ch32-p045 - 45. (a) We use the notation P(µ) for the...

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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. 32-14. By adjusting the parameters used in one’s plot, the curve in Fig. 32-14 can reliably be fit with a tanh function. ...
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