Question

The free energy of a ferromagnet is given by \[ F = F_0 + a_0 (T - T_C) M^2 + b M^4, \] where \(F_0\), \(a_0\), and \(b\) are positive constants, \(M\) is the magnetization, \(T\) is the temperature, and \(T_C\) is the Curie temperature. The relation between \(M^2\) and \(T\) is best depicted by 

• A. A
• B. B
• C. C
• D. D

Hint:

Near the Curie temperature \(T_C\), the magnetization of a ferromagnet decreases sharply as the system transitions from a ferromagnetic to a paramagnetic state.

Answer 1

The Correct Option is B

The given free energy expression for a ferromagnet involves terms that are dependent on the magnetization \(M\) and temperature \(T\): \[ F = F_0 + a_0 (T - T_C) M^2 + b M^4. \] Here:
- The first term \(F_0\) is a constant,
- The second term is linear in \(M^2\) and depends on the difference between the temperature \(T\) and the Curie temperature \(T_C\),
- The third term represents a quartic dependence on \(M\), which helps to stabilize the system at high temperatures.
At temperatures \(T > T_C\), the factor \( (T - T_C) \) is positive, and the system tends to have small magnetization. The second term in the free energy equation becomes negative as the temperature increases above \(T_C\), leading to a decrease in the magnetization squared (\(M^2\)). At temperatures \(T < T_C\), the magnetization \(M\) increases, leading to a positive \(M^2\) because the system tends to become more ordered and magnetized below the Curie temperature. The curve \(M^2\) vs \(T\) typically follows the behavior of the second term of the free energy. The graph is expected to show a decreasing curve as temperature approaches \(T_C\) from above, reflecting the fact that the system's magnetization decreases as temperature increases, particularly near the critical point \(T_C\). Thus, the correct answer is (B), which shows a linear decrease in \(M^2\) as \(T\) approaches \(T_C\) from above.