Chapter 25: Current, Resistance, and
Electromotive Force
Electric Current
• “loose” definition: a flow of electric charge.
• formal definition: the rate at which charge passes a point (in a wire,
e.g.)
Two kinds:
1. Average Current, I av : average rate at which charge passing a point
(over the whole time interval Δt ):
ΔQ
I av ≡
(1)
Δt
2. Instantaneous Current, I : rate at which charge passing a point
“right now”, at some instant in time.
ΔQ dQ
I ≡ lim
=
(2)
Δt →0 Δt
dt
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Ch. 25: Current, Resistance, and Electromotive Force
Notes:
• By convention (going all the way back to Franklin) it’s customary to
speak of current as being a flow of positive charges. Thus, the
direction of conventional current is defined to be the direction in
which positive charges would flow, even if the actual charge carriers
are negative (electrons, e.g.).
C
• Unit for I , I av :
≡ Ampere, A (“amp”, for short). Named after
s
French scientist André-Marie Ampère (1775-1836).
• Whenever the current is steady (not changing), I = I av .
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Ch. 25: Current, Resistance, and Electromotive Force
Question: How do we get a current to flow through some stuff?
Answer: Put a voltage across the stuff!
For example, consider a block of carbon connected to a battery by wires
connecting each end of the block of carbon to one terminal of the battery.
Imagine a single positive charge carrier + q inside the battery. The
battery does positive work on + q , raising it from a region of low
potential (at the negative terminal of the battery) to a region of high
potential (at the positive terminal of the battery). This charge carrier
then leaves the positive terminal of the battery, flows through the top
connecting wire, through the block of carbon, through the bottom
connecting wire, and back into the negative terminal of the battery.
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Ch. 25: Current, Resistance, and Electromotive Force
Two Models of Current Flow
There are two models of, or ways of thinking about, a flow of electric
current: the microscopic model and the macroscopic model.
In the microscopic model, we ask how the current that flows (in a wire,
e.g.) depends on properties of the material at the level of individual
charge carriers (electrons, e.g.). The microscopic model describes how
the current depends on:
• the density of charge carriers (i.e., number of
charge carriers per unit volume)
• the charge q of each charge carrier (in case
they aren’t just electrons)
• the average speed of the charge carriers as
they move through the material
In the macrosopic model, we think only about “bulk” properties of the
material. The macroscopic model deals with the question: “Suppose I
put a known voltage V across a material. How much current is going to
flow – and don’t bother me with details about how many electrons there
are per cubic meter and all of that.”
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Ch. 25: Current, Resistance, and Electromotive Force
Microscopic Model of Electric Current
Imagine a block of some material that conducts (copper, e.g.) connected
across a battery.
A Current-Carrying Wire
Is Not In Electrostatic Equilibrium
Recall that we said (Chapter 23) that a conductor in electrostatic
equilibrium is one big “blob” of equipotential. When we connect the
battery across the block of copper, however, we establish some potential
difference between the two ends of the copper block. This means that
the copper block is, strictly speaking, not an equipotential.
As a consequence, there is some Gelectric field within the conducting
block, and it is precisely this field E that pushes charge through the
conducting block.
G
In good conductors, only a very small E is required to get a current to
flow. This is like saying that only a very small voltage ΔV has to be put
across the material to get a current to flow. Therefore, a conducting wire
in a circuit is almost an equipotential.
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Ch. 25: Current, Resistance, and Electromotive Force
G
Imagine an electric field E within the copper block described above.
Under the action of this field, charge carriers within the block “drift”
through the block. The actual path of an individual charge carrier is quite
complicated, though, because of interactions with other charge carriers
and with atoms in the material. As a consequence, the charge carriers
don’t move with a single velocity. We call the average velocity with
G
which the charge carriers move the drift velocity, vd .
Let n be the (number) density of the charge carriers (number per unit
volume). Each charge carrier has (positive) charge q . Also, let A be the
cross-sectional area of the wire.
To see how the current I depends on these factors, think about an
infinitesimal time interval dt . During this time, charge carriers move a
distance
dx = vd dt
These charge carriers move from some x to x + dx . The number of
charge carriers that do this is the number of charge carriers within a
“slice” of the wire having length dx and cross-sectional area A. This
number of charge carriers is given by:
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Ch. 25: Current, Resistance, and Electromotive Force
⎛ # charge carriers ⎞
# charge carriers = ⎜
⎟ ( volume of this slice )
⎝ unit volume ⎠
# charge carriers = n ( Adx ) = nAvd dt
The total charge dQ flowing from x to x + dx in the time dt is then:
dQ = q ( nAvd dt )
The current I flowing in the wire is therefore:
dQ
I=
dt
I = nqvd A
(3)
G
The current density, J , is defined to be the current per unit crosssectional area. In magnitude:
I
J≡
(4)
A
From (3):
J = nqvd
(5)
It’s useful to give the current
density a vector representation. To do this,
G
G
we define the vector J to have the same direction as vd :
G
G
J ≡ nqvd
(6)
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Ch. 25: Current, Resistance, and Electromotive Force
Note:
• The current I is large if:
o n is large (lots of charge carriers per cubic meter of material)
o q is large (each charge carrier carries lots of charge)
o vd is large (charge carriers are moving fast)
o A is large (big wire… like fluid flow through pipe)
• Unit for J : A m 2
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Ch. 25: Current, Resistance, and Electromotive Force
Ohm’s Law (Microscopic Form)
It’s helpful to think of a cause-and-effect relationship between current
(more precisely, current density) and the electric Gfield in the copper block
discussed above: We establish an electric field E in the copper block
(by putting a voltage
across the block), and the field causes some
G
current density J within the block.
G
G
It turns out that, for some materials,
G J is
G just proportional to E :
J =σE
(7)
This simple proportionality between the current density and the field was
discovered by the German physicist Georg Simon Ohm in 1826. It is the
microscopic form of Ohm’s law.
Not all materials obey Ohm’s law (over all ranges of temperature, e.g.).
Those that do are called ohmic materials. Common materials that obey
Ohm’s law for a fairly wide range of conditions include copper, gold,
silver, and carbon. Carbon is an especially important example because
it’s quite common to use carbon to make carbon composition resistors.
These are nearly ubiquitous in everyday electronic equipment.
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Ch. 25: Current, Resistance, and Electromotive Force
The constant of proportionality σ in (7) is called the conductivity, and it
depends on the material and on temperature.
Note:
G
G
• σ large for good conductors (lots of J for a given E )
• σ small for poor conductors (good insulators)
It’s sometimes useful to express the microscopic form of Ohm’s law in
terms of a different constant of proportionality:
G
G
(8)
E = ρJ
The constant of proportionality ρ is called the resistivity of the material.
It also depends on the temperature.
Comparing (7) and (8), we see:
ρ=
1
σ
(9)
The resistivity ρ is a measure of the degree to which the material resists
a flow of current; it is large for poor conductors and small for good
conductors.
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Ch. 25: Current, Resistance, and Electromotive Force
Temperature Dependence of Resistivity
The resistivity of any block of material depends on the material, but it
also depends on temperature. As long as the range of temperatures is
not too large, it turns out that ρ varies with T as follows:
ρ (T ) = ρ 0 ⎡⎣1 + α (T − T0 ) ⎤⎦
(10)
The temperature T0 is the reference temperature, often taken to be 0D C
or 20D C . The constant ρ0 is the resistivity at T0 . The constant α is called
the temperature coefficient of resistivity. It is a measure of how much the
resistivity increases (or decreases) for a given increase in temperature.
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Ch. 25: Current, Resistance, and Electromotive Force
Macroscopic Model of Electric Current
Rather than think about what’s happening to individual charge carriers
within a current-carrying block of material (drift velocity, etc), in the
macroscopic model of electric current, we take a more global point of
view and just think about properties of the material in bulk. That is, we
ask simply, “if I put a known voltage across a material, how much current
will flow?”
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Ch. 25: Current, Resistance, and Electromotive Force
Ohm’s Law (Macroscopic Form)
Suppose a steady current I flows in a section of wire of length L and
cross-sectional areal A . The microscopic form of Ohm’s law, in the form
of Eq. (8), says the magnitude of the electric field within the wire is:
⎛I⎞
E = ρJ = ρ ⎜ ⎟
⎝ A⎠
But we also now that E is related to the voltage V across this section of
wire by:
V
E=
L
Combining the two equations immediately above, I find:
⎛ ρL ⎞
V = I⎜
⎟
⎝ A ⎠
The quantity is parentheses is called the resistance, R , of this section of
wire:
ρL
(11)
R=
A
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Ch. 25: Current, Resistance, and Electromotive Force
So we have:
V = IR
(12)
This is the macroscopic form of Ohm’s law. It answers the question “how
much current flows if I put a known voltage across a block of material.”
Note:
• Unit for R : V A ≡ Ohm, Ω
• R is a measure of the degree to which the material resists a flow of
current (hence the name, resistance):
o If R large, not much current flows (for given V ).
o If R small, lots of current flows.
• R depends on:
o the material used
o the cross-sectional area A
o the length L
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Ch. 25: Current, Resistance, and Electromotive Force
Temperature Dependence of Resistance
The resistance R of a block of material is related to the resistivity ρ of
the material by (11):
ρL
R=
A
But ρ depends on temperature according to (10):
ρ (T ) = ρ 0 ⎡⎣1 + α (T − T0 ) ⎤⎦
Substituting this into (11), we find that R also depends on temperature:
R (T ) = R0 ⎡⎣1 + α (T − T0 ) ⎤⎦ ,
(13)
in which:
ρL
R0 = 0
(14)
A
is the resistance at the reference temperature T0 .
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Ch. 25: Current, Resistance, and Electromotive Force
----------(Insert stuff on emf, internal resistance, and power here.)-----------
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