Editing Curie's law

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This is magnetization of one paramagneton, total magnetization of the solid is given by
 
This is magnetization of one paramagneton, total magnetization of the solid is given by
:{{resultbox|<math>M = N\left\langle\mu\right\rangle = N \mu \tanh\left({\mu B\over k_B T}\right)</math>}}
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<blockquote style="border: 1px solid black; padding:10px;">
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<math>M = N\left\langle\mu\right\rangle = N \mu \tanh\left({\mu B\over k T}\right)</math></blockquote>
 
The formula above is known as the [[Langevin]] [[Paramagnetic]] equation.
 
The formula above is known as the [[Langevin]] [[Paramagnetic]] equation.
 
Pierre Curie found an approximation to this law that applies to the reasonably high temperatures and low magnetic fields used in his experiments.  Let's see what happens to the magnetization as we specialize it to large <math>T</math> and small <math>B</math>.  As temperature increases and magnetic field decreases, the argument of hyperbolic tangent decreases.  Another way to say this is
 
Pierre Curie found an approximation to this law that applies to the reasonably high temperatures and low magnetic fields used in his experiments.  Let's see what happens to the magnetization as we specialize it to large <math>T</math> and small <math>B</math>.  As temperature increases and magnetic field decreases, the argument of hyperbolic tangent decreases.  Another way to say this is
  
:<math>\left({\mu B\over k_B T}\right) << 1</math>
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:<math>\left({\mu B\over k T}\right) << 1</math>
  
 
this is sometimes called the '''Curie regime'''.  We also know that if <math>|x|<<1</math>, then
 
this is sometimes called the '''Curie regime'''.  We also know that if <math>|x|<<1</math>, then
 
:<math>\tanh x \approx x</math>
 
:<math>\tanh x \approx x</math>
leading to
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so
:{{resultbox|<math>\mathbf{M}(T\rightarrow\infty)={N\mu^2\over k_B}{\mathbf{B}\over T}</math>}}
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<blockquote style="border: 1px solid black; padding:10px;">
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:<math>\mathbf{M}(T\rightarrow\infty)={N\mu^2\over k}{\mathbf{B}\over T}</math></blockquote>
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Q.E.D.
  
 
== More Involved Derivation (Statistical Mechanics) ==
 
== More Involved Derivation (Statistical Mechanics) ==

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