Combining this result with Eq. 26-7
in magnitude form, yields
(26-20)
which we can write as
(26-21)
Comparing this with Eq. 26-11
in magnitude form, leads to
(26-22)
Equation 26-22 may be taken as a statement that metals obey Ohm’s law if we
can show that, for metals, their resistivity r is a constant, independent of the
strength of the applied electric field Let’s consider the quantities in Eq. 26-22.
We can reasonably assume that n, the number of conduction electrons per volume, is independent of the field, and m and e are constants. Thus, we only need to
convince ourselves that t, the average time (or mean free time) between collisions, is a constant, independent of the strength of the applied electric field.
Indeed, t can be considered to be a constant because the drift speed v d caused by
the field is so much smaller than the effective speed v eff that the electron speed —
and thus t— is hardly affected by the field. Thus, because the right side of
Eq. 26-22 is independent of the field magnitude, metals obey Ohm’s law.
E
:
.
r ϭ
m
e
2
nt
.
(E
: ϭ r J
: ),
E ϭ
m
e
2
nt J.
v d ϭ
J
ne
ϭ
eEt
m
,
( J
: ϭ nev
:
d ),
759
26-4 OH M’S L AW
Using these results and substituting for the electron mass m,
we then have
(Answer)
(b) The mean free path l of the conduction electrons in a
conductor is the average distance traveled by an electron
between collisions. (This definition parallels that in
Module 19-5 for the mean free path of molecules in a gas.)
What is l for the conduction electrons in copper, assuming
that their effective speed v eff is 1.6 ϫ 10
6
m/s?
KEY IDEA
The distance d any particle travels in a certain time t at a
constant speed v is d ϭ vt.
Calculation: For the electrons in copper, this gives us
(26-24)
(Answer)
This is about 150 times the distance between nearestneighbor atoms in a copper lattice. Thus, on the average,
each conduction electron passes many copper atoms before
finally hitting one.
ϭ 4.0 ϫ 10
Ϫ8
m ϭ 40 nm.
ϭ (1.6 ϫ 10
6
m/s)(2.5 ϫ 10
Ϫ14
s)
l ϭ v eff t
t ϭ
9.1 ϫ 10
Ϫ31
kg
3.67 ϫ 10
Ϫ17
kg/s
ϭ 2.5 ϫ 10
Ϫ14
s.
Sample Problem 26.05 Mean free time and mean free distance
(a) What is the mean free time t between collisions for the
conduction electrons in copper?
KEY IDEAS
The mean free time t of copper is approximately constant,
and in particular does not depend on any electric field that
might be applied to a sample of the copper. Thus, we need
not consider any particular value of applied electric field.
However, because the resistivity r displayed by copper
under an electric field depends on t, we can find the mean
free time t from Eq. 26-22 (r ϭ m/e
2
nt).
Calculations: That equation gives us
(26-23)
The number of conduction electrons per unit volume in copper is 8.49 ϫ 10
28
m
Ϫ3
.We take the value of r from Table 26-1.
The denominator then becomes
where we converted units as
C
2 и⍀
m
2
ϭ
C
2 иV
m
2 иA
ϭ
C
2 иJ/C
m
2 иC/s
ϭ
kg иm
2
/s
2
m
2
/s
ϭ
kg
s
.
ϭ 3.67 ϫ 10
Ϫ17
C
2 и⍀/m
2 ϭ 3.67 ϫ 10
Ϫ17
kg/s,
(8.49 ϫ 10
28
m
Ϫ3
)(1.6 ϫ 10
Ϫ19
C)
2
(1.69 ϫ 10
Ϫ8 ⍀иm)
t ϭ
m
ne
2 r
.
Additional examples, video, and practice available at WileyPLUS
in magnitude form, yields
(26-20)
which we can write as
(26-21)
Comparing this with Eq. 26-11
in magnitude form, leads to
(26-22)
Equation 26-22 may be taken as a statement that metals obey Ohm’s law if we
can show that, for metals, their resistivity r is a constant, independent of the
strength of the applied electric field Let’s consider the quantities in Eq. 26-22.
We can reasonably assume that n, the number of conduction electrons per volume, is independent of the field, and m and e are constants. Thus, we only need to
convince ourselves that t, the average time (or mean free time) between collisions, is a constant, independent of the strength of the applied electric field.
Indeed, t can be considered to be a constant because the drift speed v d caused by
the field is so much smaller than the effective speed v eff that the electron speed —
and thus t— is hardly affected by the field. Thus, because the right side of
Eq. 26-22 is independent of the field magnitude, metals obey Ohm’s law.
E
:
.
r ϭ
m
e
2
nt
.
(E
: ϭ r J
: ),
E ϭ
m
e
2
nt J.
v d ϭ
J
ne
ϭ
eEt
m
,
( J
: ϭ nev
:
d ),
759
26-4 OH M’S L AW
Using these results and substituting for the electron mass m,
we then have
(Answer)
(b) The mean free path l of the conduction electrons in a
conductor is the average distance traveled by an electron
between collisions. (This definition parallels that in
Module 19-5 for the mean free path of molecules in a gas.)
What is l for the conduction electrons in copper, assuming
that their effective speed v eff is 1.6 ϫ 10
6
m/s?
KEY IDEA
The distance d any particle travels in a certain time t at a
constant speed v is d ϭ vt.
Calculation: For the electrons in copper, this gives us
(26-24)
(Answer)
This is about 150 times the distance between nearestneighbor atoms in a copper lattice. Thus, on the average,
each conduction electron passes many copper atoms before
finally hitting one.
ϭ 4.0 ϫ 10
Ϫ8
m ϭ 40 nm.
ϭ (1.6 ϫ 10
6
m/s)(2.5 ϫ 10
Ϫ14
s)
l ϭ v eff t
t ϭ
9.1 ϫ 10
Ϫ31
kg
3.67 ϫ 10
Ϫ17
kg/s
ϭ 2.5 ϫ 10
Ϫ14
s.
Sample Problem 26.05 Mean free time and mean free distance
(a) What is the mean free time t between collisions for the
conduction electrons in copper?
KEY IDEAS
The mean free time t of copper is approximately constant,
and in particular does not depend on any electric field that
might be applied to a sample of the copper. Thus, we need
not consider any particular value of applied electric field.
However, because the resistivity r displayed by copper
under an electric field depends on t, we can find the mean
free time t from Eq. 26-22 (r ϭ m/e
2
nt).
Calculations: That equation gives us
(26-23)
The number of conduction electrons per unit volume in copper is 8.49 ϫ 10
28
m
Ϫ3
.We take the value of r from Table 26-1.
The denominator then becomes
where we converted units as
C
2 и⍀
m
2
ϭ
C
2 иV
m
2 иA
ϭ
C
2 иJ/C
m
2 иC/s
ϭ
kg иm
2
/s
2
m
2
/s
ϭ
kg
s
.
ϭ 3.67 ϫ 10
Ϫ17
C
2 и⍀/m
2 ϭ 3.67 ϫ 10
Ϫ17
kg/s,
(8.49 ϫ 10
28
m
Ϫ3
)(1.6 ϫ 10
Ϫ19
C)
2
(1.69 ϫ 10
Ϫ8 ⍀иm)
t ϭ
m
ne
2 r
.
Additional examples, video, and practice available at WileyPLUS