P32_049 - 49. (a) We use the notation P () for the...

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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 ) = ( e µB/KT e µB/KT ) e µB/KT + e µB/KT = tanh µ µB kT . (b) ±or µB ¿ kT (that is, µB/kT ¿ 1) we have e ± µB/kT 1 ± µB/kT ,so M = tanh µ µB kT [(1 + µB/kT ) (1 µB/kT )] (1 + µB/kT )+(1
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