Question 1

In semiconductor physics, what is the electrical conductivity of an intrinsic (pure) semiconductor expressed in terms of the intrinsic carrier concentration n_i, electronic mobility μ_n, hole mobility μ_p, and the elementary charge e?

Accepted Answer

Introduction / Context: Intrinsic semiconductors conduct electricity via thermally generated electrons and holes present in equal numbers. Their macroscopic conductivity depends on how many charge carriers exist per unit volume and how easily each carrier drifts under an applied electric field, captured by mobility parameters. This question tests the fundamental expression for conductivity in a pure (undoped) semiconductor. Given Data / Assumptions: Material is intrinsic: electron concentration n and hole concentration p are equal, n = p = n_i. Elementary charge is e (Coulombs). Carrier mobilities are μ_n for electrons and μ_p for holes. Low-field, ohmic conditions so drift current dominates and mobilities are constant. Concept / Approach: In semiconductors, the drift current density is J = e (n μ_n + p μ_p) E, where E is the electric field. Conductivity σ is defined by J = σ E. For intrinsic material, substitute n = p = n_i to obtain σ = e n_i (μ_n + μ_p). Step-by-Step Solution: Start with J = e (n μ_n + p μ_p) E.Use intrinsic condition: n = p = n_i.Then J = e n_i (μ_n + μ_p) E.Hence σ = J / E = e n_i (μ_n + μ_p). Verification / Alternative check: Dimensional check: e (C) * n_i (m^-3) * μ (m^2/V·s) gives C·m^-1·V^-1·s^-1 = S/m, which is correct for conductivity. Why Other Options Are Wrong: e n_i (μ_p − μ_n): subtraction is incorrect; both carriers add to conductivity.n_i (μ_p + μ_n): missing the factor e.n_i (μ_p μ_n): wrong functional form; mobilities do not multiply.e (μ_p + μ_n)^2 / n_i: nonphysical dependence on 1/n_i. Common Pitfalls: Confusing intrinsic with doped material (where n ≠ p) or forgetting the sign of charge (magnitudes are used in σ). Final Answer: e n_i (μ_p + μ_n)

Question 2

In solid-state magnetism, materials that lack permanent magnetic dipoles (and develop only weak, negative magnetization in an applied field) are classified as which type?

Accepted Answer

Introduction / Context: Magnetic behavior of materials arises from electron spins and orbital motions. Whether a solid exhibits a net permanent moment depends on the presence of unpaired spins and cooperative ordering. This question checks the basic classification when no permanent dipoles exist in the absence of a field. Given Data / Assumptions: No permanent magnetic dipoles; all atomic or molecular moments cancel out in zero field. Consider response in a weak, applied magnetic field. Temperature is not driving long-range magnetic order (no ferromagnetic or ferrimagnetic states). Concept / Approach: Diamagnetic materials have all electrons paired; they exhibit an induced magnetic moment opposite to the external field due to Lenz-like response of orbital motion, giving a small negative susceptibility. Paramagnets have unpaired electrons that align weakly with the field (positive susceptibility) but no permanent magnetization without a field. Ferromagnets and ferrimagnets show spontaneous ordering and domain behavior leading to permanent dipoles. Step-by-Step Solution: Identify the absence of permanent dipoles → rules out ferro-, ferri-, and antiferromagnets in their ordered states.Paramagnets still possess permanent atomic moments (unpaired spins) even if disordered → not suitable.Materials with paired electrons only exhibit diamagnetism with negative susceptibility. Verification / Alternative check: Examples: bismuth, copper, and superconductors (perfect diamagnetism in the Meissner state) demonstrate diamagnetic behavior. Why Other Options Are Wrong: Paramagnetic: contains permanent dipoles from unpaired spins.Ferromagnetic/Ferrimagnetic: possess domain magnetization and remanence.Antiferromagnetic (not listed in stem but as an option): still has ordered sublattice spins, although net moment may cancel; not “no permanent dipoles.” Common Pitfalls: Equating “no net moment” with “no dipoles”; antiferromagnets have opposing sublattice dipoles, not the absence of dipoles. Final Answer: Diamagnetic

Question 3

For permanent (hard) magnetic materials used in machines and devices, which property combination is most desirable to ensure strong, stable magnetization and resistance to demagnetizing influences?

Accepted Answer

Introduction / Context: Permanent magnets must retain magnetization in service and resist demagnetizing fields, temperature excursions, and mechanical shocks. The magnetic hysteresis loop provides a direct visual of the required properties: significant remanence and coercivity, and typically a large loop area. Given Data / Assumptions: Application demands stable, long-term magnetization (hard magnetic material). Demagnetizing forces may be present (e.g., leakage fields, temperature variations). Losses from cyclic magnetization are secondary to retention for permanent magnets. Concept / Approach: Hard magnets are characterized by high coercive field H_c and high remanent induction B_r. The product (B H)_max reflects energy density available from the magnet and correlates with a “fat” loop. In contrast, soft magnetic materials (transformer steels) need low coercivity and narrow loops to minimize hysteresis loss. Step-by-Step Solution: Identify permanent magnet objective: resist demagnetization → need large H_c.Ensure strong field output → need large B_r (residual induction).Conclude that option emphasizing both large B_r and large H_c fits best. Verification / Alternative check: Materials like Nd-Fe-B and Sm-Co exhibit high coercivity and remanence; their hysteresis loops confirm the property combination. Why Other Options Are Wrong: Small remanence/coercivity suits soft magnets, not permanent ones.Small loop area minimizes core losses but implies easy demagnetization.High initial permeability is useful for soft cores, not the defining trait for permanent magnets. Common Pitfalls: Confusing design goals of soft versus hard magnetic materials. Final Answer: Large residual induction (remanence) and large coercive field (high H_c)

Question 4

A parallel-plate capacitor has its plate length and width each doubled (area ×4) and the plate separation doubled (d ×2). Neglecting fringing, to keep the capacitance unchanged, how must the dielectric constant (relative permittivity ε_r) be adjuste…

Accepted Answer

Introduction / Context: Capacitance in a parallel-plate capacitor scales directly with plate area and dielectric constant, and inversely with plate separation. When geometry is modified, compensating changes in the dielectric constant can maintain the same capacitance, an important idea in sensor design and capacitor packaging. Given Data / Assumptions: Initial capacitance: C = ε_0 ε_r A / d. New dimensions: area A' = 4 A (both length and width doubled); separation d' = 2 d. Fringing fields are neglected (valid when plate dimensions ≫ separation). Concept / Approach: Write the new capacitance in terms of ε_r' and set it equal to the original. Solve for the required ε_r' as a multiple of ε_r. This relies purely on proportionality; no numerical values are necessary. Step-by-Step Solution: Original: C = ε_0 ε_r A / d.Modified: C' = ε_0 ε_r' (4 A) / (2 d) = ε_0 ε_r' (2 A / d).Set C' = C ⇒ ε_0 ε_r' (2 A / d) = ε_0 ε_r (A / d).Cancel common factors ⇒ 2 ε_r' = ε_r ⇒ ε_r' = ε_r / 2. Verification / Alternative check: Insert sample values (e.g., ε_r = 4): original C ∝ 4 A / d; new with ε_r' = 2 gives C' ∝ 2 * 2 A / d = 4 A / d, matching. Why Other Options Are Wrong: Keeping ε_r the same doubles C; doubling or quadrupling ε_r increases C beyond target; reducing to one quarter under-compensates. Common Pitfalls: Forgetting that area increased by factor 4 (both dimensions doubled) and separation doubled by factor 2. Final Answer: Halved (ε_r → ε_r/2)

Question 5

In rotating electrical machinery and power systems, hydrogen is commonly employed in which equipment, primarily for high-efficiency cooling and reduced windage losses?

Accepted Answer

Introduction / Context: High-power turbo-generators dissipate megawatts of heat in their rotors and stators. Efficient cooling is essential to maintain insulation life and machine reliability. Hydrogen cooling has long been used in very large generators due to the gas’s high specific heat, excellent thermal conductivity, and low density (reducing windage losses). Given Data / Assumptions: Consider equipment at utility or large industrial scale. Cooling performance and parasitic losses are key design concerns. Hydrogen handling systems (seals, purging, safety) are available only in specialized installations. Concept / Approach: Hydrogen-cooled generators enclose the active parts in a sealed casing filled with hydrogen, sometimes with direct-cooled conductors. Benefits include lower gas density (reduced windage), superior heat transfer, and better acoustic performance. Transformers are generally oil-cooled, not hydrogen-cooled. Circuit breakers and MCBs use different interrupting media (air, SF6, vacuum, oil) and do not rely on hydrogen for bulk cooling. Step-by-Step Solution: Identify the equipment with extreme heat flux and rotating parts → large generators.Evaluate hydrogen properties: high thermal conductivity & low density support generator cooling.Conclude that option naming hydrogen-cooled alternators is correct. Verification / Alternative check: Compare with water-cooled stator windings (used additionally) and air-cooled designs; hydrogen systems remain common for highest ratings. Why Other Options Are Wrong: Transformers: predominantly mineral oil/ester plus air- or water-cooling radiators.MCB and typical breakers: use arc-interrupting mediums other than hydrogen. Common Pitfalls: Assuming hydrogen is used for arc interruption or small devices—it is mainly a generator cooling medium. Final Answer: Large-size generators (hydrogen-cooled alternators)

Question 6

In a standard coaxial cable construction, the braided copper sheath is primarily used as which layer?

Accepted Answer

Introduction / Context: Coaxial cables carry high-frequency signals with minimal radiation and susceptibility to external interference. Their layered construction is designed to confine electromagnetic fields within the dielectric between inner conductor and outer shield. Given Data / Assumptions: Typical coax: solid/stranded inner conductor; uniform dielectric; braided or foil outer conductor; protective outer jacket. Purpose is efficient RF transmission with controlled impedance (e.g., 50 Ω or 75 Ω). Braided copper provides both electrical and partial mechanical functions. Concept / Approach: The braided copper (sometimes with foil) forms the outer conductor connected to ground (return path) and acts as an electromagnetic shield. It prevents external fields from entering and internal fields from radiating out, preserving signal integrity. The dielectric provides spacing and sets the characteristic impedance, while the jacket offers environmental protection. Step-by-Step Solution: Identify each layer’s function: inner conductor (signal), dielectric (spacing), braid/foil (shield & return), jacket (protection).Recognize that the braid is not the central conductor or dielectric.Conclude the braid is the shield/return path. Verification / Alternative check: Check continuity from braid to connector shells and system ground in RF systems; this confirms its shielding/return role. Why Other Options Are Wrong: Central conductor is a separate solid/stranded wire.Dielectric is polymer foam/solid (e.g., PE, PTFE), not the braid.Jacket is typically PVC/PE for protection, not copper. Common Pitfalls: Assuming the shield is purely mechanical; it is an electrical conductor that defines the coaxial return path. Final Answer: Electromagnetic shield (return/ground)

Question 7

Thermionic emission: At T = 1000 K, the emission current density from a hot cathode is of the order of 0.1 A/cm². The relation I_th = S A_0 T^2 exp(−E_w / kT) (Richardson–Dushman) governs emission. Assess the statements.

Accepted Answer

Introduction / Context: Thermionic emission underpins vacuum tubes, electron guns, and certain sensors. The Richardson–Dushman equation relates temperature, work function, and emission current density, providing design guidance for cathode materials and operating temperatures. Given Data / Assumptions: Temperature T = 1000 K. Current density order ≈ 0.1 A/cm² refers to practical oxide-coated cathodes with relatively low work function. Equation: J = A_0 T^2 exp(−E_w / kT), and I_th = S J for emitting area S. Concept / Approach: For oxide cathodes (work function roughly 1–1.5 eV), the exponential term is not prohibitively small at 1000 K. Using typical A_0 values (tens to ~120 A/cm²·K²), predicted J values fall in the 0.1–1 A/cm² range, consistent with the assertion. Therefore, the reason (the Richardson–Dushman law) both states the governing physics and explains why appreciable current occurs near 1000 K. Step-by-Step Solution: Choose a representative E_w ≈ 1.1 eV and A_0 ≈ 60–120 A/cm²·K².Compute kT at 1000 K: kT ≈ 0.086 eV.Evaluate the exponential: exp(−E_w/kT) ≈ exp(−12.8) ≈ 2.7×10^-6.Estimate J ≈ A_0 T^2 exp(−E_w/kT) ≈ (60–120)×(10^6)×(2.7×10^-6) ≈ 160–320 A/cm²; practical effective A_0 for oxide cathodes is lower due to surface effects, yielding operational J on the order of 0.1–1 A/cm². Verification / Alternative check: Empirical cathode data sheets for oxide emitters list 0.1–1 A/cm² near 950–1100 K, supporting the assertion's order of magnitude. Why Other Options Are Wrong: If R were false, no physical law would justify the temperature dependence; if A were false, it would conflict with practical cathode operation. Common Pitfalls: Applying the same estimate to high work function metals (≈4–5 eV) would give negligible emission at 1000 K; material choice matters. Final Answer: Both A and R are true and R is correct explanation of A

Question 8

Dielectrics and polarization: Using a plot derived from measurements (e.g., total polarization vs. 1/T), can the permanent dipole moment of molecules be obtained from the slope? Consider that for a polyatomic gas the total polarization is P = N(a_e …

Accepted Answer

Introduction / Context: In dielectric materials, the total polarization of a gas or dilute medium can include three contributions: electronic polarization (a_e), ionic polarization (a_i), and orientation polarization caused by permanent molecular dipoles (μ_p). A standard way to extract μ_p from experimental data is to analyze how polarization varies with temperature, because the orientation term depends inversely on absolute temperature T. Given Data / Assumptions: Total polarization of a polyatomic gas: P = N(a_e + a_i + μ_p^2 / (3 k T)) * E. N is the number density, k is Boltzmann constant, E is the applied electric field. The assertion mentions determining μ_p from a graph (commonly P/E vs. 1/T or dielectric constant vs. 1/T). Concept / Approach: The orientation polarization term equals N * (μ_p^2 / (3 k T)) * E, which is proportional to 1/T. Therefore, plotting P/E against 1/T (or equivalently the dielectric susceptibility χ against 1/T) yields a straight line whose slope is proportional to N * μ_p^2 / (3 k). From this slope, the permanent dipole moment μ_p can be calculated. The electronic and ionic terms are temperature independent and contribute to the intercept of the line, not the slope. Step-by-Step Solution: Write P/E = N(a_e + a_i) + N * μ_p^2 / (3 k) * (1/T).Let Y = P/E and X = 1/T. Then Y = intercept + slope * X.Slope = N * μ_p^2 / (3 k) ⇒ μ_p = sqrt( (3 k * slope) / N ).Intercept = N(a_e + a_i), giving the temperature-independent part. Verification / Alternative check: Equivalently, the Clausius–Mossotti relation can be used to convert measured dielectric constant to molecular polarizability, and the temperature dependence again isolates μ_p via the 1/T behavior. Why Other Options Are Wrong: If R were false, the slope–dipole connection would not hold; but R states the correct formula, which explains A. Saying A is false contradicts well-established orientation polarization theory. Common Pitfalls: Not plotting against 1/T; plotting against T will not linearize the orientation term. Ignoring density N variations; number density should be known or controlled. Final Answer: Both A and R are true and R is correct explanation of A

Question 9

Polarization mechanisms and temperature: Electronic and ionic polarization in a polyatomic gas are approximately independent of temperature, whereas the orientation polarization depends on temperature. Evaluate the assertion–reason pair.

Accepted Answer

Introduction / Context: Dielectric polarization in gases and dilute media arises from multiple mechanisms. Electronic and ionic polarizations are associated with displacement of electron clouds or ions relative to nuclei and are effectively instantaneous for alternating fields at ordinary frequencies; they depend very weakly on temperature. Orientation polarization, however, is due to alignment of permanent dipoles against thermal agitation and is strongly temperature dependent. Given Data / Assumptions: Electronic and ionic polarizations (a_e and a_i) are treated as constants over moderate temperature ranges. Orientation polarization contributes μ_p^2 / (3 k T), which varies as 1/T. The assertion states that electronic and ionic polarizations are independent of temperature; the reason incorrectly says orientation polarization is independent of temperature. Concept / Approach: Orientation polarization decreases as temperature increases because thermal agitation disrupts dipole alignment. Hence, a plot of susceptibility versus 1/T yields a straight line. The assertion is correct, but the reason contradicts the well-known 1/T dependence, making it false. Step-by-Step Solution: Use P = N(a_e + a_i + μ_p^2/(3 k T)) * E.Electronic/ionic terms a_e, a_i ≈ constants ⇒ negligible T dependence.Orientation term ∝ 1/T ⇒ temperature dependent.Therefore: A true, R false. Verification / Alternative check: Curie-like behavior in polar gases confirms 1/T dependence for orientation polarization. Why Other Options Are Wrong: Options claiming R true are inconsistent with the formula and experiments. Common Pitfalls: Assuming all polarization mechanisms vary with temperature the same way. Final Answer: A is true but R is false

Question 10

Atomic spectroscopy nomenclature: In atomic physics, is a state with orbital quantum number l = 0 called a p state?

Accepted Answer

Introduction / Context: Atomic states are labeled by spectroscopic notation tied to the orbital angular momentum quantum number l. This notation is universal across atomic physics, spectroscopy, and quantum chemistry, and is essential for interpreting line spectra and selection rules. Given Data / Assumptions: Orbital quantum number l takes values 0, 1, 2, 3, … Standard letter mapping is used. The claim is that l = 0 corresponds to a p state. Concept / Approach: The accepted mapping is: l = 0 → s, l = 1 → p, l = 2 → d, l = 3 → f, and so on (g, h, … for higher l). Therefore, l = 0 is an s state, not a p state. This holds regardless of the atom (hydrogenic or many-electron) and does not depend on n or spin–orbit effects; those affect term symbols but not the s, p, d, f letter code for l. Step-by-Step Solution: Recall mapping: 0→s, 1→p, 2→d, 3→f.Compare statement: “l = 0 is p” → contradicts mapping.Conclude the statement is false. Verification / Alternative check: Any elementary spectroscopy table or periodic table electron configurations confirm s orbitals correspond to l = 0 (e.g., 1s, 2s, 3s). Why Other Options Are Wrong: Conditions about hydrogen-like atoms, high n, or neglecting spin–orbit do not change the l-to-letter mapping. Common Pitfalls: Confusing principal quantum number n with orbital quantum number l. Final Answer: False

Question 11

AC dielectrics and ideal capacitor behavior: In a perfect capacitor, current density leads the electric field by 90°. Given J = ω * epsilon_0 * epsilon_r' * E0 * cos(ωt + 90°), and that dielectric losses are zero, evaluate the assertion–reason pair.

Accepted Answer

Introduction / Context: In linear dielectrics driven by a sinusoidal field, displacement current density J_d is related to the time derivative of the electric displacement D. For an ideal (lossless) dielectric, the complex permittivity is purely real; hence current and field are in perfect quadrature (90° phase difference). Given Data / Assumptions: Perfect capacitor: no conduction loss and no dielectric loss (loss tangent = 0). epsilon_r' denotes the real (in-phase permittivity) part used in the expression. E(t) = E0 cos(ωt) is the applied field. Concept / Approach: For a perfect capacitor, D = epsilon_0 * epsilon_r' * E, and J_d = dD/dt = omega * epsilon_0 * epsilon_r' * E0 * sin(ωt). Since sin(ωt) = cos(ωt + 90°), J leads E by 90°. Zero dielectric losses imply no in-phase component of current with voltage; all current is reactive, which explains the 90° lead and the stated form of J. Step-by-Step Solution: Write D(t) = epsilon_0 * epsilon_r' * E0 * cos(ωt).Differentiate: J_d = dD/dt = −omega * epsilon_0 * epsilon_r' * E0 * sin(ωt − 90°) = omega * epsilon_0 * epsilon_r' * E0 * cos(ωt + 90°).Phase relationship: J leads E by 90° in lossless dielectrics. Verification / Alternative check: Introducing dielectric loss adds an imaginary part to permittivity; the current then has a component in phase with E, reducing the phase lead below 90°. In the perfect case (loss = 0), the 90° lead is exact. Why Other Options Are Wrong: If losses were not zero, the exact 90° relation would not hold. Common Pitfalls: Confusing conduction current (σE) with displacement current (dD/dt). Final Answer: Both A and R are true and R is correct explanation of A

Question 12

Atomic binding in metals: The attraction between the nucleus and the valence electron of a copper atom is best described as which of the following?

Accepted Answer

Introduction / Context: In metallic solids such as copper, valence electrons are delocalized and participate in metallic bonding, forming an electron sea. The individual attraction between a given nucleus and a valence electron is therefore weaker than in covalent or ionic bonds where electrons are localized between specific atoms or ions. Given Data / Assumptions: Copper has a single s-type valence electron beyond a filled d-shell. In the solid, this electron is delocalized across the lattice. We refer to effective binding of valence electrons in metals. Concept / Approach: Delocalization causes conduction electrons to respond readily to applied fields and thermal energy, evidenced by good electrical and thermal conductivity. A strong nucleus–valence electron attraction would impede conduction. Zero attraction is unphysical; electrostatic forces always exist, but in the collective metallic environment the effective binding is weak compared to localized bonding scenarios. Step-by-Step Solution: Recognize metallic bonding: electrons not tightly bound to individual nuclei.Weak effective attraction facilitates conduction and low ionization for conduction bands.Select “weak.” Verification / Alternative check: Band theory shows partially filled bands in copper; the small effective mass and high mobility align with weak localization of valence electrons. Why Other Options Are Wrong: Zero: Contradicts Coulomb interaction; cannot be exactly zero. Strong: Inconsistent with metallic conduction behavior. Either zero or strong/variable-zero: Not a physically meaningful description. Common Pitfalls: Equating core electron binding (strong) with valence electron binding (much weaker in metals). Final Answer: weak

Question 13

Semiconductor carrier transport: Given minority-carrier lifetime τ = 100 μs and diffusion constant D = 100 cm^2/s, compute the diffusion length L of the carriers.

Accepted Answer

Introduction / Context: The diffusion length L characterizes how far minority carriers can diffuse before recombining. It is a key parameter for device dimensions in photodiodes, solar cells, and bipolar transistors, linking material quality (lifetime) and transport (diffusion constant). Given Data / Assumptions: Minority-carrier lifetime τ = 100 μs = 100 × 10^-6 s = 1 × 10^-4 s. Diffusion constant D = 100 cm^2/s. One-dimensional steady-state diffusion–recombination model. Concept / Approach: The diffusion length is L = sqrt(D * τ). Substituting the given values gives L in centimeters because D is in cm^2/s and τ in seconds. Step-by-Step Solution: Compute product: D * τ = 100 * (1 × 10^-4) = 1 × 10^-2 cm^2.Take square root: L = sqrt(1 × 10^-2) = 0.1 cm.Select the matching option: 0.1 cm. Verification / Alternative check: Units: sqrt(cm^2) = cm, consistent. Orders of magnitude are typical for high-quality materials with long lifetimes. Why Other Options Are Wrong: 0.01 cm and 0.0141 cm correspond to smaller D or τ; not matching the given data. 1 cm is an order of magnitude too large for the stated parameters. 0.32 cm would require D * τ ≈ 0.1024 cm^2, not given here. Common Pitfalls: Mistaking μs for ms, which would inflate L by sqrt(10). Final Answer: 0.1 cm

Question 14

Doped semiconductors at room temperature: If germanium and silicon samples have the same impurity (dopant) density, how do their resistivities compare at room temperature?

Accepted Answer

Introduction / Context: Resistivity ρ of a doped semiconductor depends on free carrier concentration and mobility: ρ = 1 / (q (n μ_n + p μ_p)). For the same dopant density, the majority carrier concentration is similar in magnitude, so material-dependent mobility and intrinsic properties determine relative resistivity at room temperature. Given Data / Assumptions: Equal impurity concentration in germanium (Ge) and silicon (Si). Room temperature conditions. Standard mobility values: carriers in Ge generally have higher mobilities than in Si. Concept / Approach: Because Ge typically exhibits higher carrier mobility than Si, the conductivity σ = q n μ (for dominant carrier type) is greater in Ge at the same carrier concentration. Therefore, resistivity ρ = 1/σ is lower in Ge and correspondingly higher in Si. Step-by-Step Solution: Assume equal majority carrier concentration set by dopants in both samples.Use σ ∝ μ (since q and n are comparable) → Ge has higher μ.Hence ρ_Si > ρ_Ge → choose silicon higher than germanium. Verification / Alternative check: Empirical mobility data: μ_n and μ_p in Ge are larger than in Si at 300 K, matching observed lower resistivity in Ge for equal doping levels. Why Other Options Are Wrong: Equal resistivity: ignores mobility differences. Negative resistivity: unphysical in this context. Ge higher than Si contradicts mobility trends. Common Pitfalls: Confusing intrinsic carrier concentration effects with doped majority-carrier regimes; at typical doping, dopants dominate. Final Answer: resistivity of silicon will be higher than that of germanium

Question 15

Paramagnetism and temperature dependence: For a paramagnetic material, does magnetic susceptibility increase or decrease with increasing temperature?

Accepted Answer

Introduction / Context: Paramagnetic materials contain atomic or ionic moments that tend to align with an applied magnetic field, increasing the material's magnetization. Thermal agitation counteracts this alignment. The classical Curie law captures how susceptibility varies with absolute temperature for many paramagnets. Given Data / Assumptions: Material is paramagnetic (no spontaneous magnetization in zero field). Moderate temperature range where Curie or Curie–Weiss behavior holds. Weak applied field so linear susceptibility χ applies. Concept / Approach: Curie law states χ = C / T, where C is the Curie constant. Therefore, susceptibility decreases as temperature increases, approximately inversely proportional to T. In some materials with interactions, Curie–Weiss law applies: χ = C / (T − θ), but the inverse temperature relationship remains the essential trend away from ordering temperatures. Step-by-Step Solution: Adopt χ ∝ 1/T for paramagnets.As T increases, χ decreases.Hence the statement “susceptibility increases with temperature” is false. Verification / Alternative check: Plotting 1/χ vs. T yields a straight line for many paramagnets, confirming inverse dependence. Why Other Options Are Wrong: Temperature independence contradicts Curie behavior. Increase-then-decrease lacks basis for simple paramagnets outside critical regions. Common Pitfalls: Confusing paramagnetism with ferromagnetism, where domain behavior dominates and χ can exhibit complex T-dependence near Curie temperature. Final Answer: False: susceptibility decreases with temperature (approximately inversely proportional to T)

Question 16

Magnetic response of wood: When a piece of wood is placed in a magnetic field, how do magnetic lines of force behave?

Accepted Answer

Introduction / Context: Most organic materials, including dry wood, are weakly diamagnetic. Diamagnetism corresponds to a small negative magnetic susceptibility, meaning the material develops an induced magnetic moment opposing the applied field. This subtly distorts external field lines. Given Data / Assumptions: Wood sample is dry and non-conductive. Applied magnetic field is uniform at the scale of the sample. No ferromagnetic contaminants are present. Concept / Approach: In a diamagnetic medium (χ < 0), the relative permeability μ_r is slightly less than 1. Field lines prefer paths of higher permeability; hence, they are slightly repelled by the diamagnetic sample and bend away from it. The effect is very small but conceptually clear in field-line diagrams. Step-by-Step Solution: Classify wood as weakly diamagnetic.Diamagnetism ⇒ μ_r < 1 and χ < 0.Therefore, field lines bend away from the sample. Verification / Alternative check: Experiments with strong fields show slight levitation of diamagnetic materials (e.g., pyrolytic graphite); wood displays the same sign of response though much weaker. Why Other Options Are Wrong: No effect: incorrect; the effect is small but not zero. Bend towards: characteristic of paramagnets (χ > 0), not wood. Increase or decrease ambiguously: physics predicts a definite bending away for diamagnets. Field lines do not terminate inside materials; they form continuous loops. Common Pitfalls: Confusing diamagnetic and paramagnetic responses due to their subtlety at low fields. Final Answer: the magnetic lines of force will bend away from the piece

Question 17

Magnetic properties of diamond: Evaluate the assertion–reason pair—Assertion (A) For diamond, μ_r = 1. Reason (R) Diamond is a diamagnetic material.

Accepted Answer

Introduction / Context: Magnetic materials are characterized by susceptibility χ and relative permeability μ_r = 1 + χ. Diamagnetic materials have small negative χ, leading to μ_r slightly less than 1. Diamond, an sp^3-bonded carbon crystal, is strongly covalent and exhibits diamagnetism with very small magnitude. Given Data / Assumptions: Assertion claims μ_r equals exactly 1 for diamond. Reason states that diamond is diamagnetic. Concept / Approach: Because diamond is diamagnetic (χ < 0 but |χ| ≪ 1), its μ_r is close to but not exactly 1 (μ_r = 1 + χ < 1). Thus the assertion is strictly false, while the reason is true. The reason does not support the assertion; if anything, it contradicts it by implying μ_r is slightly less than 1. Step-by-Step Solution: Use μ_r = 1 + χ.For diamagnetic diamond, χ is small and negative ⇒ μ_r ≈ 1 − |χ| < 1.Therefore A false, R true. Verification / Alternative check: Measured susceptibilities of many diamagnetic insulators are on the order of 10^-6 to 10^-5 (negative), confirming μ_r is slightly below 1. Why Other Options Are Wrong: Any option implying A true is inconsistent with the definition μ_r = 1 + χ and χ < 0 for diamagnetics. Common Pitfalls: Rounding μ_r to 1 for convenience; exact statements should distinguish “≈1” from “=1”. Final Answer: A is false but R is true

Question 18

In solid-state physics of metals, are the valence-electron wave functions strongly perturbed by neighboring atoms, leading to energy-band formation rather than isolated atomic levels?

Accepted Answer

Introduction / Context: Band theory explains why metals conduct electricity so well. In a crystal, each atom is surrounded by many neighbors. The overlap of their valence-electron wave functions means electrons can no longer be described as belonging to single atoms. Instead, their allowed energies spread into bands. This question checks your conceptual grasp of how interatomic interactions in a lattice transform discrete atomic levels into continuous energy bands that govern metallic behavior. Given Data / Assumptions: Periodic crystalline metal with closely spaced atoms (typical metallic bond distances). Focus on valence electrons (s, p, or d depending on metal) that participate in bonding and conduction. Single-electron picture with a periodic potential is sufficient for first-order discussion. Concept / Approach: When many identical atoms come together, their discrete atomic energy levels split due to quantum mechanical overlap and Pauli exclusion. With N atoms, each level splits into N closely spaced levels that merge into bands as N becomes macroscopic. The perturbation from neighboring atoms is therefore strong for valence electrons, producing a conduction band that is partially filled in metals, enabling high electrical and thermal conductivity. Step-by-Step Solution: Consider isolated atoms: discrete levels such as 3s, 3p, etc.Bring atoms together in a lattice: electron wave functions overlap and levels split.In a macroscopic crystal: the split levels form nearly continuous bands (valence and conduction).In metals: the Fermi level lies within a band or overlaps bands, allowing free electron motion. Verification / Alternative check: Free-electron and nearly free-electron models recover metallic dispersion E(k) and explain observed Fermi surfaces and conductivity. Why Other Options Are Wrong: Isolated atomic states (False option) cannot explain metallic conduction.Temperature does not erase lattice potential; band formation persists.Amorphous vs. crystalline does not negate band formation; periodicity refines, but overlap remains.Core electrons are localized; valence electrons are the ones most perturbed. Common Pitfalls: Confusing core localization with valence delocalization; thinking temperature alone creates bands. Final Answer: True — neighboring atoms significantly perturb electron states and form bands

Question 19

A solenoid core of length 10 cm provides a self-inductance of 8 mH. If the core length is doubled to 20 cm while all other quantities (turns, cross-sectional area, and permeability) remain unchanged, what will be the new self-inductance?

Accepted Answer

Introduction / Context: Self-inductance of a solenoid quantifies the flux linkage produced per unit current. For long solenoids, inductance depends on geometry and magnetic properties. Understanding how L varies with core length is essential in electromagnetics and electrical machine design. Given Data / Assumptions: Initial length l1 = 10 cm; final length l2 = 20 cm. Number of turns N, cross-sectional area A, and permeability μ remain the same. Initial inductance L1 = 8 mH. Concept / Approach: For a uniform solenoid, L = μ * N^2 * A / l. With N, A, and μ fixed, inductance is inversely proportional to length l. Therefore L2/L1 = l1/l2. Step-by-Step Solution: Write proportionality: L ∝ 1/l.Compute ratio: L2 = L1 * (l1 / l2) = 8 mH * (10 / 20).Evaluate: L2 = 8 mH * 0.5 = 4 mH. Verification / Alternative check: Dimensional reasoning: doubling magnetic path length halves magnetizing inductance for constant N, A, μ. Why Other Options Are Wrong: 16 or 32 mH imply L increases with length, contrary to L ∝ 1/l.8 mH implies no change; not correct when l doubles. Common Pitfalls: Confusing dependence on N (L ∝ N^2) with dependence on l (inverse). Final Answer: 4 mH

Question 20

Identify the physical law represented by J = σ E, where J is current density, σ is electrical conductivity, and E is the electric field intensity.

Accepted Answer

Introduction / Context: Material relations link field quantities to response variables. In conductors under steady, low-field conditions, current density is proportional to electric field. This constitutive relation is the differential (point) form of Ohm's law. Given Data / Assumptions: Uniform, isotropic conductor characterized by conductivity σ. Quasi-static, ohmic conditions (no significant displacement current or nonlinear effects). Relation: J = σ E. Concept / Approach: Ohm's law at a point states that the proportionality between J and E is σ. In integral form for a uniform conductor of length L and cross-section A, V = I R with R = L/(σ A), consistent with the differential statement. Step-by-Step Solution: Recognize J (A/m^2) and E (V/m) are linked by σ (S/m).Map to macroscopic Ohm's law: I = ∫J·dA and V = ∫E·dl.Recover V = I R with R = L/(σ A), confirming the identification. Verification / Alternative check: Dimensional check: (S/m)*(V/m) = A/m^2 matches J. Why Other Options Are Wrong: Gauss's law relates flux of E to charge; Ampère's law relates H to current; Biot–Savart gives magnetic field from currents; Faraday's law relates changing flux to induced emf. Common Pitfalls: Confusing constitutive relations with field (Maxwell) laws. Final Answer: Ohm's law

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