{"id":974,"date":"2025-12-15T02:09:23","date_gmt":"2025-12-15T02:09:23","guid":{"rendered":"https:\/\/vracademy.in\/?p=974"},"modified":"2025-12-15T02:15:09","modified_gmt":"2025-12-15T02:15:09","slug":"superconductivity-a-comprehensive-study-guide","status":"publish","type":"post","link":"https:\/\/vracademy.in\/?p=974","title":{"rendered":"Superconductivity: A Comprehensive Study Guide"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">BSc Physics MCQ TGT Non medical <\/h2>\n\n\n\n<p> <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/vracademy.in\/wp-content\/uploads\/2025\/12\/physics-superconductivity-1024x576.png\" alt=\"\" class=\"wp-image-975\" srcset=\"https:\/\/vracademy.in\/wp-content\/uploads\/2025\/12\/physics-superconductivity-1024x576.png 1024w, https:\/\/vracademy.in\/wp-content\/uploads\/2025\/12\/physics-superconductivity-300x169.png 300w, https:\/\/vracademy.in\/wp-content\/uploads\/2025\/12\/physics-superconductivity-768x432.png 768w, https:\/\/vracademy.in\/wp-content\/uploads\/2025\/12\/physics-superconductivity.png 1280w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">1. Fundamental Concepts<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe title=\"Physics BSc level MCQ For TGT Non medical \/\/Superconductivity Physics BSc\/\/HP TGT Non medical exam\" width=\"525\" height=\"295\" src=\"https:\/\/www.youtube.com\/embed\/1c75GjgLgDs?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">1.1 Critical Temperature ($T_c$)<\/h3>\n\n\n\n<p>The <strong>Critical Temperature (<\/strong>$T_c$<strong>)<\/strong> 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).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Significance:<\/strong> Above $T_c$, the material behaves as a normal metal. Below $T_c$, charge carriers move without scattering, resulting in no energy loss.<\/li>\n\n\n\n<li><strong>Examples:<\/strong> Mercury ($T_c \\approx 4.2$ K), Lead ($T_c \\approx 7.2$ K), Niobium ($T_c \\approx 9.3$ K).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">1.2 Critical Magnetic Field ($H_c$)<\/h3>\n\n\n\n<p>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 <strong>Critical Magnetic Field (<\/strong>$H_c$<strong>)<\/strong>.<\/p>\n\n\n\n<p>The dependence of $H_c$ on temperature $T$ is generally parabolic: $$H_c(T) = H_c(0) \\left[ 1 &#8211; \\left( \\frac{T}{T_c} \\right)^2 \\right]$$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$H_c(0)$ is the critical field at absolute zero (0 K).<\/li>\n\n\n\n<li>$T_c$ is the critical temperature in zero field.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">1.3 Meissner Effect<\/h3>\n\n\n\n<p>Discovered by Meissner and Ochsenfeld in 1933, this effect is the <strong>complete expulsion of magnetic field lines<\/strong> from the interior of a superconductor when it is cooled below $T_c$ in the presence of a magnetic field.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Perfect Diamagnetism:<\/strong> Inside the superconductor, the magnetic induction $B = 0$.<\/li>\n\n\n\n<li>Since $B = \\mu_0 (H + M)$, where $M$ is magnetization, setting $B=0$ yields $M = -H$, or magnetic susceptibility $\\chi = M\/H = -1$.<\/li>\n\n\n\n<li><strong>Distinction:<\/strong> This proves superconductivity is a thermodynamic state, not just perfect conductivity. A perfect conductor (zero resistance) would <em>trap<\/em> magnetic flux (Lenz&#8217;s law), whereas a superconductor <em>expels<\/em> it.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">2. Classification of Superconductors<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">2.1 Type I Superconductors (Soft Superconductors)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Behavior:<\/strong> Exhibit a sharp transition from superconducting to normal state at a single critical field $H_c$.<\/li>\n\n\n\n<li><strong>Meissner Effect:<\/strong> They exhibit the perfect Meissner effect (complete flux expulsion) up to $H_c$.<\/li>\n\n\n\n<li><strong>Materials:<\/strong> Typically pure elemental metals like Aluminum, Lead, and Mercury.<\/li>\n\n\n\n<li><strong>Applications:<\/strong> Limited practical use because $H_c$ is usually very low.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2.2 Type II Superconductors (Hard Superconductors)<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Behavior:<\/strong> Have two critical magnetic fields, Lower ($H_{c1}$) and Upper ($H_{c2}$).\n<ul class=\"wp-block-list\">\n<li><strong>Below <\/strong>$H_{c1}$<strong>:<\/strong> Perfectly superconducting (Meissner state).<\/li>\n\n\n\n<li><strong>Between <\/strong>$H_{c1}$<strong> and <\/strong>$H_{c2}$<strong>:<\/strong> The <strong>Mixed State<\/strong> (or Vortex State). Magnetic flux penetrates the material in quantized filaments (fluxoids) while the bulk remains superconducting with zero resistance.<\/li>\n\n\n\n<li><strong>Above <\/strong>$H_{c2}$<strong>:<\/strong> Superconductivity is destroyed.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Materials:<\/strong> Alloys and compounds like Niobium-Titanium (NbTi) and Niobium-Tin (Nb$_3$Sn).<\/li>\n\n\n\n<li><strong>Applications:<\/strong> Used in high-field superconducting magnets (MRI, Particle Accelerators) because $H_{c2}$ can be very high.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">3. Electrodynamics: London Equations<\/h2>\n\n\n\n<p>Fritz and Heinz London (1935) proposed two equations to govern the microscopic electric and magnetic fields in a superconductor.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.1 First London Equation (Acceleration Equation)<\/h3>\n\n\n\n<p>Describes resistance-less current. In a normal conductor, current is proportional to electric field (Ohm&#8217;s Law). In a superconductor, the <strong>rate of change<\/strong> of current is proportional to the electric field (electrons accelerate freely). $$\\frac{\\partial \\mathbf{J}_s}{\\partial t} = \\frac{n_s e^2}{m} \\mathbf{E}$$<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\mathbf{J}_s$: Superconducting current density<\/li>\n\n\n\n<li>$n_s$: Number density of superconducting electrons<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3.2 Second London Equation (Meissner Equation)<\/h3>\n\n\n\n<p>Describes the expulsion of the magnetic field. $$\\nabla \\times \\mathbf{J}_s = -\\frac{n_s e^2}{m} \\mathbf{B}$$<\/p>\n\n\n\n<p>Taking the curl of Maxwell&#8217;s equation $\\nabla \\times \\mathbf{B} = \\mu_0 \\mathbf{J}_s$ and combining it with the Second London Equation leads to: $$\\nabla^2 \\mathbf{B} = \\frac{\\mu_0 n_s e^2}{m} \\mathbf{B} = \\frac{1}{\\lambda_L^2} \\mathbf{B}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3.3 Penetration Depth ($\\lambda_L$)<\/h3>\n\n\n\n<p>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.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition:<\/strong> 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.<\/li>\n\n\n\n<li><strong>Formula:<\/strong> $$\\lambda_L = \\sqrt{\\frac{m}{\\mu_0 n_s e^2}}$$<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">4. Microscopic Theory: BCS Theory<\/h2>\n\n\n\n<p>Proposed by Bardeen, Cooper, and Schrieffer in 1957.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4.1 The Isotope Effect<\/h3>\n\n\n\n<p>Experiments showed that the critical temperature $T_c$ varies with the isotopic mass $M$ of the element: $$T_c \\cdot M^{\\alpha} = \\text{Constant} \\quad (\\text{where } \\alpha \\approx 0.5)$$<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Significance:<\/strong> 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.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">4.2 Cooper Pairs<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Mechanism:<\/strong> 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.<\/li>\n\n\n\n<li><strong>Result:<\/strong> This creates an effective weak attraction between two electrons, mediated by lattice vibrations (<strong>Phonons<\/strong>).<\/li>\n\n\n\n<li><strong>Properties:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Cooper pairs consist of two electrons with <strong>opposite momenta (<\/strong>$k, -k$<strong>)<\/strong> and <strong>opposite spins (<\/strong>$\\uparrow, \\downarrow$<strong>)<\/strong>.<\/li>\n\n\n\n<li>The net spin is 0 (Singlet state), making the pair behave like a <strong>Boson<\/strong>.<\/li>\n\n\n\n<li>Bosons can condense into the same ground quantum state (Bose-Einstein Condensation), allowing coherent motion without scattering.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">4.3 BCS Theory Key Points<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Energy Gap (<\/strong>$\\Delta$<strong>):<\/strong> 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.<\/li>\n\n\n\n<li><strong>Stability:<\/strong> 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.<\/li>\n\n\n\n<li><strong>Coherence Length (<\/strong>$\\xi$<strong>):<\/strong> The characteristic distance over which the two electrons in a Cooper pair are correlated.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">5. 40 MCQs on BCS Theory &amp; Superconductivity (With Solutions)<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Section A: General Properties &amp; Definitions<\/strong><\/h3>\n\n\n\n<p><strong>1. The temperature at which a conductor becomes a superconductor is called:<\/strong><\/p>\n\n\n\n<p>A) Curie Temperature B) Debye Temperature C) Critical Temperature D) Weiss Temperature<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> Critical Temperature ($T_c$) is the transition point for superconductivity.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>2. Which of the following is NOT a property of a superconductor in the superconducting state?<\/strong><\/p>\n\n\n\n<p>A) Zero electrical resistance B) Perfect diamagnetism C) Infinite electrical conductivity D) Strong ferromagnetic behavior<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: D)<\/strong> Superconductors are perfect diamagnets ($\\chi = -1$), not ferromagnets.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>3. The value of magnetic susceptibility (<\/strong>$\\chi$<strong>) for a perfect superconductor is:<\/strong><\/p>\n\n\n\n<p>A) 0 B) 1 C) -1 D) Infinity<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> From$B = \\mu_0(H+M) = 0$, we get$M = -H$, so$\\chi = M\/H = -1$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>4. The critical magnetic field <\/strong>$H_c$<strong> varies with temperature as:<\/strong><\/p>\n\n\n\n<p>A) $H_c = H_0 (1 &#8211; T\/T_c)$ B) $H_c = H_0 (1 &#8211; (T\/T_c)^2)$ C) $H_c = H_0 (1 + (T\/T_c)^2)$ D) $H_c = H_0 (1 &#8211; \\sqrt{T\/T_c})$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> The relationship is parabolic.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>5. Superconductivity is a phenomenon of:<\/strong><\/p>\n\n\n\n<p>A) High resistance at low temperature B) Zero resistance at high temperature C) Zero resistance at low temperature D) Zero voltage at high current<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> It generally occurs near absolute zero (except for High-$T_c$oxides).<\/p>\n<\/blockquote>\n\n\n\n<p><strong>6. The transition from the normal state to the superconducting state is:<\/strong><\/p>\n\n\n\n<p>A) Irreversible B) Reversible C) Adiabatic D) Isenthalpic<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> The transition is thermodynamically reversible (properties return when T is raised).<\/p>\n<\/blockquote>\n\n\n\n<p><strong>7. In a superconductor, the entropy:<\/strong><\/p>\n\n\n\n<p>A) Increases below $T_c$ B) Remains constant C) Decreases below $T_c$ D) Becomes zero immediately<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> The superconducting state is more ordered (Cooper pair condensation), so entropy decreases.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>8. The specific heat of a superconductor shows a\/an ______ at <\/strong>$T_c$<strong>.<\/strong><\/p>\n\n\n\n<p>A) Exponential decay B) Linear increase C) Discontinuous jump D) Constant value<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> There is a specific heat jump at$T_c$, characteristic of a second-order phase transition (in zero field).<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Section B: Magnetic Properties &amp; Types<\/strong><\/h3>\n\n\n\n<p><strong>9. The Meissner effect refers to:<\/strong><\/p>\n\n\n\n<p>A) Generation of magnetic field B) Expulsion of magnetic flux C) Penetration of magnetic flux D) Stabilization of Cooper pairs<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Expulsion of magnetic flux from the interior.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>10. A Type I superconductor is also known as a:<\/strong><\/p>\n\n\n\n<p>A) Hard superconductor B) Soft superconductor C) High-$T_c$ superconductor D) Alloy superconductor<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Type I are essentially pure metals and are called &#8220;soft&#8221;.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>11. Which material is an example of a Type I superconductor?<\/strong><\/p>\n\n\n\n<p>A) NbTi B) Pb (Lead) C) YBCO D) Nb$_3$Sn<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Lead is a classic pure metal Type I superconductor.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>12. Type II superconductors are characterized by:<\/strong><\/p>\n\n\n\n<p>A) A single critical field B) Two critical fields ($H_{c1}$ and $H_{c2}$) C) Zero critical field D) Low critical temperature<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> They exhibit a mixed state between$H_{c1}$and$H_{c2}$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>13. In the &#8220;Mixed State&#8221; or &#8220;Vortex State&#8221; of a Type II superconductor:<\/strong><\/p>\n\n\n\n<p>A) The material is fully normal B) The material is fully diamagnetic C) Magnetic flux penetrates in quantized filaments D) Resistance is infinite<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> Flux penetrates as vortices (fluxoids) while the bulk remains superconducting.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>14. Hard superconductors are preferred for making powerful electromagnets because:<\/strong> 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 $T_c$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> They can sustain superconductivity in very high magnetic fields.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>15. The region between <\/strong>$H_{c1}$<strong> and <\/strong>$H_{c2}$<strong> in a Type II superconductor is called:<\/strong><\/p>\n\n\n\n<p>A) Meissner state B) Intermediate state C) Vortex (Mixed) state D) Normal state<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> The mixed\/vortex state.<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Section C: London Equations &amp; Electrodynamics<\/strong><\/h3>\n\n\n\n<p><strong>16. The London penetration depth <\/strong>$\\lambda_L$<strong> represents the distance where magnetic field decays to:<\/strong><\/p>\n\n\n\n<p>A) 0 B) $1\/2$ of surface value C) $1\/e$ of surface value D) $1\/10$ of surface value<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> It is an exponential decay constant ($e^{-1}$).<\/p>\n<\/blockquote>\n\n\n\n<p><strong>17. The London penetration depth is proportional to:<\/strong><\/p>\n\n\n\n<p>A) $n_s$ B) $n_s^2$ C) $n_s^{-1\/2}$ D) $n_s^{-1}$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong>$\\lambda_L = \\sqrt{m \/ \\mu_0 n_s e^2}$, so it is proportional to$1\/\\sqrt{n_s}$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>18. As temperature approaches <\/strong>$T_c$<strong> from below (<\/strong>$T \\to T_c$<strong>), the penetration depth <\/strong>$\\lambda_L$<strong>:<\/strong><\/p>\n\n\n\n<p>A) Approaches zero B) Approaches infinity C) Remains constant D) Decreases linearly<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> As$T \\to T_c$, the density of superconducting electrons$n_s \\to 0$, so$\\lambda_L \\propto 1\/\\sqrt{n_s} \\to \\infty$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>19. The First London Equation describes:<\/strong><\/p>\n\n\n\n<p>A) Flux quantization B) Perfect diamagnetism C) Acceleration of superconducting electrons without resistance D) The energy gap<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong>$\\frac{\\partial \\mathbf{J}}{\\partial t} \\propto \\mathbf{E}$. E causes acceleration, not velocity, implying zero drag (resistance).<\/p>\n<\/blockquote>\n\n\n\n<p><strong>20. The quantity <\/strong>$\\frac{h}{2e}$<strong> is known as:<\/strong><\/p>\n\n\n\n<p>A) Bohr magneton B) Magnetic flux quantum ($\\Phi_0$) C) Permeability constant D) Penetration depth<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> It is the basic unit of quantized magnetic flux in a superconductor.<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Section D: BCS Theory &amp; Microscopic Mechanisms<\/strong><\/h3>\n\n\n\n<p><strong>21. The attractive force between electrons in a Cooper pair is mediated by:<\/strong><\/p>\n\n\n\n<p>A) Photons B) Phonons C) Magnons D) Gluons<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Lattice vibrations (phonons) mediate the attraction.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>22. A Cooper pair consists of two electrons with:<\/strong><\/p>\n\n\n\n<p>A) Same spin, same momentum B) Opposite spin, same momentum C) Opposite spin, opposite momentum D) Same spin, opposite momentum<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> Momentum$\\mathbf{k}$and$-\\mathbf{k}$; Spin$\\uparrow$and$\\downarrow$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>23. The total spin of a Cooper pair is:<\/strong><\/p>\n\n\n\n<p>A) $1\/2$ B) $1$ C) $0$ (Singlet) D) $3\/2$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> Since spins are antiparallel ($+1\/2, -1\/2$), the total spin is 0.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>24. Electrons in a Cooper pair obey which statistics?<\/strong><\/p>\n\n\n\n<p>A) Fermi-Dirac B) Bose-Einstein C) Maxwell-Boltzmann D) Rayleigh-Jeans<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> The pair acts as a composite Boson (integer spin).<\/p>\n<\/blockquote>\n\n\n\n<p><strong>25. The Isotope Effect states that <\/strong>$T_c \\propto M^{-\\alpha}$<strong>. The value of <\/strong>$\\alpha$<strong> predicted by simple BCS theory is:<\/strong><\/p>\n\n\n\n<p>A) 0.2 B) 0.5 C) 1.0 D) 2.0<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> BCS theory predicts$\\alpha = 0.5$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>26. The energy gap <\/strong>$\\Delta$<strong> in BCS theory varies with temperature. At <\/strong>$T = T_c$<strong>, the gap is:<\/strong><\/p>\n\n\n\n<p>A) Maximum B) Equal to Fermi energy C) Zero D) Infinite<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> The gap closes and vanishes exactly at$T_c$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>27. The energy required to break a Cooper pair is:<\/strong><\/p>\n\n\n\n<p>A) $\\Delta$ B) $2\\Delta$ C) $k_B T_c$ D) $E_F$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> You need$2\\Delta$(typically denoted as$2\\Delta(0)$at 0K) to separate the two electrons.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>28. According to BCS theory, the critical temperature <\/strong>$T_c$<strong> is related to the Debye frequency <\/strong>$\\omega_D$<strong> and interaction potential <\/strong>$V$<strong> by:<\/strong><\/p>\n\n\n\n<p>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) $T_c \\approx N(0)V$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: A)<\/strong> The standard BCS formula is$k_B T_c = 1.13 \\hbar \\omega_D e^{-1\/N(0)V}$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>29. The coherence length <\/strong>$\\xi$<strong> represents:<\/strong><\/p>\n\n\n\n<p>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<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> It characterizes the spatial extent of the correlation between the two paired electrons.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>30. The Isotope Effect provides direct evidence for:<\/strong><\/p>\n\n\n\n<p>A) Electronic contribution to specific heat B) Electron-Phonon interaction mechanism C) Electron-Magnon interaction D) Impurity scattering<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Dependence on ion mass proves lattice vibrations (phonons) are involved.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>31. At <\/strong>$T=0$<strong> K, the Fermi level in a superconductor lies:<\/strong><\/p>\n\n\n\n<p>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<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> The gap opens up symmetrically around the Fermi level.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>32. BCS theory is most successful in explaining:<\/strong><\/p>\n\n\n\n<p>A) High-$T_c$ ceramic superconductors B) Type I (Low-$T_c$) superconductors C) Heavy Fermion superconductors D) Iron-based superconductors<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Standard BCS works well for conventional, low-temperature metallic superconductors.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>33. The frequency of lattice vibrations (phonons) is proportional to mass <\/strong>$M$<strong> as:<\/strong> A) $M$ B) $M^{-1}$ C) $M^{1\/2}$ D) $M^{-1\/2}$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: D)<\/strong> Frequency$\\omega \\propto \\sqrt{k\/M}$, hence$M^{-1\/2}$. This drives the Isotope Effect.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>34. The density of states <\/strong>$N(0)$<strong> in the BCS formula refers to:<\/strong><\/p>\n\n\n\n<p>A) Density of phonons B) Density of electrons at the Fermi level C) Density of Cooper pairs D) Density of lattice ions<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Electronic density of states at the Fermi energy.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>35. If the electron-phonon coupling constant is increased, <\/strong>$T_c$<strong> will:<\/strong><\/p>\n\n\n\n<p>A) Decrease B) Increase C) Remain constant D) Become zero<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Since$T_c \\propto \\exp(-1\/\\lambda_{eff})$, increasing coupling$\\lambda_{eff}$increases$T_c$.<\/p>\n<\/blockquote>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Section E: Advanced &amp; Miscellaneous<\/strong><\/h3>\n\n\n\n<p><strong>36. The ratio of the energy gap to critical temperature <\/strong>$2\\Delta(0) \/ k_B T_c$<strong> in BCS theory is approximately:<\/strong><\/p>\n\n\n\n<p>A) 1.5 B) 2.0 C) 3.53 D) 10.5<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> This is a universal constant derived in weak-coupling BCS theory.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>37. High-<\/strong>$T_c$<strong> superconductors (like YBCO) are generally:<\/strong><\/p>\n\n\n\n<p>A) Type I B) Type II C) Semiconductors D) Insulators at all temperatures<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> They are extreme Type II superconductors with very high$H_{c2}$.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>38. Flux quantization means magnetic flux through a superconducting ring is an integer multiple of:<\/strong><\/p>\n\n\n\n<p>A) $e\/h$ B) $h\/e$ C) $h\/2e$ D) $2e\/h$<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong>$\\Phi_0 = h\/2e \\approx 2.07 \\times 10^{-15}$Wb. The &#8216;2&#8217; comes from the electron pair.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>39. In a superconductor, the specific heat at very low temperatures (<\/strong>$T \\ll T_c$<strong>) varies as:<\/strong> A) $T$ B) $T^3$ C) $\\exp(-\\Delta\/k_B T)$ D) Constant<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: C)<\/strong> It decays exponentially due to the energy gap.<\/p>\n<\/blockquote>\n\n\n\n<p><strong>40. Which of the following destroys the Cooper pairing?<\/strong>&nbsp;.<\/p>\n\n\n\n<p>A) Lowering the temperature B) Thermal agitation above $T_c$ C) Removing magnetic field D) Removing impurities<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><strong>Answer: B)<\/strong> Thermal energy$k_B T &gt; \\Delta$breaks the pair binding energy.<\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>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&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-974","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/vracademy.in\/index.php?rest_route=\/wp\/v2\/posts\/974","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vracademy.in\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vracademy.in\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vracademy.in\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vracademy.in\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=974"}],"version-history":[{"count":2,"href":"https:\/\/vracademy.in\/index.php?rest_route=\/wp\/v2\/posts\/974\/revisions"}],"predecessor-version":[{"id":977,"href":"https:\/\/vracademy.in\/index.php?rest_route=\/wp\/v2\/posts\/974\/revisions\/977"}],"wp:attachment":[{"href":"https:\/\/vracademy.in\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=974"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vracademy.in\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=974"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vracademy.in\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=974"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}