Question 4.42: Check the Clausius-Mossotti relation (Eq. 4.72) for the gase...

Check the Clausius-Mossotti relation (Eq. 4.72) for the gases listed in Table 4.1. (Dielectric constants are given in Table 4.2.) (The densities here are so small that Eqs. 4.70 and 4.72 are indistinguishable. For experimental data that confirm the Clausius-Mossotti correction term see, for instance, the first edition of Purcell’s Electricity and Magnetism, Problem 9.28.)^{23}

 

\alpha=\frac{3 \epsilon_{0}}{N}\left(\frac{\epsilon_{r}-1}{\epsilon_{r}+2}\right)                      (4.72)

\chi_{e}=\frac{N \alpha}{\epsilon_{0}}                           (4.70)

H He Li Be C Ne Na Ar K Cs
0.667 0.205 24.3 5.60 1.67 0.396 24.1 1.64 43.4 59.4

TABLE 4.1

Material Dielectric Constant Material  Dielectric Constant
Vacuum 1 Benzene 2.28
Helium 1.000065 Diamond 5.7-5.9
Neon 1.00013 Salt 5.9
Hydrogen \left( H _{2}\right) 1.000254 Silicon 11.7
Argon 1.000517 Methanol 33.0
Air (dry) 1.000536 Water 80.1
Nitrogen \left( N _{2}\right) 1.000548 Ice (-30° C) 104
Water vapor (100° C) 1.00589 KTaNbO _{3}\left(0^{\circ} C \right) 34,000

TABLE 4.2

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For an ideal gas, N = Avagadro’s number/22.4 liters = \left(6.02 \times 10^{23}\right) /\left(22.4 \times 10^{-3}\right)=2.7 \times 10^{25} . N \alpha / \epsilon_{0}=\left(2.7 \times 10^{25}\right)\left(4 \pi \epsilon_{0} \times 10^{-30}\right) \beta / \epsilon_{0}=3.4 \times 10^{-4} \beta , where \beta is the number listed in Table 4.1.

\left.\begin{array}{l} H : \beta=0.667, N \alpha / \epsilon_{0}=\left(3.4 \times 10^{-4}\right)(0.67)=2.3 \times 10^{-4}, \chi_{e}=2.5 \times 10^{-4} \\He : \beta=0.205, N \alpha / \epsilon_{0}=\left(3.4 \times 10^{-4}\right)(0.21)=7.1 \times 10^{-5}, \chi_{e}=6.5 \times 10^{-5} \\Ne : \beta=0.396, N \alpha / \epsilon_{0}=\left(3.4 \times 10^{-4}\right)(0.40)=1.4 \times 10^{-4}, \chi_{e}=1.3 \times 10^{-4} \\Ar : \beta=1.64, \quad N \alpha / \epsilon_{0}=\left(3.4 \times 10^{-4}\right)(1.64)=5.6 \times 10^{-4}, \chi_{e}=5.2 \times 10^{-4} \end{array}\right\} \text { agreement is quite good. }

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