phys 369 09 hw 5 solution

phys 369 09 hw 5 solution - Phys. 369, Fall 2009 Homework...

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1 Phys. 369, Fall 2009 Homework #5 Solutions 1. (12 points) The total magnetic moment (Schroeder’s “magnetization”) of the gas is given by Schroeder, eq. 3.22 tanh B MN kT μ ⎛⎞ = ⎜⎟ ⎝⎠ ( 1 ) Let’s estimate the ratio / B kT for a typical laboratory field of 1 Telsa: 24 23 J 2 9.27 10 1 T 2 T 0.0045 J 1.381 10 298 K K B B B kT kT ×× × == = ( 2 ) This shows that for laboratory fields and room temperature, we can safely assume /1 . Bk T ± For small x , ( ) ( ) () 11 sinh( ) tanh( ) cosh( ) 1 1 xx xe e x x e x x +−− ==≈ = ++ + ( 3 ) Therefore tanh B B N kT kT μμ =≈ ( 4 ) The magnetization (usual definition) of the sample is then 22 0 B N N B H V V kT VkT VkT = M ( 5 ) where we have assumed that the magnetic susceptibility m χ is going to be small, from which it follows that 0 . B H The magnetic susceptibility is defined by the relation m H = ( 6 ) Comparing eqs. (5) and (6) yields 2 2 2 00 0 2 4 B B m N NN VkT VkT V kT χμ = ( 7 ) The number of oxygen molecules per unit volume can be determined from the ideal gas law:
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2 5 25 3 23 1.013 10 Pa 2.46 10 m J 1.381 10 298 K K NP Vk T × == = × ×× ( 8 ) From this we obtain () ( ) ( ) 2 24 2 72 5 6 0 23 9.27 10 4
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phys 369 09 hw 5 solution - Phys. 369, Fall 2009 Homework...

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