? 40,000 Watts
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Batteries
A device that stores chemical energy and converts
it to electrical energy.
Emf of a battery is the amount of increase of
electrical potential of the charge when it flows
from negative to positive in the battery. (Emf
stands for electromotive force.)
Carbon-zinc Emf 1.5V Lead-acid in car Emf
2V per cell (large areas of cells give lots of
current)
Car battery has 6 cells or 12 volts.
Power of a battery P
P ?I
? is the Emf
Batteries are rated by their energy content.
Normally they give an equivalent measure such as
the charge content in
mA-Hrs
milliamp-Hours Charge (coulomb/seconds) x
seconds
Internal Resistance As the battery runs out of
chemical energy the internal resistance increases.
Terminal Voltage decreases quickly. How do you
visualize this?
What is terminal voltage?
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What is the relationship between Emf, resistance,
current, and terminal voltage?
Circuit model looks like this
I

r
R
Terminal voltage V V IR (decrease in PE)
?

• Ir IR ? ? I (r R)
• I ?/(r R)

? - Ir V IR
The terminal voltage decrease ? - Ir as the
internal resistance r increases or when I
increases.
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Example This is called impedance matching. The
question is what value of load resistor R do you
want to maximize power transfer from the battery
to the load.
current from battery P I2R power
dissipated in load
P
R
?
Solve for R
R r
You get max. power when load resistor equals
internal resistance of battery. (battery doesnt
last long)
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Demo show lemon or apple batteries Estimate
internal resistance by adjusting R until the
terminal voltage is half of the open circuit
voltage. ? For lemons rlemons ? 3600 ?.
r R
? - Ir V
terminal voltage
open circuit voltage
Note When
, r R
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Combination of resistors
Resistors in series
V R1I R2I (R1 R2)I
Requiv R1 R2
Resistors in parallel
Voltages are the same, currents add.
I I1 I2
V/R V/R1 V/R2
? 1/R 1/R1 1/R2
Requiv R1R2 /(R1 R2)
Demo 3 lightbulb resistor puzzle
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Equivalence of two versions of Ohms Law
E ?J ? V RI
LE L ?J
V L ?J
R ? (L/A)
?L AR
V ARJ RJA
I
? V RI
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Warm-up Set 6
1. HRW6 27.TB.08. 119812 Current is a measure
of amount of charge that moves past a point
per unit time force that moves a charge past
a point energy used to move a charge past a
point speed with which a charge moves past a
point resistance to the movement of a charge
past a point 2. HRW6 27.TB.14. 119818 In a
conductor carrying a current we expect the
electron drift speed to be about the same as
the average electron speed much less than the
average electron speed less than the electron
speed at high temperature and greater than the
electron speed at low temperature less than
the electron speed at low temperature and greater
than the electron speed at high temperature
much greater than the average electron speed 3.
HRW6 27.TB.49. 119853 You buy a "75 W" light
bulb. The label means that the bulb is
expected to "burn out" after you use up its 75
watts none of these no matter how you use
the bulb, the power will be 75 W the bulb was
filled with 75 W at the factory the actual
power dissipated will be much higher than 75 W
since most of the power appears as heat
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What is the electric field in a sphere of uniform
distribution of positive charge. (nucleus of
protons)
R
E
r
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S
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Find the capacitance of a ordinary piece of
coaxial cable (TV cable)
For a long wire we found that
where r is radial to the wire.
a b
ds - dr because path of integration is radially
inward
a 0.5 mm b 2.0 mm ? ? 2
or
Va is higher than Vb
?0 (air)

? 2
? air
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Capacitance of two concentric spherical shells
- q
Integration path
E
q
a
ds - dr
b
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Model of coaxial cable for calculation of
capacitance
Outer metal braid
Signal wire