HW2_chapter 17:
17-8:
Q.
If
N
w
is the number of protons aligned with a magnetic field
B
z
and
N
0
is the number of
protons opposed to the field, show that
/
0
z
B
B
k T
w
N
e
N
Given that
7
1
1
26.7522 10
rad
s
for a proton, calculate
N
0
/
N
w
as a function of
temperature for a field strength of 5.0 T. At what temperature is
N
0
=
N
w
? Interpret this result
physically.
A.
The energy of the protons aligned with the magnetic field is
1
2
w
z
E
B
, and the energy of
the protons aligned against the magnetic field is
1
2
o
z
E
B
. Using Equation 17.10, we can
write
1
1
exp
exp
2
2
o
z
z
z
w
B
N
B
B
B
N
k T
For a field strength of 5.0 T,
34
7
1
1
23
1
(1.05459 10
)(26.7522 10
)(5.0 )
0.010
exp
exp
(1.38066 10
)
o
w
N
J s
rad
s
N
J
T
T
Let
N
0
=
N
w
, then
0.010
1
exp
o
w
N
N
T
,
0.010
0
T
And
T
is undefined. Therefore, we can never attain the condition
N
0
=
N
w
. ( See section 15-3 for
a similar problem, where we find that the population of the ground state is always greater than
the population of the excited state in a two-level system.)

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