Use the Bohr magneton μB = eh/2m = 9.27 × 10^-24 J/T as a typical magnetic moment and B = 0.50 T to do the following: (a) Find the temperature T at which μB = 0.1kT. (b) For μB = 0.1kT compare tanh (βμB) with βμB and thereby check the suitability of this value of T as a “classical” temperature. (c)

Question

Use the Bohr magneton [latex]\mu_{ B }=e \hbar / 2 m=9.27 imes 10^{-24}[/latex] J/T as a typical magnetic moment and B = 0.50 T to do the following:

(a) Find the temperature T at which [latex]\mu B=0.1 k T[/latex].

(b) For [latex]\mu B=0.1 k T[/latex] compare [latex] anh (\beta \mu B) ext { with } \beta \mu B[/latex] and thereby check the suitability of this value of T as a “classical” temperature.

(c) Make the same comparison in the expression [latex] anh (\beta \mu B) \approx \beta \mu B ext { at } T=100 K[/latex].

Strategy With the numerical values of the magnetic moment and magnetic field given, the computations are all straightforward. For the comparisons called for in parts (b) and (c), the given values suggest using two significant figures.

Accepted Answer

(a) We are given that [latex]\mu B=0.1 k T[/latex]. We let [latex]\mu=\mu_{ B }[/latex] and solve for T:

[latex]T=\frac{10 \mu B}{k}=\frac{10\left(9.27 imes 10^{-24} J / T ight)(0.5 T )}{1.38 imes 10^{-23} J / K }=3.36 K[/latex]

(b) Now

[latex]\begin{gathered}\beta \mu B=\frac{\mu B}{k T}=0.10 \ anh (\beta \mu B)= anh (0.10)=0.10\end{gathered}[/latex]

to two significant digits. Therefore we conclude that the approximation is a good one for most purposes, even at this low temperature.

(c) Now

[latex]\begin{aligned}\beta \mu B=\frac{\mu B}{k T} &=\frac{\left(9.27 imes 10^{-24} J / T ight)(0.5 T )}{\left(1.38 imes 10^{-23} J / K ight)(100 K )} \&=3.36 imes 10^{-3} \ anh (\beta \mu B) &= anh \left(3.36 imes 10^{-3} ight)=3.36 imes 10^{-3}\end{aligned}[/latex]

so that [latex]\beta \mu B ext { and } anh (\beta \mu B)[/latex] are the same to three significant figures. At these relatively high temperatures the approximation is an excellent one.

Question 10.6: Use the Bohr magneton μB = eh/2m = 9.27 × 10^-24 J/T as a ty...

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