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Unformatted text preview: 45. (a) We use the notation P(µ) for the probability of a dipole being parallel to B , and
P(–µ) for the probability of a dipole being antiparallel to the field. The magnetization
may be thought of as a “weighted average” in terms of these probabilities:
M= N µ P ( µ ) − N µ P ( −µ )
P ( µ ) + P ( −µ ) = ( N µ e µ B KT − e − µ B KT
e µ B KT +e − µ B KT ) = N µ tanh ⎛ µ B ⎞ .
⎜
⎟
⎝ kT ⎠ <
<
(b) For µB < kT (that is, µB / kT < 1) we have e±µB/kT ≈ 1 ± µB/kT, so
M = Nµ tanh FG µB IJ ≈ Nµ b1 + µB kT g − b1 − µB kT g = Nµ B .
H kT K b1 + µB kT g + b1 − µB kT g kT (c) For µB > kT we have tanh (µB/kT) ≈ 1, so M = Nµ tanh
> 2 FG µB IJ ≈ Nµ .
H kT K (d) One can easily plot the tanh function using, for instance, a graphical calculator. One
can then note the resemblance between such a plot and Fig. 3214. By adjusting the
parameters used in one’s plot, the curve in Fig. 3214 can reliably be fit with a tanh
function. ...
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 Spring '08
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 Physics

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