Dielectric, Magnetic Properties & Superconductivity
Subtopic: Meissner Effect, Type I and Type II Superconductors
Question Statement:
Which of the following option(s) is/are correct for a Type I superconductor?
Options:
(A) The phase transition to the normal state in the absence of a magnetic field is o f
second order
(B) With increase in temperature, the critical magnetic field decreases linearly to zero
(C) Below the critical temperature, the entropy in the superconducting state is less than
that in the normal state
(D) The phase transition to the normal state in the presence of a magnetic field is of first
order
Official Answer Key:
A;C;D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.47)**
**1. Thermodynamics of Type I Superconductors**:
- Critical magnetic field relation: $B_c(T) = B_c(0)[1 - (T/T_c)^2]$.
**2. Detailed Analysis of Statements (MSQ)**:
- **(A) Second-order phase transition at $B = 0$ [TRUE]**:
- In zero external magnetic field ($B = 0$), the latent heat $L = -T \frac{B_c}{\mu_0}\frac{dB_c}{dT} = 0$.
- However, the specific heat exhibits a finite jump discontinuity at $T_c$ (Rutgers formula $\Delta C = \frac{T_c}{\mu_0}\left(\frac{dB_c}{dT}\right)^2 > 0$). Hence, it is a second-order phase transition.
- **(B) False**: $B_c(T)$ decreases parabolically, not linearly.
- **(C) Entropy of superconducting state is less than normal state [TRUE]**:
- Below $T_c$, electrons condense into macroscopic coherent Cooper pairs, reducing disorder ($S_{\text{super}} < S_{\text{normal}}$).
- **(D) First-order phase transition in the presence of magnetic field [TRUE]**:
- When $B \ne 0$, the transition occurs at a reduced temperature with non-zero critical field $B_c(T) > 0$.
- The entropy discontinuity $\Delta S = S_n - S_s = -\frac{B_c}{\mu_0}\frac{dB_c}{dT} > 0$ produces a non-zero latent heat $L = T\Delta S > 0$, making it a first-order phase transition.
**3. Official Verification**:
The verified official IIT answer is **Options (A), (C), and (D)** (`['A', 'C', 'D']`).
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