Electricity and Magnetism
Current and Resistance
Resistance and Resistivity
Lana Sheridan
De Anza College
Oct 16, 2015
Last time
• current
• current density
• drift velocity
:
:
J " (ne)v
d.
Warm Up Question
(26-7)
3
I unit The
is the
coulomb
cubic meter
(C/m
), is the
figure
shows per
conduction
electrons
moving
leftward in a wire.
sitive Are
carriers,
ne
is
positive
and
Eq.
26-7
predicts
the following leftward or rightward:
:
direction.
negative carriers,
ne I,
is negative and J
(a) For
the (conventional)
current
ns.
(b) the current density J,
(c) the electric field E in the wire?
lectrons moving lefting leftward or right:
current density J , (c)
(A) all leftwards
(B) all rightwards
(C) leftward, leftward, rightward
(D) rightward, rightward, leftward
:
:
J " (ne)v
d.
Warm Up Question
(26-7)
3
I unit The
is the
coulomb
cubic meter
(C/m
), is the
figure
shows per
conduction
electrons
moving
leftward in a wire.
sitive Are
carriers,
ne
is
positive
and
Eq.
26-7
predicts
the following leftward or rightward:
:
direction.
negative carriers,
ne I,
is negative and J
(a) For
the (conventional)
current
ns.
(b) the current density J,
(c) the electric field E in the wire?
lectrons moving lefting leftward or right:
current density J , (c)
(A) all leftwards
(B) all rightwards
←
(C) leftward, leftward, rightward
(D) rightward, rightward, leftward
Overview
• resistance
• resistivity
• conductivity
• Ohm’s Law
electric field can
ns Resistance
are available,
cting loop is no
When aup
potential difference is applied across a conductor, current
rial making
(a )
begins to flow.
them to move
e electron flow
i
t does not vary
ducting loop in
h a hypothetical
e is defined as
i
i
Battery
i +
–
i
(b )
However, different amounts of current will flow in different
Fig.
26-1
(a) A loop
of copper
in
(26-1) even
conductors,
when
the applied
potential
difference
is the same.
electrostatic equilibrium. The entire
loop is at a single potential, and the
a timeWhat
interval
is the characteristic of the conductor which determines the
electric field is zero at all points inamount of current
thatcopper.
will flow?
side the
(b) Adding a battery
imposes an electric potential difference between
ends9th
of the
Figure from Halliday,
Resnick,the
Walker,
ed. loop
(26-2)
1
Resistance
Resistance
The resistance of a conductor is given by the ratio of the applied
potential to the current that flows through the conductor at that
potential:
∆V
R=
I
The units of resistance are Ohms, Ω, symbol is the capital Greek
letter “Omega”. 1 Ω = 1 V/A
We can think of a high resistance as resisting, or impeding, the
flow of current.
Resistivity
An individual conductor or circuit component has a resistance.
The resistance is based on
• the material it is made of,
• its geometry, and
• the temperature
The material that a component is made from affects the
resistance, because different materials have different resistivities.
Resistor
A resistor is a component that can be incorporated into a circuit.
It has a particular resistance at a given voltage.
1
Image from thomasnet.com
Resistor in a Circuit diagram
Resistivity
resistivity, ρ
the ratio of the electric field strength in a material to the current
density this field causes in the material:
ρ=
E
J
Resistivity is a property of a material. Its symbol is the Greek letter
ρ, pronounced “rho”.
The units of resistivity are Ω m.
1Ωm=1
V
A
m=1
V/m
A/m2
which agrees with the definition of ρ = E /J.
it diagram, we represent a resistor and
rite Eq. 26-8 as
V
,
R
Resistivity
Table 26-1
esistance, the smaller the current.
s on the manner in which the potential
r example, shows a given potential difo the same conductor. As the current
n the two cases — hence the measured
erwise stated, we shall assume that any
Fig. 26-8b.
difference to a conducting rod. The gray
stance. When they are arranged as in
red resistance is larger than when they
nd.
er connections, we often wish to take a
bjects but with materials. Here we do so
V across a particular resistor but on the
rial. Instead of dealing with the current i
:
rent density J at the point in question.
deal with the resistivity r of the material:
efinition of r).
(26-10)
according to Eq. 26-10, we get, for the
V
Resistivities of Some Materials at Room
Temperature (20°C)
Material
Resistivity, r
(! # m)
Silver
Copper
Gold
Aluminum
Manganina
Tungsten
Iron
Platinum
Typical Metals
1.62 % 10 $8
1.69 % 10 $8
2.35 % 10 $8
2.75 % 10 $8
4.82 % 10 $8
5.25 % 10 $8
9.68 % 10 $8
10.6 % 10 $8
Temperature
Coefficient
of Resistivity,
a (K$1)
4.1 % 10 $3
4.3 % 10 $3
4.0 % 10 $3
4.4 % 10 $3
0.002 % 10 $3
4.5 % 10 $3
6.5 % 10 $3
3.9 % 10 $3
Typical
Semiconductors
Silicon,
pure
Silicon,
n-typeb
Silicon,
p-typec
Glass
Fused
quartz
2.5 % 10 3
8.7 % 10 $4
2.8 % 10 $3
Typical
Insulators
10 10 $ 10 14
!10 16
$70 % 10 $3
Resistivity
Together with the geometry of the component made of that
material, we can predict the resistance of the component.
Resistivity
Together with the geometry of the component made of that
material,
we can predict the resistance
of the
component.
halliday_c26_682-704hr.qxd
7-12-2009
14:30
Page 690
For a wire, cylinder, or anything with uniform cross-section A,
made of material with resistivity ρ:
R=
ρL
A
CHAPTER 26 CURRENT AND RESISTANCE
690
where A is the cross-sectional area of the wire, and L is the length
of the wire.
We can write E
Current is driven by
a potential difference.
L
i
i
A
V
A potential difference
V
is
applied
between
the ends of a
from the definition of ρ.)
Fig. 26-9
(This follows
Equations 26-10 a
electrical propertie
We often speak
cal of its resistivity,
can be applied only to a homogeneous isotropic conductor of
tion,
with the potential difference applied as in Fig. 26-8b.
Question
opic quantities V, i, and R are of greatest interest when we are
measurements on specific conductors. They are the quantities
Rank the three cylindrical copper conductors according to the
ctly on meters. We turn to the microscopic quantities E, J, and r
current through them, greatest first, when the same potential
rested in the fundamental electrical properties of materials.
difference V is placed across their lengths.
NT 3
L
e shows three
1.5L
A
er conductors
A
A
_
_
2
2
face areas and
em according to
(a)
(b)
gh them, greatsame potential difference V is placed across their lengths.
(A) a, b, c
(B) c, b, a
(C) b, (a and c)
(D) (a and c), b
L/2
(c)
can be applied only to a homogeneous isotropic conductor of
tion,
with the potential difference applied as in Fig. 26-8b.
Question
opic quantities V, i, and R are of greatest interest when we are
measurements on specific conductors. They are the quantities
Rank the three cylindrical copper conductors according to the
ctly on meters. We turn to the microscopic quantities E, J, and r
current through them, greatest first, when the same potential
rested in the fundamental electrical properties of materials.
difference V is placed across their lengths.
NT 3
L
e shows three
1.5L
A
er conductors
A
A
_
_
2
2
face areas and
em according to
(a)
(b)
gh them, greatsame potential difference V is placed across their lengths.
(A) a, b, c
(B) c, b, a
(C) b, (a and c)
(D) (a and c), b
←
L/2
(c)
Resistivity can depend on Temperature
Room temperature
Resistivity (10–8 Ω .m)
The recopper as a
temperaot on the
s a conveence point at
e T0 " 293
tivity r0 "
8
$ % m.
0
10
8
6
4
2
0
0
Resistivity c
on tempera
(T0, ρ0)
200 400 600 800 1000 1200
Temperature (K)
n with Temperature
Resistivity can depend on Temperature
The relationship between resistivity and temperature is close to
linear.
For most engineering purposes, a linear model is good enough.
The model:
ρ − ρ0 = ρ0 α(T − T0 )
The resistivity varies linearly with the difference in temperature
from some reference value T0 .
ρ0 is the resistivity at T0 .
Resistivity can depend on Temperature
ρ − ρ0 = ρ0 α(T − T0 )
α is just a constant, however it takes different values for different
materials.
α is called the temperature coefficient of resistivity. It has
units K−1 .
For example for copper:
ρ0 = 1.62 × 10−8 Ω m
α = 4.3 × 10−3 K−1
Conductivity
Sometimes it is useful to represent how conductive a material is:
how readily it permits current to flow, as opposed to how much it
resists the flow of current.
conductivity, σ
a measure of what the current density is in a material for a
particular electric field; the inverse of resistivity:
σ=
1
J
=
ρ
E
Conductivity
Sometimes it is useful to represent how conductive a material is:
how readily it permits current to flow, as opposed to how much it
resists the flow of current.
conductivity, σ
a measure of what the current density is in a material for a
particular electric field; the inverse of resistivity:
σ=
1
J
=
ρ
E
This is different than surface charge density (also written σ). This
is just an unfortunate coincidence of notation.
Conductivity
conductivity, σ
a measure of what the current density is in a material for a
particular electric field; the inverse of resistivity:
σ=
J
1
=
ρ
E
The units of conductivity are (Ω m)−1 .
We can use conductivity to relate the current density to the
electric field in a material:
J = σE
Resistance of Resistors with Non-Uniform Area
For a resistor with uniform cross-section A, made of material with
resistivity ρ:
ρL
R=
A
What if the cross section isn’t uniform?
Resistance of Resistors with Non-Uniform Area
For a resistor with uniform cross-section A, made of material with
resistivity ρ:
ρL
R=
A
What if the cross section isn’t uniform?
Integrate.
Use:
dR =
ρ
d`
A(`)
tential
difference
10 V.
Calculate
theofresis-
b
Example: Coaxial Cable
Inner
Outer
Find the resistance between
the two conducting
layers.
conductor
conductor
ofpli-the problem. The
The
ors. The undesired
c as
rection
is radial.
recPolyethylene
a
L
dr
but
astic are known, we
ctor
a
gth
of the plastic from
sisb
r
esistance of a block
uation.Inner
BecauseOuter
the
osition,conductor
we mustconductor
use
The
red
a
Current
direction
End view
b
dr
Current
direction
Figure 27.8
(Example 27.3) A
Example: Coaxial Cable
Find the resistance between the two conducting layers.
At radius r the area a current can pass through is A(r ) = 2πrL
Zb
R =
a
ρ
dr
2πrL
Example: Coaxial Cable
Find the resistance between the two conducting layers.
At radius r the area a current can pass through is A(r ) = 2πrL
Zb
R =
a
=
=
ρ
dr
2πrL
ρ
[ln b − ln a]
2πL
ρ
b
ln
2πL
a
Ohm’s Law
Ohm’s Law
The current through a device is directly proportional to the
potential difference applied across the device.
∆V ∝ I
Not all devices obey Ohm’s Law!
In fact, for all materials, if ∆V is large enough, Ohm’s law fails.
They only obey Ohm’s law when the resistance of the device is
independent of the applied potential difference and its polarity
(that is, which side is the higher potential).
26-5 Ohm’s Law
Current (mA)
Current (mA)
Current (mA)
Ohm’s Law
Curr
V is
high
V
–2
(fro
+
–
(wit
As we just
26-4, a resisto
?
–4 discussed
–2
0in Section
+2
+4
cau
i
i
resistance.
It
has
that
same
resistance
no
matter
Potential
difference
(V)
Obeys Ohm’s
law:
Does not obey Ohm’s law:
(a )
(b ) potential difference ar
(polarity) of the applied
ever, might have resistances that change withpas
th
Figure 26-11a shows how to distinguishissut
+2
dev
V is applied
across the device being tested, and
+4
diff
device is measured as V is varied in both magn
0
+2
V is arbitrarily
taken to be positive when the
dev
higher potential than the right terminal. The
0
is m
–2
(from left to right) is arbitrarily assigned a pl
not
(with the
–2 right terminal at a higher potential
–4
–2
0
+2
+4
–4
–2
0
+2
+4
causes is assigned
a minus
sign.(V)
Potential
difference
Potential difference (V)
Ohm
Figure 26-11b is(ca) plot of i versus V for one
(b )
passing through the origin, so the ratio i/V (whi
(a)
A
difference
We can write this linear relationship
as26-11
∆V
IR
ifvalues
and only
RV means tO
isFig.
the
same=
for
allpotential
of V. ifThis
pr
is applied to the terminals of a device,
is independent of the magnitude and
is constant
and independent of ∆Vdevice
.
+4
establishing
a current i. (b) A plot of curdifference
(Th
rent i versusV.applied potential difference V
Figure
26-11c
a plot
for another
+2
term
when
the device
is ais1000
!
resistor.
(c) A conducti
∆V
However, notice that we can always
define
R(∆V
)is=apolarity
device
only
the
of V is positive
an
I even
plot when
thewhen
device
semiconducting
resi
0 resistance does depend on ∆V
when
.junction
ispnmore
thandiode.
about 1.5 V. When current doestion
ex
not linear; it depends on the value of the applied
–2
We distinguish between the two types ofA
–4
–2
0
+2
+4
Potential difference (V)
potential differen
junction diode, we
Ohm’s Law Question
The following table gives the current i (in
of Ohm’s law, ho
amperes) through two devices for sevThe following table
gives
the
current
i
(in
amperes)
through
two of V.
pendent
eral values of potential difference V (in
We
can expre
devices for several
values
of
potential
difference
V
(in
volts).
volts). From these data, determine which
device
does
not
obey
Ohm’s
law.
materials
rather
Which of the devices obeys Ohm’s law?
:
Eq. 26-11 (E " #
CHECKPOINT 4
Device 1
Device 2
V
i
V
i
2.00
3.00
4.00
4.50
6.75
9.00
2.00
3.00
4.00
1.50
2.20
2.80
(A) 1 only
(B) 2 only
(C) both
(D) neither
1
Halliday, Resnick, Walker, page 692.
A conducting m
independent of th
All homogeneous
ductors like pure
within some rang
there are departu
potential differen
junction diode, we
Ohm’s Law Question
The following table gives the current i (in
of Ohm’s law, ho
amperes) through two devices for sevThe following table
gives
the
current
i
(in
amperes)
through
two of V.
pendent
eral values of potential difference V (in
We
can expre
devices for several
values
of
potential
difference
V
(in
volts).
volts). From these data, determine which
device
does
not
obey
Ohm’s
law.
materials
rather
Which of the devices obeys Ohm’s law?
:
Eq. 26-11 (E " #
CHECKPOINT 4
Device 1
(A) 1 only
Device 2
V
i
V
i
2.00
3.00
4.00
4.50
6.75
9.00
2.00
3.00
4.00
1.50
2.20
2.80
←
(B) 2 only
(C) both
(D) neither
1
Halliday, Resnick, Walker, page 692.
A conducting m
independent of th
All homogeneous
ductors like pure
within some rang
there are departu
Summary
• Resistance
• resistivity
• conductance
• Ohm’s Law
Homework
• Collected homework 2, posted online, due on Monday, Oct 26.
Serway & Jewett:
• PREVIOUS: Ch 27, onward from page 824. Problems: 1, 5, 7
• NEW: Ch 27, Problems: 15, 23, 25, 29, 33, 71