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49.
(a) We use the notation
P
(
µ
) for the probability of a dipole being parallel to
~
B
,and
P
(
−
µ
)forthe
probability of a dipole being antiparallel to the Feld. The magnetization may be thought of as a
“weighted average” in terms of these probabilities:
M
=
NµP
(
µ
)
−
NµP
(
−
µ
)
P
(
µ
)+
P
(
−
µ
)
=
Nµ
(
e
µB/KT
−
e
−
µB/KT
)
e
µB/KT
+
e
−
µB/KT
=
Nµ
tanh
µ
µB
kT
¶
.
(b) ±or
µB
¿
kT
(that is,
µB/kT
¿
1) we have
e
±
µB/kT
≈
1
±
µB/kT
,so
M
=
Nµ
tanh
µ
µB
kT
¶
≈
Nµ
[(1 +
µB/kT
)
−
(1
−
µB/kT
)]
(1 +
µB/kT
)+(1
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 Fall '08
 SPRUNGER
 Physics

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