Chapter 11
Resistivity, Power
84
11
Resistivity, Power
In chapter 9 we discussed resistors that conform to Ohm’s Law.
From the discussion, one could
deduce that the resistance of such a resistor depends on the nature of the material of which the
resistor is made and on the size and shape of the resistor.
In fact, for resistors made out of a
single kind of material, in the shape of a wire
1
with a terminal at each end,
the resistance is given by:
A
L
R
r
=
(111)
where:
R
is the resistance of the resistor as measured between the ends,
r
is the resistivity of the substance of which the resistor is made,
A
is the crosssectional area of the wireshaped resistor, and
L
is the length of the resistor.
The values of resistivity for several common materials are provided in the following table:
Material
Resistivity
r
Silver
1
.
6
×
10
−
8
Ω⋅
m
Copper
1
.
7
×
10
−
8
Ω⋅
m
Gold
2
.
4
×
10
−
8
Ω⋅
m
Aluminum
3
×
10
−
8
Ω⋅
m
Tungsten
5
.
6
×
10
−
8
Ω⋅
m
Nichrome
1
.
0
×
10
−
6
Ω⋅
m
Seawater
0
.
25
Ω⋅
m
Rubber
1
×
10
13
Ω⋅
m
Glass
1
×
10
10
to
1
×
10
14
Ω⋅
m
Quartz
5
×
10
15
to 7
.
5
×
10
17
Ω⋅
m
1
The resistor can have any shape such that one linear dimension can be identified as the length of the resistor, and,
such that the intersection of a plane perpendicular to the length of the resistor, at any position along the length of the
resistor, has one and the same area (the crosssectional area of the resistor).
I am calling the shape “the shape of a
wire” for ease in identification of what we mean by the “alongthelength” dimension.
Crosssectional Area
A
Length
L
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Chapter 11
Resistivity, Power
85
In the expression
A
L
R
r
=
, the resistivity
r
depends on the charge carrier
2
density, that is, the
numberofchargecarrierspervolume.
The more charge carriers per volume, the smaller the
resistance since, for a given velocity of the charge carriers, more of them will be passing any
point along the length of the resistor every second for a given voltage across the resistor.
The
resistivity also depends on the retarding force factor. We said that the retarding force on each
charge carrier is proportional to the velocity of that charge carrier.
Retarding Force
=
−
(factor) times (charge carrier velocity)
(The minus sign is there because the retarding force is in the direction opposite that of the
chargecarrier velocity.)
The bigger the retarding force factor, the greater the resistivity of the
material for which the factor applies.
The charge carrier density and the retarding force factor determine the value of
r
.
The effect of
r
on the resistance is evident in the expression
A
L
R
r
=
.
The bigger
r
is, the greater the
resistance is.
Why the factor of
L
in
A
L
R
r
=
?
It’s saying that the greater the length of the singlesubstance
resistor in the shape of a wire, the greater the resistance of the resistor, all other things being
equal (same substance, same crosssectional area).
It means, for instance, that if you have two
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 Fall '08
 RABE
 Physics, Electric Potential, Resistance, Energy, Potential Energy, Power, Resistor, Electric charge, power supply

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