A simple theory of the thermodynamics of a ferromagnet uses...

A simple theory of the thermodynamics of a ferromagnet uses the free energy F written as a function of the magnetization M in the form: F = ?HM + F_0 + A(T ? T_c)M^2 + BM^4, where H is the applied magnetic field, F_0, A and B are positive constants, T is the temperature and T_c is the critical temperature. (a) What are the conditions on the free energy F that determines the equilibrium magnetization M of the system? (b) Determine the equilibrium value of M for T > T_c and sketch a graph of M versus T for small constant H. (c) Sketch F as a function of M for zero-applied field H = 0. What is the physical significance of the temperature dependence of M as T gets close to T_c at H = 0.

Transcript

00:01 Hi, in this question, we need to use a simple theory of thermodynamic ferromagnet in which the free energy f is given by minus h times m plus f naught plus a multiplied by t minus pc multiplied by m square and then plus b times m raised to the power 4.

00:31 Now, in the first part of the question, free energy f determines the equilibrium magnetization.

00:43 So for this, we will differentiate f with respect to m and we will equate this with 0.

00:53 So there if you differentiate this f with respect to m, this will be minus h plus 0 plus 2a multiplied by t minus tc times m plus 4 times bm cube equal to 0.

01:10 So from here we can write h minus 2a t minus tc times m equal to 4 bm cube.

01:26 So m cube can be written as 1 divided by 4b of h minus 2a t minus tc multiplied by m and then m will be equal to 1 divided by 4b h minus 2a times t minus tc multiplied by m whole raised to the power 1 divided by 3.

01:53 So this is the answer for the first part of the question.

01:59 Now in the second part, since p is greater than tc, then equilibrium value m is equal to 1 divided by 4b multiplied by h plus 2a times t minus tc and whole raised to the power 1 divided by 3.

02:24 So, if we plot the graph between m and t, so this axis is t, this is m.

02:37 So this temperature is let us consider tc.

02:39 So this is tc.

02:42 So m will be varying like this and this curve will be m proportional to t raised to the power 1 divided by 3 and at this point t equal to tc, this m will be equal to h divided by 4b whole raised to the power 1 divided by 3 and this will be constant...

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