Q: Hall effect for semiconductor type identification — assertion–reason Assertion (A): The Hall effect can determine whether a semiconductor sample is p-type or n-type. Reason (R): In a magnetic field, electrons and holes deflect in opposite transverse…

Introduction / Context: The Hall effect is a fundamental transport phenomenon used to extract carrier sign, concentration, and mobility in semiconductors. Measuring the polarity of the Hall voltage is the simplest, most direct way to distinguish p-type from n-type material. Given Data / Assumptions: Rectangular sample with current along x, magnetic field along z. Steady, uniform fields; negligible magnetoresistance for simplicity. Conventional definitions: positive Hall coefficient indicates hole dominance. Concept / Approach: Charge carriers moving in a magnetic field experience a Lorentz force q * (v × B), deflecting them transverse to both current and field. Because electrons and holes have opposite charge, they deflect to opposite sample edges, charging the sides differently. The resulting Hall electric field builds until magnetic and electric forces balance, establishing a measurable Hall voltage with a sign that identifies the majority carrier type. Step-by-Step Solution: Drive current I along x in a sample; apply magnetic field B along z.Carriers experience transverse force along ±y depending on charge sign.Accumulated charges create Hall voltage V_H across y; its polarity corresponds to p-type (positive R_H) or n-type (negative R_H). Verification / Alternative check: Hall coefficient R_H = E_y / (J_x * B_z) has opposite signs for holes and electrons. Experimental setups routinely use this to classify doping type before device processing. Why Other Options Are Wrong: Option (b) denies the explanatory link. Options (c) and (d) contradict well-established Hall physics. Option (e) rejects both truths. Common Pitfalls: Confusing conventional current direction with electron motion; sign conventions must be applied carefully. Misinterpreting contact placement leading to an apparent sign error. Final Answer: Both A and R are true and R is correct explanation of A

Q: Quantization of orbital angular momentum in the Bohr model In early quantum theory (Bohr’s atomic model), the orbital angular momentum L of an electron is quantized in integral multiples of Planck’s constant divided by 2π. What is the smallest allow…

Introduction / Context: The Bohr model postulates that electrons revolve in stationary circular orbits around the nucleus with quantized angular momentum. Although superseded by modern quantum mechanics, this idea leads to correct spectral predictions for hydrogen and is a standard starting point for understanding angular momentum quantization. Given Data / Assumptions: Bohr postulate: angular momentum is quantized. Planck’s constant is h, and ħ = h / (2π). Ground state corresponds to principal quantum number n = 1. Concept / Approach: Bohr’s hypothesis states L = n * h / (2π) = n * ħ, where n is a positive integer. The smallest allowed angular momentum occurs for n = 1 and equals ħ = h / (2π). This discrete set of values explains line spectra via transitions between allowed orbits. Step-by-Step Solution: Write quantization rule: L = n * h / (2π).For ground state, n = 1.Thus L = h / (2π). Verification / Alternative check: Modern quantum mechanics yields the same magnitude for the orbital angular momentum eigenvalues through ħ and quantum numbers, showing Bohr’s value aligns with ħ for n = 1 in the simple picture. Why Other Options Are Wrong: Values like h, 2πh, h/π, or h/4π do not match Bohr’s postulate for the ground state. Only h/(2π) equals ħ and is correct. Common Pitfalls: Confusing h with ħ, or forgetting the factor 2π that converts between them. Final Answer: h / (2π)

Q: Effect of magnetic field on superconductivity When a material is in the superconducting state and an external magnetic field is applied, what is the correct outcome regarding its superconducting property?

Introduction / Context: Superconductors exhibit zero DC resistance and the Meissner effect (expulsion of magnetic flux). However, these properties hold only below certain critical parameters: temperature, magnetic field, and current density. Recognizing the critical field concept is essential for magnet design, power devices, and cryogenic systems. Given Data / Assumptions: Material is superconducting initially (T below critical temperature Tc). A magnetic field is applied externally. Critical field Hc (or Hc1, Hc2 for type-II) limits exist. Concept / Approach: For type-I superconductors, superconductivity is destroyed when the applied field exceeds the thermodynamic critical field Hc. For type-II superconductors, superconductivity persists in a mixed state between Hc1 and Hc2, but is destroyed above Hc2. Hence, in general, an applied field can terminate superconductivity if it is strong enough, making option (c) correct. Step-by-Step Solution: Start in superconducting state at T = critical field (Hc for type-I or Hc2 for type-II), superconductivity vanishes. Verification / Alternative check: Experimental magnetization curves show flux expulsion and sudden loss of superconductivity beyond critical fields; practical superconducting magnets operate well below Hc2 to avoid quenching. Why Other Options Are Wrong: (a) ignores critical fields; (b) introduces a “small resistivity” concept inconsistent with the definition; (d) insulator behavior is unrelated; (e) superconductors are not paramagnets in the Meissner state. Common Pitfalls: Confusing type-II mixed states with finite resistance (they still carry DC without loss below Jc), and ignoring the role of pinning and quench behavior. Final Answer: The superconducting property may be destroyed (above a critical field)

Q: Force on an electron due to a time-varying magnetic flux density through its circular orbit An electron orbits a proton in a circle of radius R. If a spatially uniform magnetic flux density B threading the orbit is increased at a rate dB/dt, what is…

Introduction / Context: A time-varying magnetic field induces a circulating electric field according to Faraday’s law. A charge on a circular path then experiences a tangential electric force. This concept underlies electromagnetic induction, eddy currents, and betatron acceleration. Given Data / Assumptions: Orbit is circular with radius R and threaded by a uniform B(t). Magnetic flux through the orbit is Φ = π R^2 B. Rate of change dB/dt is spatially uniform across the loop. Electron charge magnitude is e. Concept / Approach: Faraday’s law in integral form gives the induced emf around a closed path: emf = − dΦ/dt. For a circle, the induced tangential electric field E_t is constant around the circumference, so emf = ∮ E · dl = E_t * 2π R. Substituting Φ = π R^2 B relates E_t to dB/dt. Step-by-Step Solution: Compute induced emf: emf = − d(π R^2 B)/dt = − π R^2 (dB/dt).Set emf = E_t * 2π R → E_t = [− π R^2 (dB/dt)] / (2π R) = − (R/2) (dB/dt).Magnitude of tangential E: |E_t| = (R/2) (dB/dt).Tangential force on the electron: F = e |E_t| = (e R / 2) (dB/dt). Verification / Alternative check: Dimensional check: [e]·[R]·[dB/dt] → C·m·(T/s) = N, which is correct for force. The direction is azimuthal, consistent with Lenz’s law (sign depends on increasing or decreasing B). Why Other Options Are Wrong: (b) misses the factor 1/2; (c) uses B instead of dB/dt; (d) has wrong dependence on R; (e) magnetic fields themselves do no work, but the induced electric field does. Common Pitfalls: Confusing the roles of B and the induced E, or forgetting the circumference factor 2πR in Faraday’s integral law. Final Answer: (e R / 2) * (dB/dt)

Q: Estimating copper resistivity at elevated temperature (700 K) A copper specimen has resistivity ρ ≈ 1.8 × 10^-8 Ω·m at room temperature (~293 K). Using a linear temperature coefficient model with α ≈ 3.9 × 10^-3 K^-1, estimate the resistivity at 700…

Introduction / Context: Metallic resistivity increases with temperature due to enhanced electron–phonon scattering. For engineering estimates far from phase transitions, a linear temperature coefficient model is widely used. This question checks your ability to apply ρ(T) = ρ0 [1 + α (T − T0)]. Given Data / Assumptions: Base resistivity: ρ0 = 1.8 × 10^-8 Ω·m at T0 ≈ 293 K. Target temperature: T = 700 K. Temperature coefficient: α ≈ 3.9 × 10^-3 K^-1 (typical for copper near room temperature). Linear model valid over this range for an estimate. Concept / Approach: Use the linear relation ρ(T) = ρ0 [1 + α (T − T0)]. Compute the temperature rise and scale ρ0 accordingly to get the elevated-temperature resistivity. Select the closest option to the computed value. Step-by-Step Solution: Compute ΔT = T − T0 = 700 − 293 = 407 K.Compute the factor: 1 + α ΔT = 1 + (3.9 × 10^-3) * 407 ≈ 1 + 1.5873 ≈ 2.5873.Multiply: ρ(T) ≈ 1.8 × 10^-8 * 2.5873 ≈ 4.66 × 10^-8 Ω·m.Closest choice: 4.7 × 10^-8 Ω·m. Verification / Alternative check: Using α in the range 3.8–4.1 × 10^-3 K^-1 yields ρ(T) ≈ 4.6–4.8 × 10^-8 Ω·m, consistent with the selected value. Why Other Options Are Wrong: 2.0 and 1.6 × 10^-8 Ω·m are near or below room-temperature values; 3.0 × 10^-8 Ω·m underestimates the increase; 6.0 × 10^-8 Ω·m overestimates typical linear scaling to 700 K. Common Pitfalls: Forgetting to convert temperature to Kelvin for ΔT or using α with the wrong sign; assuming constant α far outside the fitted range without sanity checks. Final Answer: 4.7 × 10^-8 Ω·m

Q: Basic atomic number knowledge: silicon How many protons are present in the nucleus of a neutral silicon atom (i.e., what is the atomic number of silicon)?

Introduction / Context: The atomic number Z equals the number of protons in the nucleus, which also equals the number of electrons in a neutral atom. Knowing Z for common semiconductor elements is foundational in electronics and materials science. Given Data / Assumptions: Element: silicon (Si). Neutral atom: electrons = protons = Z. Concept / Approach: Silicon is in group 14 of the periodic table and has atomic number Z = 14. Therefore, a neutral silicon atom contains 14 protons (and 14 electrons). Step-by-Step Solution: Identify element: Si.Look up periodic position: group 14 → Z = 14.Conclude: number of protons = 14. Verification / Alternative check: Silicon isotopes (e.g., Si-28) differ in neutron count, not proton count; Z remains 14 for all silicon. Why Other Options Are Wrong: 4, 8, 12, and 28 correspond to other elements or mass number confusion, not Si’s proton count. Common Pitfalls: Mixing atomic number with mass number (e.g., Si-28 has 28 nucleons, not 28 protons). Final Answer: 14

Question 1

Room-temperature conductivity versus superconductivity at low temperature Elements that become superconducting only at extremely low temperatures are typically:

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Introduction / Context: Superconductivity is characterized by zero DC resistivity and flux expulsion below a critical temperature Tc. Curiously, the best normal-state conductors at room temperature (Cu, Ag, Au) typically do not exhibit superconductivity at ambient pressure, whereas many poorer conductors do at cryogenic temperatures. Given Data / Assumptions: We consider elemental superconductors under standard pressure (no high-pressure phases). “Poor conductor at room temperature” means higher resistivity compared with noble metals. Examples include Pb, Sn, Hg, Al (moderate conductivity but not the very best), which become superconducting at low Tc. Concept / Approach: Electron–phonon coupling strength and electronic structure govern Tc in conventional (BCS) superconductors. Strong coupling and favorable density of states can occur in materials that are not the very best normal conductors. Meanwhile, the noble metals have electronic structures that disfavor the Cooper pairing required for superconductivity. Step-by-Step Solution: Observe empirical trend: elements with very low room-temperature resistivity (Cu, Ag, Au) lack superconductivity at ambient pressure.Note many elements of moderate/poorer conductivity (Pb ≈ 7.2 K, Sn ≈ 3.7 K, Hg ≈ 4.2 K, Al ≈ 1.2 K) become superconducting at low T.Therefore, the best general statement is: they are relatively poor conductors at room temperature. Verification / Alternative check: Handbooks listing elemental Tc values corroborate the pattern: noble metals are not superconductors; many other metals are, but only at cryogenic temperatures. Why Other Options Are Wrong: Good conductors at room temperature: Typically not superconducting at 1 atm. Ferromagnetic: Long-range ferromagnetism is generally antagonistic to conventional superconductivity. Insulators/semiconductors: Elements that are insulators at room temperature are not the usual low-T elemental superconductors. Common Pitfalls: Assuming “good conductor” at 300 K implies superconductivity at low T; the mechanisms are distinct. Final Answer: relatively poor conductors at room temperature

Question 2

Dielectric hysteresis: does it require AC cycling? Dielectric hysteresis refers to a loop in the polarization–electric field (P–E) relationship. Under which type of excitation is hysteresis observed in practice?

Accepted Answer

Introduction / Context: Dielectric (ferroelectric) hysteresis is the path-dependent relationship between polarization P and applied electric field E, producing a closed loop when the field is cycled. This behavior is fundamental for nonvolatile memories, capacitors with voltage-dependent capacitance, and electroceramic actuators. Given Data / Assumptions: We consider materials exhibiting nonlinear, history-dependent polarization (e.g., ferroelectrics like BaTiO3). “Observed hysteresis” means tracing a closed P–E loop. Field is varied cyclically through positive and negative values. Concept / Approach: A static dc field sets a single operating point on the P–E curve; without changing the field, there is no loop—just a point (possibly with remanent polarization if the field is then removed). Hysteresis is revealed when E is swept through a cycle (typically sinusoidal or triangular), i.e., an ac or time-varying field, so that polarization follows a different path on increasing and decreasing E, forming the loop. Step-by-Step Solution: Apply dc only: polarization shifts to a steady value; no loop is traced.Apply cyclic (ac) field: increase and decrease E across coercive values.Observe remanence and coercive field leading to a closed P–E loop—this is hysteresis.Hence, hysteresis is practically observed with ac (time-varying) excitation. Verification / Alternative check: Standard Sawyer–Tower circuits drive samples with an alternating waveform to record P–E loops; commercial ferroelectric testers use ac cycling to characterize hysteresis. Why Other Options Are Wrong: dc: A constant field does not trace a loop. Both ac and dc: Misleading; the loop requires cycling. None: Contradicts well-known ferroelectric behavior. Only pulsed above breakdown: Unrelated to the fundamental requirement of cyclic excitation. Common Pitfalls: Confusing remanent polarization (measured after dc poling) with hysteresis loop measurement, which requires varying the field. Final Answer: ac

Question 3

Magnetic classification of materials in physics and electrical engineering Evaluate the statement: “Wood is a diamagnetic material.” State whether this is true or false, considering how organic, non-conductive solids interact with external magnetic …

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Introduction / Context: This question probes your understanding of magnetic material classification—diamagnetic, paramagnetic, and ferromagnetic—and asks you to place wood, a common organic solid, into the correct category. Correctly identifying diamagnetism is important in materials science and electrical engineering because it influences how materials behave in magnetic fields and in devices like transformers, magnetic sensors, and MRI environments. Given Data / Assumptions: Wood is an organic, largely non-conductive composite of cellulose, hemicellulose, and lignin. No intentional magnetic dopants or metallic inclusions are present. External fields are moderate (laboratory scale), not strong enough to cause exotic effects. Concept / Approach: Diamagnetism is a universal, weak magnetic response arising from induced orbital currents that oppose the applied magnetic field (Lenz’s law at the atomic scale). In diamagnetic materials, magnetic susceptibility chi is negative and very small in magnitude. Paramagnetism (positive, small chi) requires unpaired electron spins. Ferromagnetism (large, positive chi with hysteresis) requires cooperative spin alignment and magnetic domains, which are absent in organic materials like wood. Step-by-Step Solution: Identify electronic structure of wood’s constituents: closed-shell molecular orbitals with no persistent unpaired spins → no paramagnetism.Absence of magnetic exchange interactions and crystal structures that support domain formation → no ferromagnetism.Remaining possibility is diamagnetism: tiny, negative susceptibility due to field-induced electron motion opposing the applied field.Therefore, the correct magnetic classification for clean, dry wood is diamagnetic. Verification / Alternative check: Handbooks list organic polymers and most biological tissues as diamagnetic. Simple bench demonstrations show negligible attraction to magnets; if any effect is detectable, it is a very slight repulsion in precision balances. Why Other Options Are Wrong: “False” contradicts reference data. “True only at cryogenic temperatures” is incorrect; diamagnetism persists at all temperatures. “True only when saturated with water” confuses conductivity with magnetism. “Ferromagnetic” is categorically incorrect for wood. Common Pitfalls: Confusing electrical conductivity with magnetic response. Assuming any attraction to a magnet implies ferromagnetism; most materials are weakly dia- or paramagnetic. Final Answer: True

Question 4

Quantum numbers and angular momentum components Decide whether the statement is correct: “The magnetic quantum number determines the component of angular momentum along a prescribed (quantization) direction.”

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Introduction / Context: Atomic states are labeled by quantum numbers. Among them, the magnetic quantum number (commonly m_l for orbital angular momentum) is directly connected to the projection of angular momentum along a chosen axis, usually the z-axis. Understanding this mapping is fundamental for topics like Zeeman splitting and selection rules in spectroscopy. Given Data / Assumptions: Single-electron or many-electron atoms described by orbital quantum number l and magnetic quantum number m_l. A fixed quantization axis (typically z) is defined by an external magnetic field or convention. Non-relativistic angular momentum algebra applies. Concept / Approach: The magnetic quantum number m_l takes integer values from −l to +l. It specifies the eigenvalue of the angular momentum operator's z-component. The total magnitude of orbital angular momentum depends only on l, while the component along the quantization axis depends on m_l. This distinction explains spectral line splitting under magnetic fields (Zeeman effect), where different m_l states shift differently. Step-by-Step Solution: Angular momentum magnitude L depends on l via L = sqrt(l(l+1)) * ħ (conceptual relation).The quantized component along z is L_z = m_l * ħ.Therefore, m_l determines the specific allowed component of angular momentum along the chosen axis and explains directional properties of atomic orbitals in fields. Verification / Alternative check: Experimental Zeeman splitting patterns confirm that energy shifts are proportional to m_l (and g-factors), matching the interpretation that m_l labels the axial component. Why Other Options Are Wrong: “False” ignores core quantum principles. “True only for spin” is incorrect; both orbital (m_l) and spin (m_s) have axial projections. Statements claiming it sets only the magnitude or only the energy misunderstand the roles of l and m_l. Common Pitfalls: Confusing l (magnitude) with m_l (projection). Forgetting that external fields define the preferred axis for projection quantization. Final Answer: True

Question 5

Carrier generation–recombination balance in an intrinsic semiconductor at steady temperature At a fixed temperature T, which statement correctly describes the rates of thermal electron–hole pair generation and recombination in an intrinsic (undoped)…

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Introduction / Context: Understanding intrinsic semiconductors requires grasping the dynamic equilibrium between thermal generation of carriers and their recombination. Steady-state conditions at a fixed temperature mean that average carrier concentrations remain constant in time, which constrains the relationship between these opposing processes. Given Data / Assumptions: Material is intrinsic (no doping, no illumination). Temperature T is fixed and uniform. No external injection currents or optical generation; only thermal processes operate. Concept / Approach: In steady state, carrier concentrations n and p are time-invariant. Therefore, the net rate of change dn/dt and dp/dt must be zero. This implies that the thermal generation rate G_th equals the recombination rate R for both electrons and holes. Any persistent imbalance would produce a time evolution of n and p, contradicting steady state. Step-by-Step Solution: Let G_th be thermal generation rate and R be recombination rate.Steady state condition: dn/dt = G_th − R = 0 and dp/dt = G_th − R = 0.Thus, G_th = R; carrier concentrations remain constant at their intrinsic values n_i and p_i with n_i = p_i. Verification / Alternative check: Perturbation from steady state (e.g., sudden illumination) creates a transient where G − R ≠ 0, but as the system relaxes, recombination and generation re-balance and steady concentrations are re-established at the given temperature. Why Other Options Are Wrong: “Always lower” or “always higher” would lead to monotonic decay or growth of carrier density, violating steady-state conditions. “Any of the above” disregards the equilibrium requirement. “Generation ceases” is unphysical at finite temperature; thermal generation is continuous. Common Pitfalls: Confusing steady state (time-independent) with thermal equilibrium; both enforce G = R for intrinsic conditions. Assuming recombination dominates at low temperature; while absolute rates change, equality holds at steady state. Final Answer: Generation rate equals recombination rate at steady state

Question 6

Curie–Weiss law for paramagnetism and ferromagnetism above the Curie point Which of the following expresses the Curie–Weiss law relating magnetic susceptibility to temperature?

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Introduction / Context: The Curie–Weiss law generalizes Curie’s law by incorporating internal (molecular) fields via the Weiss temperature θ. It describes the temperature dependence of magnetic susceptibility χ for paramagnets and for ferromagnets when measured above their Curie temperature. This law is widely used to extract key parameters from high-temperature susceptibility data. Given Data / Assumptions: Linear response regime in modest applied fields. Temperatures well above any ordering temperature for the material under test. Material behaves as an assembly of local moments with mean-field coupling. Concept / Approach: Curie’s law states χ = C / T for non-interacting moments. Weiss introduced an internal field proportional to the magnetization, shifting the effective temperature scale. The result is the Curie–Weiss law: χ = C / (T − θ), where C is the Curie constant and θ (Weiss temperature) captures the strength and sign of spin–spin interactions (positive for ferromagnetic-like correlations, negative for antiferromagnetic-like correlations). Step-by-Step Solution: Start with Curie law: χ = C / T.Introduce molecular field H_m = λ M, giving an effective field H_eff = H + λ M.Mean-field treatment leads to χ = M / H = C / (T − θ) with θ proportional to λ. Verification / Alternative check: Plotting 1/χ versus T yields a straight line with slope 1/C and intercept θ on the temperature axis, a standard experimental diagnostic for magnetic materials. Why Other Options Are Wrong: χ = C / T is Curie’s law (no interaction). χ = C * T and χ = (T − θ) / C invert the correct dependence. χ = C / (T + θ) gives the wrong intercept sign and fails to describe ferromagnetic precursors correctly. Common Pitfalls: Applying Curie–Weiss below the Curie temperature where spontaneous magnetization invalidates the linear relation. Misinterpreting the sign of θ; positive θ usually indicates ferromagnetic correlations. Final Answer: χ = C / (T − θ)

Question 7

Material requirements for electrical contact applications (switches, brushes, relays) Which combination of properties is most desirable for electrical contact materials used in practical devices?

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Introduction / Context: Contacts in switches, brushes, and relays experience local hot spots, arcing, and mechanical wear. Selecting suitable materials is crucial for reliability and safety. This question emphasizes thermal and thermomechanical considerations that dominate contact performance. Given Data / Assumptions: Contacts carry appreciable current and may break/make under load. Joule heating and possible arcing occur at asperities. We focus on thermal properties and melting behavior; mechanical hardness and oxidation resistance also matter but are secondary here. Concept / Approach: High thermal conductivity spreads heat away from the microscopic junction quickly, limiting temperature rise and reducing welding. A high melting point resists softening and erosion during arcing. Common solutions include silver, copper, and refractory composites (e.g., silver–tungsten), which combine excellent conductivity with high softening temperatures and arc-erosion resistance. Step-by-Step Solution: Identify failure modes: local overheating, metal transfer, and contact welding.Mitigation: maximize thermal conduction to sink heat; select materials that remain solid and structurally sound at high temperatures.Therefore choose materials with both high thermal conductivity and high melting point. Verification / Alternative check: Catalogs list Ag, Cu, Ag-W, Ag-CdO (legacy), and newer Ag-SnO2 as contact materials, reflecting the combined need for thermal performance and arc endurance. Why Other Options Are Wrong: Low thermal conductivity promotes hot spots. Low melting point fosters welding and rapid wear. Mixing high conductivity with low melting point fails under arcing conditions; conversely, high melting point with low conductivity still overheats locally. Common Pitfalls: Overlooking oxidation/tarnish behavior, which also affects contact resistance. Assuming mechanical hardness alone ensures durability; thermal factors are equally critical. Final Answer: High thermal conductivity and high melting point

Question 8

SI unit of magnetomotive force (mmf) in electromagnetic circuits Identify the correct SI unit for magnetomotive force, considering mmf = N * I for a coil with N turns carrying current I.

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Introduction / Context: Magnetomotive force (mmf) drives magnetic flux in a magnetic circuit analogous to how electromotive force drives current in an electric circuit. Understanding correct SI units prevents confusion when relating coil parameters, reluctance, and flux. Given Data / Assumptions: Definition: mmf = N * I, where N is number of turns and I is current in amperes. Reluctance R_m relates mmf and flux via mmf = Φ * R_m. We use SI base units. Concept / Approach: Since N is a pure count (dimensionless), mmf carries the units of current. Thus, the SI unit for mmf is ampere (A). In practice, engineers often speak of ampere-turn (A·t) to emphasize that increasing turns multiplies the mmf, but this is not a distinct SI base unit. Step-by-Step Solution: Start from mmf = N * I.Dimensional analysis: N is dimensionless, I has unit A.Therefore, unit of mmf is A. Verification / Alternative check: Textbook tables list ampere (or ampere-turn as a descriptive derivative) for mmf; standards agree that the base SI unit reduces to ampere. Why Other Options Are Wrong: Weber (Wb) measures magnetic flux; tesla (T) measures flux density; volt (V) measures electrical potential difference. None represent mmf. Common Pitfalls: Believing that ampere-turn is a separate SI unit; it is a convenient expression but dimensionally equivalent to ampere. Confusing mmf with magnetic field strength H, which has units A/m. Final Answer: A (ampere)

Question 9

Effect of impurity alloying on metallic resistivity — Cu added to Ni A small amount of copper is added to a nickel conductor. Which statement best describes the change in resistivity and why?

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Introduction / Context: Alloying modifies electron scattering in metals. Even small amounts of substitutional impurities introduce lattice disorder, which affects the electrical resistivity. This question focuses on how adding copper to nickel influences low-temperature (residual) resistivity and the general behavior explained by Matthiessen’s rule. Given Data / Assumptions: Dilute substitutional alloy: Cu atoms replace some Ni sites. Phonon spectrum and carrier density remain broadly similar for small additions. We distinguish residual resistivity (impurity-limited, low T) from temperature-dependent phonon scattering. Concept / Approach: Matthiessen’s rule states that total resistivity ρ(T) ≈ ρ_residual + ρ_phonon(T). Impurity atoms break translational symmetry and act as scattering centers even at T → 0, giving a finite ρ_residual. Therefore, adding Cu increases ρ_residual. At ordinary temperatures, total resistivity typically increases as well because the added residual term adds to the phonon term. Step-by-Step Solution: Introduce impurities → enhanced electron scattering independent of temperature.This raises ρ_residual (T → 0 limit).Therefore, compared with pure Ni, Ni-Cu alloy exhibits higher resistivity, with the clearest signature at low temperature. Verification / Alternative check: Experimental curves of ρ(T) for dilute Ni-Cu alloys show an upward shift of the entire ρ(T) curve and a finite intercept at T = 0 proportional to impurity concentration. Why Other Options Are Wrong: “Resistivity decreases” ignores defect scattering. “Remains the same” contradicts alloy scattering physics. “Thermal resistivity decreases” is unsupported. “Becomes zero” is impossible in normal metals. Common Pitfalls: Assuming a purer element always conducts better; alloy disorder can dominate scattering. Confusing mobility changes with carrier concentration changes; small alloying mainly alters scattering, not electron count. Final Answer: Residual (low-temperature) resistivity increases because Cu atoms act as defect scatterers

Question 10

Magnetic force on a current-carrying conductor in a uniform field (direction and magnitude) A straight conductor carries 10 A along the +x direction within a uniform magnetic flux density of 0.2 T directed along +z. For 1 m of conductor, find the ma…

Accepted Answer

Introduction / Context: The Lorentz force on a current-carrying segment in a magnetic field is foundational in electromagnetics and electric machines. This problem reinforces right-hand rule vector products and the linear dependence of force on current, field, and length. Given Data / Assumptions: Current I = 10 A along +x. Magnetic flux density B = 0.2 T along +z. Conductor length considered L = 1 m oriented along +x. Concept / Approach: Use the vector form of magnetic force on a current element: F = I * (L × B). With L along +x and B along +z, evaluate the cross product. Magnitude is |F| = I * L * B * sin(90°) because L is perpendicular to B. Direction follows the right-hand rule for L × B. Step-by-Step Solution: Compute magnitude: |F| = 10 * 1 * 0.2 = 2 N.Determine direction: x × z = −y (since z × x = +y, then x × z = −y).Therefore, F = 2 N in the −y direction. Verification / Alternative check: Apply Fleming’s left-hand rule: First finger (B) along +z, second finger (I) along +x, thumb points to −y, confirming the direction. Why Other Options Are Wrong: “+y” reverses the cross-product orientation. Options with 0.02 N miscalculate magnitude by a factor of 100. “Zero force” is incorrect because current is perpendicular to the field, which maximizes force, not minimizes it. Common Pitfalls: Swapping the order in the cross product (L × B ≠ B × L). Using degrees instead of radians improperly in vector formulas; here the sine factor is 1. Final Answer: 2 N, −y direction

Question 11

Magnetic field in an air cavity within a magnetic core (uniform field assumption) A solenoid core of relative permeability μr contains a small cylindrical cavity of area dA and length dl. If the desired flux density inside the cavity equals the core…

Accepted Answer

Introduction / Context: Air cavities in magnetic materials are common in sensors and magnetic circuit analysis. Because flux density B relates to field strength H through material permeability, ensuring a specified B inside a cavity requires understanding how H differs across materials. Given Data / Assumptions: Core region: B_core = μ0 * μr * H_core. Cavity region (air): B_cav = μ0 * H_cav. We desire B_cav = B0, equal to the core’s B0 (assumed uniform for the argument). Linear, isotropic materials and small cavity so fringing is neglected. Concept / Approach: Since permeability in air is μ0 and in core is μ0 * μr (μr > 1), the same B requires larger H in the lower-permeability region. Thus, to keep B_cav equal to B0, the magnetic field strength in the cavity must be larger than that in the core by a factor approximately μr. Step-by-Step Solution: Set equality of desired flux densities: B_cav = B_core = B0.Use constitutive relations: μ0 * H_cav = μ0 * μr * H_core.Cancel μ0 and solve: H_cav = μr * H_core.Conclusion: The cavity requires a stronger H than the surrounding core to achieve the same B. Verification / Alternative check: Finite-element simulations of small air gaps in magnetic circuits show elevated H inside the gap relative to the core for continuous B across the interface (normal component of B is continuous neglecting surface currents). Why Other Options Are Wrong: Equal fields would yield B_cav < B_core. Weaker field inside the cavity would reduce B further. “Arbitrary” and “zero” ignore constitutive physics. Common Pitfalls: Confusing continuity: normal B is continuous, while tangential H may change with permeability jump. Forgetting that air has far lower permeability, demanding higher H to reach the same B. Final Answer: Field inside the cavity must be stronger than the field in the core

Question 12

Hall effect for semiconductor type identification — assertion–reason Assertion (A): The Hall effect can determine whether a semiconductor sample is p-type or n-type. Reason (R): In a magnetic field, electrons and holes deflect in opposite transverse…

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Introduction / Context: The Hall effect is a fundamental transport phenomenon used to extract carrier sign, concentration, and mobility in semiconductors. Measuring the polarity of the Hall voltage is the simplest, most direct way to distinguish p-type from n-type material. Given Data / Assumptions: Rectangular sample with current along x, magnetic field along z. Steady, uniform fields; negligible magnetoresistance for simplicity. Conventional definitions: positive Hall coefficient indicates hole dominance. Concept / Approach: Charge carriers moving in a magnetic field experience a Lorentz force q * (v × B), deflecting them transverse to both current and field. Because electrons and holes have opposite charge, they deflect to opposite sample edges, charging the sides differently. The resulting Hall electric field builds until magnetic and electric forces balance, establishing a measurable Hall voltage with a sign that identifies the majority carrier type. Step-by-Step Solution: Drive current I along x in a sample; apply magnetic field B along z.Carriers experience transverse force along ±y depending on charge sign.Accumulated charges create Hall voltage V_H across y; its polarity corresponds to p-type (positive R_H) or n-type (negative R_H). Verification / Alternative check: Hall coefficient R_H = E_y / (J_x * B_z) has opposite signs for holes and electrons. Experimental setups routinely use this to classify doping type before device processing. Why Other Options Are Wrong: Option (b) denies the explanatory link. Options (c) and (d) contradict well-established Hall physics. Option (e) rejects both truths. Common Pitfalls: Confusing conventional current direction with electron motion; sign conventions must be applied carefully. Misinterpreting contact placement leading to an apparent sign error. Final Answer: Both A and R are true and R is correct explanation of A

Question 13

Quantization of orbital angular momentum in the Bohr model In early quantum theory (Bohr’s atomic model), the orbital angular momentum L of an electron is quantized in integral multiples of Planck’s constant divided by 2π. What is the smallest allow…

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Introduction / Context: The Bohr model postulates that electrons revolve in stationary circular orbits around the nucleus with quantized angular momentum. Although superseded by modern quantum mechanics, this idea leads to correct spectral predictions for hydrogen and is a standard starting point for understanding angular momentum quantization. Given Data / Assumptions: Bohr postulate: angular momentum is quantized. Planck’s constant is h, and ħ = h / (2π). Ground state corresponds to principal quantum number n = 1. Concept / Approach: Bohr’s hypothesis states L = n * h / (2π) = n * ħ, where n is a positive integer. The smallest allowed angular momentum occurs for n = 1 and equals ħ = h / (2π). This discrete set of values explains line spectra via transitions between allowed orbits. Step-by-Step Solution: Write quantization rule: L = n * h / (2π).For ground state, n = 1.Thus L = h / (2π). Verification / Alternative check: Modern quantum mechanics yields the same magnitude for the orbital angular momentum eigenvalues through ħ and quantum numbers, showing Bohr’s value aligns with ħ for n = 1 in the simple picture. Why Other Options Are Wrong: Values like h, 2πh, h/π, or h/4π do not match Bohr’s postulate for the ground state. Only h/(2π) equals ħ and is correct. Common Pitfalls: Confusing h with ħ, or forgetting the factor 2π that converts between them. Final Answer: h / (2π)

Question 14

Induced versus permanent magnetic dipole moments Evaluate the statement: “The induced magnetic dipole moment of a material is independent of its permanent magnetic dipole moment.” Decide whether this statement is true or false in the context of magn…

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Introduction / Context: Magnetization arises from two broad mechanisms: (i) field-induced moments (e.g., diamagnetism and the linear part of paramagnetism) and (ii) permanent magnetic dipoles associated with unpaired electron spins and domain structures (paramagnets, ferromagnets). Understanding their independence clarifies magnetic responses across materials. Given Data / Assumptions: Total magnetic moment is the sum of permanent and induced contributions. Induced moment depends primarily on the applied field and the material’s magnetic polarizability/susceptibility. Permanent moment exists irrespective of the external field (though it can reorient under the field). Concept / Approach: The induced magnetic dipole moment is created by the field’s action on electronic orbits/spins and is proportional to the applied field for small fields: m_induced ∝ H (or B). The permanent moment, m_perm, is intrinsic to the material (e.g., unpaired spins, domains). To first order, m_induced is independent of m_perm; the total moment is m_total = m_perm + m_induced. Couplings can exist at large fields or in nonlinear regimes, but the basic statement about independence of the induced component from the inherent permanent moment is correct in standard linear-response theory. Step-by-Step Solution: Identify contributions: m_total = m_perm + m_induced.For small fields, m_induced = χ H (linear response).Therefore m_induced is set by χ and H, not by m_perm; hence independence. Verification / Alternative check: Materials with zero permanent moments (pure diamagnets) still exhibit induced magnetization; ferromagnets with large permanent moments also show an additional field-induced component (differential susceptibility). Why Other Options Are Wrong: Stating “False” assumes dependence; options restricting to special materials or extreme conditions ignore the general linear regime. Common Pitfalls: Confusing vector alignment effects of permanent moments with the creation of induced moments; mixing nonlinear saturation behavior with small-signal response. Final Answer: True

Question 15

Effect of magnetic field on superconductivity When a material is in the superconducting state and an external magnetic field is applied, what is the correct outcome regarding its superconducting property?

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Introduction / Context: Superconductors exhibit zero DC resistance and the Meissner effect (expulsion of magnetic flux). However, these properties hold only below certain critical parameters: temperature, magnetic field, and current density. Recognizing the critical field concept is essential for magnet design, power devices, and cryogenic systems. Given Data / Assumptions: Material is superconducting initially (T below critical temperature Tc). A magnetic field is applied externally. Critical field Hc (or Hc1, Hc2 for type-II) limits exist. Concept / Approach: For type-I superconductors, superconductivity is destroyed when the applied field exceeds the thermodynamic critical field Hc. For type-II superconductors, superconductivity persists in a mixed state between Hc1 and Hc2, but is destroyed above Hc2. Hence, in general, an applied field can terminate superconductivity if it is strong enough, making option (c) correct. Step-by-Step Solution: Start in superconducting state at T < Tc.Increase external field H.When H >= critical field (Hc for type-I or Hc2 for type-II), superconductivity vanishes. Verification / Alternative check: Experimental magnetization curves show flux expulsion and sudden loss of superconductivity beyond critical fields; practical superconducting magnets operate well below Hc2 to avoid quenching. Why Other Options Are Wrong: (a) ignores critical fields; (b) introduces a “small resistivity” concept inconsistent with the definition; (d) insulator behavior is unrelated; (e) superconductors are not paramagnets in the Meissner state. Common Pitfalls: Confusing type-II mixed states with finite resistance (they still carry DC without loss below Jc), and ignoring the role of pinning and quench behavior. Final Answer: The superconducting property may be destroyed (above a critical field)

Question 16

Magnetic classification of nickel at room temperature Decide whether the statement “Nickel is a non-magnetic material” is true or false. Base your answer on the fundamental magnetic behavior of nickel at room temperature.

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Introduction / Context: Elemental nickel is a classic ferromagnetic metal alongside iron and cobalt. Recognizing ferromagnetic materials is essential for core selection, magnetic shielding, and actuator design. Given Data / Assumptions: Room-temperature behavior is considered. Nickel’s Curie temperature is well above room temperature. Bulk nickel exhibits strong magnetic ordering. Concept / Approach: Ferromagnetic materials have spontaneous magnetization due to exchange interactions and aligned domains. Nickel remains ferromagnetic at room temperature, becoming paramagnetic only above its Curie temperature. Therefore, the statement calling nickel “non-magnetic” at room temperature is false. Step-by-Step Solution: Identify nickel’s magnetic class: ferromagnetic.Check operating temperature vs. Curie temperature.Since room temperature < Curie temperature, nickel is magnetic → statement false. Verification / Alternative check: Simple magnet tests show strong attraction; nickel alloys are widely used in magnetic components (e.g., permalloy). Why Other Options Are Wrong: Nickel does not require sub-zero temperatures to be magnetic and is not non-magnetic merely due to form factor. Common Pitfalls: Confusing “non-magnetic” with “weakly magnetic” or assuming all stainless steels behave like nickel (many stainless grades are austenitic and weakly magnetic, but that does not generalize to elemental nickel). Final Answer: False

Question 17

Force on an electron due to a time-varying magnetic flux density through its circular orbit An electron orbits a proton in a circle of radius R. If a spatially uniform magnetic flux density B threading the orbit is increased at a rate dB/dt, what is…

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Introduction / Context: A time-varying magnetic field induces a circulating electric field according to Faraday’s law. A charge on a circular path then experiences a tangential electric force. This concept underlies electromagnetic induction, eddy currents, and betatron acceleration. Given Data / Assumptions: Orbit is circular with radius R and threaded by a uniform B(t). Magnetic flux through the orbit is Φ = π R^2 B. Rate of change dB/dt is spatially uniform across the loop. Electron charge magnitude is e. Concept / Approach: Faraday’s law in integral form gives the induced emf around a closed path: emf = − dΦ/dt. For a circle, the induced tangential electric field E_t is constant around the circumference, so emf = ∮ E · dl = E_t * 2π R. Substituting Φ = π R^2 B relates E_t to dB/dt. Step-by-Step Solution: Compute induced emf: emf = − d(π R^2 B)/dt = − π R^2 (dB/dt).Set emf = E_t * 2π R → E_t = [− π R^2 (dB/dt)] / (2π R) = − (R/2) (dB/dt).Magnitude of tangential E: |E_t| = (R/2) (dB/dt).Tangential force on the electron: F = e |E_t| = (e R / 2) (dB/dt). Verification / Alternative check: Dimensional check: [e]·[R]·[dB/dt] → C·m·(T/s) = N, which is correct for force. The direction is azimuthal, consistent with Lenz’s law (sign depends on increasing or decreasing B). Why Other Options Are Wrong: (b) misses the factor 1/2; (c) uses B instead of dB/dt; (d) has wrong dependence on R; (e) magnetic fields themselves do no work, but the induced electric field does. Common Pitfalls: Confusing the roles of B and the induced E, or forgetting the circumference factor 2πR in Faraday’s integral law. Final Answer: (e R / 2) * (dB/dt)

Question 18

Estimating copper resistivity at elevated temperature (700 K) A copper specimen has resistivity ρ ≈ 1.8 × 10^-8 Ω·m at room temperature (~293 K). Using a linear temperature coefficient model with α ≈ 3.9 × 10^-3 K^-1, estimate the resistivity at 700…

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Introduction / Context: Metallic resistivity increases with temperature due to enhanced electron–phonon scattering. For engineering estimates far from phase transitions, a linear temperature coefficient model is widely used. This question checks your ability to apply ρ(T) = ρ0 [1 + α (T − T0)]. Given Data / Assumptions: Base resistivity: ρ0 = 1.8 × 10^-8 Ω·m at T0 ≈ 293 K. Target temperature: T = 700 K. Temperature coefficient: α ≈ 3.9 × 10^-3 K^-1 (typical for copper near room temperature). Linear model valid over this range for an estimate. Concept / Approach: Use the linear relation ρ(T) = ρ0 [1 + α (T − T0)]. Compute the temperature rise and scale ρ0 accordingly to get the elevated-temperature resistivity. Select the closest option to the computed value. Step-by-Step Solution: Compute ΔT = T − T0 = 700 − 293 = 407 K.Compute the factor: 1 + α ΔT = 1 + (3.9 × 10^-3) * 407 ≈ 1 + 1.5873 ≈ 2.5873.Multiply: ρ(T) ≈ 1.8 × 10^-8 * 2.5873 ≈ 4.66 × 10^-8 Ω·m.Closest choice: 4.7 × 10^-8 Ω·m. Verification / Alternative check: Using α in the range 3.8–4.1 × 10^-3 K^-1 yields ρ(T) ≈ 4.6–4.8 × 10^-8 Ω·m, consistent with the selected value. Why Other Options Are Wrong: 2.0 and 1.6 × 10^-8 Ω·m are near or below room-temperature values; 3.0 × 10^-8 Ω·m underestimates the increase; 6.0 × 10^-8 Ω·m overestimates typical linear scaling to 700 K. Common Pitfalls: Forgetting to convert temperature to Kelvin for ΔT or using α with the wrong sign; assuming constant α far outside the fitted range without sanity checks. Final Answer: 4.7 × 10^-8 Ω·m

Question 19

Basic atomic number knowledge: silicon How many protons are present in the nucleus of a neutral silicon atom (i.e., what is the atomic number of silicon)?

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Introduction / Context: The atomic number Z equals the number of protons in the nucleus, which also equals the number of electrons in a neutral atom. Knowing Z for common semiconductor elements is foundational in electronics and materials science. Given Data / Assumptions: Element: silicon (Si). Neutral atom: electrons = protons = Z. Concept / Approach: Silicon is in group 14 of the periodic table and has atomic number Z = 14. Therefore, a neutral silicon atom contains 14 protons (and 14 electrons). Step-by-Step Solution: Identify element: Si.Look up periodic position: group 14 → Z = 14.Conclude: number of protons = 14. Verification / Alternative check: Silicon isotopes (e.g., Si-28) differ in neutron count, not proton count; Z remains 14 for all silicon. Why Other Options Are Wrong: 4, 8, 12, and 28 correspond to other elements or mass number confusion, not Si’s proton count. Common Pitfalls: Mixing atomic number with mass number (e.g., Si-28 has 28 nucleons, not 28 protons). Final Answer: 14

Question 20

Polarization of a rare gas in an electric field A rare (noble) gas with number density N (atoms per m^3) is subjected to a uniform electric field E. In the linear regime, the macroscopic electric dipole moment per unit volume (polarization P) is pro…

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Introduction / Context: In dielectric media, polarization P gives the dipole moment per unit volume. For dilute gases such as noble gases, induced dipoles arise via electronic polarizability, and the Clausius–Mossotti/Lorentz–Lorenz relations connect microscopic polarizability to macroscopic P. Given Data / Assumptions: Dilute gas, non-interacting atoms with electronic polarizability αe. Linear response (small fields). Number density N (atoms/m^3) and applied field E. Concept / Approach: Each atom acquires an induced dipole p_induced = αe E (in SI, αe may be written with ε0 depending on convention). The polarization is P = N * p_induced, so P ∝ N * E. Hence, doubling the density or the field doubles P in the linear regime. Step-by-Step Solution: Write single-atom response: p = αe E.Sum over N atoms in unit volume: P = N αe E.Conclude P ∝ N and P ∝ E. Verification / Alternative check: Macroscopic dielectric constant εr relates to microscopic αe via Lorentz–Lorenz relation; for small αe N, the linear dependence is clear. Why Other Options Are Wrong: Independence from N or E contradicts linear polarization theory; E^2 dependence appears in nonlinear optics, not in this small-signal regime. Common Pitfalls: Confusing microscopic polarizability with relative permittivity, or assuming strong-field nonlinear effects at ordinary field strengths. Final Answer: Proportional to N and to E (P ∝ N E)

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