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Comment on exam scores…
It just goes to show, you do not have to be faster than the
monsters… you just have to be faster than your slowest
companion.
Titan Quest screenshot, just after escaping a monster that ate a slow-footed companion.
Comment on exam scores… Exam 1 grade distribution (regrades not included)…
I will provide the Exam 1 grade distribution during this lecture.
Today’s agenda:
Electric Current.You must know the definition of current, and be able to use it in solving problems.
Current Density.You must understand the difference between current and current density, and be able to use current density in solving problems.
Ohm’s Law and Resistance.You must be able to use Ohm’s Law and electrical resistance in solving circuit problems.
Resistivity.You must understand the relationship between resistance and resistivity, and be able to calculate resistivity and associated quantities.
Temperature Dependence of Resistivity.You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.
Electric Current
Definition of Electric Current
The average current that passes any point in a conductor during a time t is defined as
where Q is the amount of charge passing the point.
One ampere of current is one coulomb per second:
av
QI
t
The instantaneous current isdQ
I = .dt
.1C
1A=1s
Here’s a really simple circuit:
+-
current
Don’t try that at home! (Why not?)
The current is in the direction of flow of positive charge… …opposite to the flow of electrons, which are usually the charge carriers.
Currents in battery-operated devices are often in the milliamp range: 1 mA = 10-3 A.
“m” for milli—another abbreviation to remember!
+ -
current electrons
An electron flowing from – to +
“Conventional” refers to our convention, which is always to consider the effect of + charges (for example, electric field direction is defined relative to + charges).
An electron flowing from – to + gives rise to the same “conventional current” as a proton flowing from + to -.
“Hey, that figure you just showed me is confusing.
+ -
current electrons
Good question.
“Hey, that figure you just showed me is confusing. Why don’t electrons flow like this?”
Chemical reactions (or whatever energy mechanism the battery uses) “force” electrons to the negative terminal. The battery won’t “let” electrons flow the wrong way inside it. So electrons pick the easiest path—through the external wires towards the + terminal.
Of course, real electrons don’t “want” anything.
Electrons “want” to get away from - and go to +.
+ -
current electrons
Note!
Current is a scalar quantity, and it has a sign associated with it.
In diagrams, assume that a current indicated by a symbol and an arrow is the conventional current.
I1
If your calculation produces a negative value for the current, that means the conventional current actually flows opposite to the direction indicated by the arrow.
Example: 3.8x1021 electrons pass through a point in a wire in 4 minutes. What was the average current?
av
Q NeI
t t
21 19
av
3.8 10 1.6 10I
4 60
avI 2.53A
Today’s agenda:
Electric Current.You must know the definition of current, and be able to use it in solving problems.
Current Density.You must understand the difference between current and current density, and be able to use current density in solving problems.
Ohm’s Law and Resistance.You must be able to use Ohm’s Law and electrical resistance in solving circuit problems.
Resistivity.You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities.
Temperature Dependence of Resistivity.You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.
Current Density
When we study details of charge transport, we use the concept of current density.
Current density is the amount of charge that flows across a unit of area in a unit of time.
Current density: charge per area per time (current / area).
A current density J flowing through an infinitesimal area dA produces an infinitesimal current dI.
dA
J
dI J dA
The total current passing through A is just
surface
I J dA
Current density is a vector. Its direction is the direction of the velocity of positive charge carriers.
Current density: charge per area per time.
No OSE’s on this page.
Simpler, less-general OSE on next page.
surface surface
II J dA J dA JA J
A
If J is constant and parallel to dA (like in a wire), then
A
J
Now let’s take a “microscopic” view of current and calculate J.
Avvt
q
If n is the number of charges per volume, then the number of charges that pass through a surface A in a time t is
numbervolume n v t A
volume
The total amount of charge passing through A is the number of charges times the charge of each.
Avvt
qQ nqv t A
Divide by t to get the current…
QI nqv A
t
…and by A to get J:
J nqv .
To account for the vector nature of the current density,
J nqv
and if the charge carriers are electrons, q=-e so that
eJ n e v.
The – sign demonstrates that the velocity of the electrons is antiparallel to the conventional current direction.
Not quite “official”
yet.
Not quite “official”
yet.
Currents in Materials
Metals are conductors because they have “free” electrons, which are not bound to metal atoms.
In a cubic meter of a typical conductor there roughly 1028 free electrons, moving with typical speeds of 1,000,000 m/s…
Thanks to Dr. Yew San Hor for this slide.
…but the electrons move in random directions, and there is no net flow of charge, until you apply an electric field.
-
E electron “drift” velocity
The electric field accelerates the electron, but only until the electron collides with a “scattering center.” Then the electron’s velocity is randomized and the acceleration begins again.Some predictions based on this model are off by a factor or 10 or so, but with the inclusion of some quantum mechanics it becomes accurate. The “scattering” idea is useful.
A greatly oversimplified model, but the “idea” is useful.
just one electron shown, for simplicity
inside a conductor
Even though some details of the model on the previous slide are wrong, it points us in the right direction, and works when you take quantum mechanics into account.
In particular, the velocity that should be used in
J n q v.
is not the charge carrier’s velocity (electrons in this example).
Instead, we should the use net velocity of the collection of electrons, the net velocity caused by the electric field.
This “net velocity” is like the terminal velocity of a parachutist; we call it the “drift velocity.”
dJ n q v .
Quantum mechanics shows us how to deal correctly with the collection of electrons.
It’s the drift velocity that we should use in our equations for current and current density in conductors:
dJ n q v
dI nqv A
d
Iv
nqA
Example: the 12-gauge copper wire in a home has a cross-sectional area of 3.31x10-6 m2 and carries a current of 10 A. The conduction electron density in copper is 8.49x1028 electrons/m3. Calculate the drift speed of the electrons.
d
Iv
nqA
d
Iv
neA
d 28 -3 19 6 2
10C/sv
(8.49 10 m )(1.60 10 C)(3.31 10 m )
4dv 2.22 10 m/s
Today’s agenda:
Electric Current.You must know the definition of current, and be able to use it in solving problems.
Current Density.You must understand the difference between current and current density, and be able to use current density in solving problems.
Ohm’s Law and Resistance.You must be able to use Ohm’s Law and electrical resistance in solving circuit problems.
Resistivity.You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities.
Temperature Dependence of Resistivity.You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.
Resistance
The resistance of a material is a measure of how easily a charge flows through it.
Resistance: how much “push” is needed to get a given current to flow. V
RI
The unit of resistance is the ohm:1 V
1 .1 A
Resistances of kilohms and megohms are common:3 61 k 10 , 1 M =10 .
This is the symbol we use for a “resistor:”
All wires have resistance. Obviously, for efficiency in carrying a current, we want a wire having a low resistance. In idealized problems, we will consider wire resistance to be zero.Lamps, batteries, and other devices in circuits have resistance.
Every circuit component has resistance.
Resistors are often intentionally used in circuits. The picture shows a strip of five resistors (you tear off the paper and solder the resistors into circuits).
The little bands of color on the resistors have meaning. Here are a couple of handy web links:1. http://www.dannyg.com/examples/res2/resistor.htm2. http://www.digikey.com/en/resources/conversion-calculators/conversion-calculator-resistor-color-code-4-band
Ohm’s Law
In some materials, the resistance is constant over a wide range of voltages.
For such materials, we write and call the equation “Ohm’s Law.”
V IR,
In fact, Ohm’s Law is not a “Law” in the same sense as Newton’s Laws…
Newton’s Laws demand; Ohm’s Law suggests.
… and in advanced Physics classes you will write something other than V=IR when you write Ohm’s Law.
Materials that follow Ohm’s Law are called “ohmic” materials, and have linear I vs. V graphs.
I
V
slope=1/R
I
V
Materials that do not follow Ohm’s Law are called “nonohmic” materials, and have curved I vs. V graphs.
Materials that follow Ohm’s Law are called “ohmic” materials, and have linear I vs. V graphs.
I
V
slope=1/R
I
V
Materials that do not follow Ohm’s Law are called “nonohmic” materials, and have curved I vs. V graphs.
Today’s agenda:
Electric Current.You must know the definition of current, and be able to use it in solving problems.
Current Density.You must understand the difference between current and current density, and be able to use current density in solving problems.
Ohm’s Law and Resistance.You must be able to use Ohm’s Law and electrical resistance in solving circuit problems.
Resistivity.You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities.
Temperature Dependence of Resistivity.You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.
This makes sense: a longer wire or higher-resistivity wire should have a greater resistance. A larger area means more “space” for electrons to get through, hence lower resistance.
It is also experimentally observed (and justified by quantum mechanics) that the resistance of a metal wire is well-described by
Resistivity
LR ,
A
where is a “constant” called the resistivity of the wire material, L is the wire length, and A its cross-sectional area.
L
The longer a wire, the “harder” it is to push electrons through it.
R = L / A,
The greater the resistivity, the “harder” it is to push electrons through it.
The greater the cross-sectional area, the “easier” it is to push electrons through it.
A
Resistivity is a useful tool in physics because it depends on the properties of the wire material, and not the geometry.
units of are m
Resistivities range from roughly 10-8 ·m for copper wire to 1015 ·m for hard rubber. That’s an incredible range of 23 orders of magnitude, and doesn’t even include superconductors (we might talk about them some time).
R = L / A
A = L / R
A = (d/2)2 geometry!
(d/2)2 = L / R
Example (will not be worked in class): Suppose you want to connect your stereo to remote speakers.(a) If each wire must be 20 m long, what diameter copper wire should you use to make the resistance 0.10 per wire.
(d/2)2 = L / R
d/2= ( L / R )½ don’t skip steps!
d = 2 ( L / R )½
d = 2 [ (1.68x10-8) (20) / (0.1) ]½
d = 0.0021 m = 2.1 mm
V = I R
(b) If the current to each speaker is 4.0 A, what is the voltage drop across each wire?
V = (4.0) (0.10)
V = 0.4 V
Homework hint you can look up the resistivity of copper in a table in your text.
Ohm’s Law Revisited
The equation for resistivity I introduced five slides back is a semi-empirical one. Here’s almost how we define resistivity:
E.
J
Our equation relating R and follows from the above equation.
We define conductivity as the inverse of the resistivity: 1 1
, or .
NOT an official starting
equation!
With the above definitions,
E J,
J E.
The “official” Ohm’s law, valid for non-ohmic materials.
Cautions!
In this context: is not volume density! is not surface density!
Think of this as our definition of resistivity.
In anisotropic materials, and are tensors. A tensor is like a matrix,
only worse.
Example: the 12-gauge copper wire in a home has a cross-sectional area of 3.31x10-6 m2 and carries a current of 10 A. Calculate the magnitude of the electric field in the wire.
IE J
A
8
6 2
(1.72 10 m) 10C/sE
(3.31 10 m )
2E 5.20 10 V/m
Homework hint (not needed in this particular example): in this chapter it is safe to use V=Ed.
Question: are we still assuming the electrostatic case?
Today’s agenda:
Electric Current.You must know the definition of current, and be able to use it in solving problems.
Current Density.You must understand the difference between current and current density, and be able to use current density in solving problems.
Ohm’s Law and Resistance.You must be able to use Ohm’s Law and electrical resistance in solving circuit problems.
Resistivity.You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities.
Temperature Dependence of Resistivity.You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.
Temperature Dependence of Resistivity
Many materials have resistivities that depend on temperature. We can model* this temperature dependence by an equation of the form 0 01 T T ,
where 0 is the resistivity at temperature T0, and is the temperature coefficient of resistivity.
*T0 is a reference temperature, often taken to be 0 °C or 20 °C. This approximation can be used if the temperature range is “not too great;” i.e. 100 °C or so.
Resistance thermometers made of carbon (inexpensive) and platinum (expensive) are widely used to measure very low temperatures.
Example: a carbon resistance thermometer in the shape of a cylinder 1 cm long and 4 mm in diameter is attached to a sample. The thermometer has a resistance of 0.030 . What is the temperature of the sample?
0 01 T T
This is the starting equation:
We use the thermometer dimensions to calculate the resistivity when the resistance is 0.03 , and use the above equation directly.
We can look up the resistivity of carbon at 20 C.
Or we can rewrite the equation in terms or R. Let’s first do the calculation using resistivity.
Example: a carbon resistance thermometer in the shape of a cylinder 1 cm long and 4 mm in diameter is attached to a sample. The thermometer has a resistance of 0.030 . What is the temperature of the sample?
The resistivity of carbon at 20 C is 5
0 3.519 10 m
RA(R)
L
LR
A
2
50.03 0.002
(R 0.03) 3.7699 10 m0.01
Example: a carbon resistance thermometer in the shape of a cylinder 1 cm long and 4 mm in diameter is attached to a sample. The thermometer has a resistance of 0.030 . What is the temperature of the sample?
0 01 T T
00
T T 1
00
1T T 1
-10.0005 C
5
5
1 3.7699 10T 20 1 122.6 C
0.0005 3.519 10
Example: a carbon resistance thermometer in the shape of a cylinder 1 cm long and 4 mm in diameter is attached to a sample. The thermometer has a resistance of 0.030 . What is the temperature of the sample?
Alternatively, we can use the resistivity of carbon at 20 C to calculate the resistance at 20 C.
50 0T 20 C 3.519 10 m L = 0.01 m r = 0.002 m
00 2
LR 0.02800
r
This is the resistance at 20 C.
0 01 T T
Example: a carbon resistance thermometer in the shape of a cylinder 1 cm long and 4 mm in diameter is attached to a sample. The thermometer has a resistance of 0.030 . What is the temperature of the sample?
0 00
0
R ARA1 T T
L L
0 0R R 1 T T
If we assume A/L = A0/L0, then
00
RT T 1
R
00
1 RT T 1
R
Example: a carbon resistance thermometer in the shape of a cylinder 1 cm long and 4 mm in diameter is attached to a sample. The thermometer has a resistance of 0.030 . What is the temperature of the sample?
0 0R R 1 T T
1 .030T 20 1 122.9 C
0.0005 .028
The result is very sensitive to significant figures in resistivity and .
