BSc Physics MCQ TGT Non medical
1. Fundamental Concepts
1.1 Critical Temperature ($T_c$)
The Critical Temperature ($T_c$) is the specific temperature below which a material undergoes a phase transition from a normal conducting state (with finite resistance) to a superconducting state (with zero DC electrical resistance).
- Significance: Above $T_c$, the material behaves as a normal metal. Below $T_c$, charge carriers move without scattering, resulting in no energy loss.
- Examples: Mercury ($T_c \approx 4.2$ K), Lead ($T_c \approx 7.2$ K), Niobium ($T_c \approx 9.3$ K).
1.2 Critical Magnetic Field ($H_c$)
Superconductivity can be destroyed not just by raising the temperature, but also by applying a strong external magnetic field. The minimum magnetic field strength required to destroy the superconducting state at a given temperature is called the Critical Magnetic Field ($H_c$).
The dependence of $H_c$ on temperature $T$ is generally parabolic: $$H_c(T) = H_c(0) \left[ 1 – \left( \frac{T}{T_c} \right)^2 \right]$$
Where:
- $H_c(0)$ is the critical field at absolute zero (0 K).
- $T_c$ is the critical temperature in zero field.
1.3 Meissner Effect
Discovered by Meissner and Ochsenfeld in 1933, this effect is the complete expulsion of magnetic field lines from the interior of a superconductor when it is cooled below $T_c$ in the presence of a magnetic field.
- Perfect Diamagnetism: Inside the superconductor, the magnetic induction $B = 0$.
- Since $B = \mu_0 (H + M)$, where $M$ is magnetization, setting $B=0$ yields $M = -H$, or magnetic susceptibility $\chi = M/H = -1$.
- Distinction: This proves superconductivity is a thermodynamic state, not just perfect conductivity. A perfect conductor (zero resistance) would trap magnetic flux (Lenz’s law), whereas a superconductor expels it.
2. Classification of Superconductors
2.1 Type I Superconductors (Soft Superconductors)
- Behavior: Exhibit a sharp transition from superconducting to normal state at a single critical field $H_c$.
- Meissner Effect: They exhibit the perfect Meissner effect (complete flux expulsion) up to $H_c$.
- Materials: Typically pure elemental metals like Aluminum, Lead, and Mercury.
- Applications: Limited practical use because $H_c$ is usually very low.
2.2 Type II Superconductors (Hard Superconductors)
- Behavior: Have two critical magnetic fields, Lower ($H_{c1}$) and Upper ($H_{c2}$).
- Below $H_{c1}$: Perfectly superconducting (Meissner state).
- Between $H_{c1}$ and $H_{c2}$: The Mixed State (or Vortex State). Magnetic flux penetrates the material in quantized filaments (fluxoids) while the bulk remains superconducting with zero resistance.
- Above $H_{c2}$: Superconductivity is destroyed.
- Materials: Alloys and compounds like Niobium-Titanium (NbTi) and Niobium-Tin (Nb$_3$Sn).
- Applications: Used in high-field superconducting magnets (MRI, Particle Accelerators) because $H_{c2}$ can be very high.
3. Electrodynamics: London Equations
Fritz and Heinz London (1935) proposed two equations to govern the microscopic electric and magnetic fields in a superconductor.
3.1 First London Equation (Acceleration Equation)
Describes resistance-less current. In a normal conductor, current is proportional to electric field (Ohm’s Law). In a superconductor, the rate of change of current is proportional to the electric field (electrons accelerate freely). $$\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m} \mathbf{E}$$
- $\mathbf{J}_s$: Superconducting current density
- $n_s$: Number density of superconducting electrons
3.2 Second London Equation (Meissner Equation)
Describes the expulsion of the magnetic field. $$\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{B}$$
Taking the curl of Maxwell’s equation $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s$ and combining it with the Second London Equation leads to: $$\nabla^2 \mathbf{B} = \frac{\mu_0 n_s e^2}{m} \mathbf{B} = \frac{1}{\lambda_L^2} \mathbf{B}$$
3.3 Penetration Depth ($\lambda_L$)
The solution to the differential equation above is $B(x) = B_0 e^{-x/\lambda_L}$. This shows the magnetic field does not abruptly drop to zero at the surface but decays exponentially.
- Definition: The London Penetration Depth ($\lambda_L$) is the distance inside the superconductor at which the magnetic field decays to $1/e$ of its surface value.
- Formula: $$\lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}}$$
4. Microscopic Theory: BCS Theory
Proposed by Bardeen, Cooper, and Schrieffer in 1957.
4.1 The Isotope Effect
Experiments showed that the critical temperature $T_c$ varies with the isotopic mass $M$ of the element: $$T_c \cdot M^{\alpha} = \text{Constant} \quad (\text{where } \alpha \approx 0.5)$$
- Significance: This indicates that the crystal lattice ions (whose mass determines vibrational frequency) play a crucial role in superconductivity, providing strong evidence for electron-phonon interaction.
4.2 Cooper Pairs
- Mechanism: An electron moving through the lattice distorts it (attracts positive ions), creating a region of excess positive charge. A second electron is attracted to this positive region.
- Result: This creates an effective weak attraction between two electrons, mediated by lattice vibrations (Phonons).
- Properties:
- Cooper pairs consist of two electrons with opposite momenta ($k, -k$) and opposite spins ($\uparrow, \downarrow$).
- The net spin is 0 (Singlet state), making the pair behave like a Boson.
- Bosons can condense into the same ground quantum state (Bose-Einstein Condensation), allowing coherent motion without scattering.
4.3 BCS Theory Key Points
- Energy Gap ($\Delta$): The formation of Cooper pairs lowers the energy of the electron system relative to the Fermi Fermi energy. An energy gap $\Delta$ separates the superconducting ground state from the excited (normal) states.
- Stability: To break a pair and create resistance, thermal energy must exceed this gap ($E \ge 2\Delta$). This explains why superconductivity exists only at low temperatures.
- Coherence Length ($\xi$): The characteristic distance over which the two electrons in a Cooper pair are correlated.
5. 40 MCQs on BCS Theory & Superconductivity (With Solutions)
Section A: General Properties & Definitions
1. The temperature at which a conductor becomes a superconductor is called:
A) Curie Temperature B) Debye Temperature C) Critical Temperature D) Weiss Temperature
Answer: C) Critical Temperature ($T_c$) is the transition point for superconductivity.
2. Which of the following is NOT a property of a superconductor in the superconducting state?
A) Zero electrical resistance B) Perfect diamagnetism C) Infinite electrical conductivity D) Strong ferromagnetic behavior
Answer: D) Superconductors are perfect diamagnets ($\chi = -1$), not ferromagnets.
3. The value of magnetic susceptibility ($\chi$) for a perfect superconductor is:
A) 0 B) 1 C) -1 D) Infinity
Answer: C) From$B = \mu_0(H+M) = 0$, we get$M = -H$, so$\chi = M/H = -1$.
4. The critical magnetic field $H_c$ varies with temperature as:
A) $H_c = H_0 (1 – T/T_c)$ B) $H_c = H_0 (1 – (T/T_c)^2)$ C) $H_c = H_0 (1 + (T/T_c)^2)$ D) $H_c = H_0 (1 – \sqrt{T/T_c})$
Answer: B) The relationship is parabolic.
5. Superconductivity is a phenomenon of:
A) High resistance at low temperature B) Zero resistance at high temperature C) Zero resistance at low temperature D) Zero voltage at high current
Answer: C) It generally occurs near absolute zero (except for High-$T_c$oxides).
6. The transition from the normal state to the superconducting state is:
A) Irreversible B) Reversible C) Adiabatic D) Isenthalpic
Answer: B) The transition is thermodynamically reversible (properties return when T is raised).
7. In a superconductor, the entropy:
A) Increases below $T_c$ B) Remains constant C) Decreases below $T_c$ D) Becomes zero immediately
Answer: C) The superconducting state is more ordered (Cooper pair condensation), so entropy decreases.
8. The specific heat of a superconductor shows a/an ______ at $T_c$.
A) Exponential decay B) Linear increase C) Discontinuous jump D) Constant value
Answer: C) There is a specific heat jump at$T_c$, characteristic of a second-order phase transition (in zero field).
Section B: Magnetic Properties & Types
9. The Meissner effect refers to:
A) Generation of magnetic field B) Expulsion of magnetic flux C) Penetration of magnetic flux D) Stabilization of Cooper pairs
Answer: B) Expulsion of magnetic flux from the interior.
10. A Type I superconductor is also known as a:
A) Hard superconductor B) Soft superconductor C) High-$T_c$ superconductor D) Alloy superconductor
Answer: B) Type I are essentially pure metals and are called “soft”.
11. Which material is an example of a Type I superconductor?
A) NbTi B) Pb (Lead) C) YBCO D) Nb$_3$Sn
Answer: B) Lead is a classic pure metal Type I superconductor.
12. Type II superconductors are characterized by:
A) A single critical field B) Two critical fields ($H_{c1}$ and $H_{c2}$) C) Zero critical field D) Low critical temperature
Answer: B) They exhibit a mixed state between$H_{c1}$and$H_{c2}$.
13. In the “Mixed State” or “Vortex State” of a Type II superconductor:
A) The material is fully normal B) The material is fully diamagnetic C) Magnetic flux penetrates in quantized filaments D) Resistance is infinite
Answer: C) Flux penetrates as vortices (fluxoids) while the bulk remains superconducting.
14. Hard superconductors are preferred for making powerful electromagnets because: A) They are soft and malleable B) They have high critical magnetic fields ($H_{c2}$) C) They exhibit complete Meissner effect D) They have low $T_c$
Answer: B) They can sustain superconductivity in very high magnetic fields.
15. The region between $H_{c1}$ and $H_{c2}$ in a Type II superconductor is called:
A) Meissner state B) Intermediate state C) Vortex (Mixed) state D) Normal state
Answer: C) The mixed/vortex state.
Section C: London Equations & Electrodynamics
16. The London penetration depth $\lambda_L$ represents the distance where magnetic field decays to:
A) 0 B) $1/2$ of surface value C) $1/e$ of surface value D) $1/10$ of surface value
Answer: C) It is an exponential decay constant ($e^{-1}$).
17. The London penetration depth is proportional to:
A) $n_s$ B) $n_s^2$ C) $n_s^{-1/2}$ D) $n_s^{-1}$
Answer: C)$\lambda_L = \sqrt{m / \mu_0 n_s e^2}$, so it is proportional to$1/\sqrt{n_s}$.
18. As temperature approaches $T_c$ from below ($T \to T_c$), the penetration depth $\lambda_L$:
A) Approaches zero B) Approaches infinity C) Remains constant D) Decreases linearly
Answer: B) As$T \to T_c$, the density of superconducting electrons$n_s \to 0$, so$\lambda_L \propto 1/\sqrt{n_s} \to \infty$.
19. The First London Equation describes:
A) Flux quantization B) Perfect diamagnetism C) Acceleration of superconducting electrons without resistance D) The energy gap
Answer: C)$\frac{\partial \mathbf{J}}{\partial t} \propto \mathbf{E}$. E causes acceleration, not velocity, implying zero drag (resistance).
20. The quantity $\frac{h}{2e}$ is known as:
A) Bohr magneton B) Magnetic flux quantum ($\Phi_0$) C) Permeability constant D) Penetration depth
Answer: B) It is the basic unit of quantized magnetic flux in a superconductor.
Section D: BCS Theory & Microscopic Mechanisms
21. The attractive force between electrons in a Cooper pair is mediated by:
A) Photons B) Phonons C) Magnons D) Gluons
Answer: B) Lattice vibrations (phonons) mediate the attraction.
22. A Cooper pair consists of two electrons with:
A) Same spin, same momentum B) Opposite spin, same momentum C) Opposite spin, opposite momentum D) Same spin, opposite momentum
Answer: C) Momentum$\mathbf{k}$and$-\mathbf{k}$; Spin$\uparrow$and$\downarrow$.
23. The total spin of a Cooper pair is:
A) $1/2$ B) $1$ C) $0$ (Singlet) D) $3/2$
Answer: C) Since spins are antiparallel ($+1/2, -1/2$), the total spin is 0.
24. Electrons in a Cooper pair obey which statistics?
A) Fermi-Dirac B) Bose-Einstein C) Maxwell-Boltzmann D) Rayleigh-Jeans
Answer: B) The pair acts as a composite Boson (integer spin).
25. The Isotope Effect states that $T_c \propto M^{-\alpha}$. The value of $\alpha$ predicted by simple BCS theory is:
A) 0.2 B) 0.5 C) 1.0 D) 2.0
Answer: B) BCS theory predicts$\alpha = 0.5$.
26. The energy gap $\Delta$ in BCS theory varies with temperature. At $T = T_c$, the gap is:
A) Maximum B) Equal to Fermi energy C) Zero D) Infinite
Answer: C) The gap closes and vanishes exactly at$T_c$.
27. The energy required to break a Cooper pair is:
A) $\Delta$ B) $2\Delta$ C) $k_B T_c$ D) $E_F$
Answer: B) You need$2\Delta$(typically denoted as$2\Delta(0)$at 0K) to separate the two electrons.
28. According to BCS theory, the critical temperature $T_c$ is related to the Debye frequency $\omega_D$ and interaction potential $V$ by:
A) $T_c \approx \theta_D \exp(-1/N(0)V)$ B) $T_c \approx \theta_D \exp(N(0)V)$ C) $T_c \approx V \exp(-1/\theta_D)$ D) $T_c \approx N(0)V$
Answer: A) The standard BCS formula is$k_B T_c = 1.13 \hbar \omega_D e^{-1/N(0)V}$.
29. The coherence length $\xi$ represents:
A) The distance between two Cooper pairs B) The size of the Cooper pair (correlation distance) C) The depth of magnetic penetration D) The lattice constant
Answer: B) It characterizes the spatial extent of the correlation between the two paired electrons.
30. The Isotope Effect provides direct evidence for:
A) Electronic contribution to specific heat B) Electron-Phonon interaction mechanism C) Electron-Magnon interaction D) Impurity scattering
Answer: B) Dependence on ion mass proves lattice vibrations (phonons) are involved.
31. At $T=0$ K, the Fermi level in a superconductor lies:
A) At the bottom of the energy gap B) At the top of the energy gap C) In the middle of the energy gap D) Inside the conduction band
Answer: C) The gap opens up symmetrically around the Fermi level.
32. BCS theory is most successful in explaining:
A) High-$T_c$ ceramic superconductors B) Type I (Low-$T_c$) superconductors C) Heavy Fermion superconductors D) Iron-based superconductors
Answer: B) Standard BCS works well for conventional, low-temperature metallic superconductors.
33. The frequency of lattice vibrations (phonons) is proportional to mass $M$ as: A) $M$ B) $M^{-1}$ C) $M^{1/2}$ D) $M^{-1/2}$
Answer: D) Frequency$\omega \propto \sqrt{k/M}$, hence$M^{-1/2}$. This drives the Isotope Effect.
34. The density of states $N(0)$ in the BCS formula refers to:
A) Density of phonons B) Density of electrons at the Fermi level C) Density of Cooper pairs D) Density of lattice ions
Answer: B) Electronic density of states at the Fermi energy.
35. If the electron-phonon coupling constant is increased, $T_c$ will:
A) Decrease B) Increase C) Remain constant D) Become zero
Answer: B) Since$T_c \propto \exp(-1/\lambda_{eff})$, increasing coupling$\lambda_{eff}$increases$T_c$.
Section E: Advanced & Miscellaneous
36. The ratio of the energy gap to critical temperature $2\Delta(0) / k_B T_c$ in BCS theory is approximately:
A) 1.5 B) 2.0 C) 3.53 D) 10.5
Answer: C) This is a universal constant derived in weak-coupling BCS theory.
37. High-$T_c$ superconductors (like YBCO) are generally:
A) Type I B) Type II C) Semiconductors D) Insulators at all temperatures
Answer: B) They are extreme Type II superconductors with very high$H_{c2}$.
38. Flux quantization means magnetic flux through a superconducting ring is an integer multiple of:
A) $e/h$ B) $h/e$ C) $h/2e$ D) $2e/h$
Answer: C)$\Phi_0 = h/2e \approx 2.07 \times 10^{-15}$Wb. The ‘2’ comes from the electron pair.
39. In a superconductor, the specific heat at very low temperatures ($T \ll T_c$) varies as: A) $T$ B) $T^3$ C) $\exp(-\Delta/k_B T)$ D) Constant
Answer: C) It decays exponentially due to the energy gap.
40. Which of the following destroys the Cooper pairing? .
A) Lowering the temperature B) Thermal agitation above $T_c$ C) Removing magnetic field D) Removing impurities
Answer: B) Thermal energy$k_B T > \Delta$breaks the pair binding energy.