Q: Magnetic–Electric Analogy – Permeability Correspondence In electromagnetic–electrical analogies, magnetic permeability μ is most closely analogous to which electrical property?

Introduction / Context: Field analogies help engineers intuit equations by mapping magnetic quantities to electrical counterparts. In simple linear media, relationships B = μ H and J = σ E have similar forms, allowing permeability μ to be compared with electrical conductivity σ in terms of how strongly the medium responds to an applied field. Given Data / Assumptions: Linear, isotropic media for both electric and magnetic cases. Constitutive relations: B = μ H; J = σ E. Analogy focuses on proportionality constants governing response strength. Concept / Approach: The medium “gain” from field to flux/current is set by μ (magnetic) and σ (electric). Higher μ yields more magnetic flux density per unit magnetic field, just as higher σ yields more current density per unit electric field. By this mapping, μ ↔ σ and, correspondingly, magnetic reluctivity (1/μ) ↔ electrical resistivity (ρ = 1/σ). Step-by-Step Solution: Write constitutive pairs: B = μ H; J = σ E.Identify analog coefficients: μ corresponds to σ.Therefore, permeability is analogous to conductivity. Verification / Alternative check: In magnetic circuit analogies: Φ = F / ℜ with ℜ ∝ 1/μ mirrors Ohm’s law I = V / R with R ∝ 1/σ, again mapping μ to σ. Why Other Options Are Wrong: Resistivity corresponds to magnetic reluctivity (inverse of μ), not μ itself. Retentivity and coercivity are hysteresis characteristics without a direct linear analogue to σ. Common Pitfalls: Mixing up direct and inverse analogies; conflating hysteresis properties with constitutive constants. Final Answer: Conductivity

Q: Bohr–de Broglie Standing-Wave Condition (True/False) In the Bohr–Sommerfeld / de Broglie picture for quantized circular orbits, the circumference of an allowed electron orbit equals an integer multiple of the electron’s wavelength (2πR = nλ). True o…

Introduction / Context: De Broglie associated a wavelength λ = h/p with matter particles. The Bohr–Sommerfeld quantization condition can be viewed as a standing-wave requirement: only those orbits are stable where an integer number of wavelengths fits around the circumference, ensuring constructive interference and a stationary state. Given Data / Assumptions: Circular orbit of radius R. Wavelength λ = h/p for the electron. Old quantum theory framework (pre-Schroedinger), used as an intuitive model. Concept / Approach: The standing-wave condition is 2πR = nλ for integer n. This is equivalent to quantized angular momentum L = nħ because p = m v and L = m v R; substituting λ = h/(m v) gives m v R = nħ. Thus, both the de Broglie and Bohr views are consistent for circular orbits in hydrogen-like systems. Step-by-Step Solution: Impose standing-wave condition: 2πR = nλ.Use λ = h/p, p = m v ⇒ 2πR = n h / (m v).Rearrange to get m v R = n h / (2π) = nħ, the Bohr condition. Verification / Alternative check: The derived energy levels E_n from this quantization match the Rydberg series for hydrogen, validating the physical picture within its domain. Why Other Options Are Wrong: “False” choices contradict the established equivalence in the old quantum model. Ground-state-only or spin–orbit caveats are not part of the standing-wave criterion. Common Pitfalls: Confusing the heuristic Bohr–de Broglie model with exact quantum mechanical orbitals; the standing-wave idea is an analogy but correctly leads to L = nħ for circular motion. Final Answer: True

Q: Multi-plate parallel-plate capacitor – dependence of capacitance on number of plates For a multi-plate capacitor with n identical plates (all of area A) arranged so that alternate plates are connected together to form two interleaved sets, the total…

Introduction / Context: Interleaved multi-plate capacitors are used to obtain large capacitance in compact volumes. Understanding how total capacitance scales with the number of plates helps in preliminary design and quick estimation for filters, timing circuits, and energy storage modules. Given Data / Assumptions: There are n equally spaced, identical conducting plates of area A. Alternate plates are connected together to form two terminals. Dielectric is uniform; edge effects (fringing) are neglected. Concept / Approach: With alternate plates strapped, each adjacent pair of plates constitutes one elementary capacitor. The number of such gaps equals the number of inter-plate spaces between oppositely connected plates. For n plates, the count of effective capacitors in parallel is (n − 1). Capacitors in parallel add directly, so the total capacitance C_total is proportional to (n − 1) times the capacitance of one gap, and also proportional to the plate area A (with all other parameters fixed). Step-by-Step Solution: Number of effective dielectric gaps = n − 1.Each gap has capacitance C_gap ∝ ε * A / d (ε and spacing d fixed).Parallel addition: C_total = (n − 1) * C_gap → C_total ∝ (n − 1) * A. Verification / Alternative check: Check small cases: for n = 2 (a single parallel-plate capacitor), (n − 1) = 1 → expected. For n = 3, there are two gaps in parallel → doubling the capacitance, consistent with the formula. Why Other Options Are Wrong: n A counts plates rather than gaps. (n + 1) A and A / n have no basis in the standard interleaved topology with alternate plates shorted. Common Pitfalls: Accidentally wiring adjacent plates together (which reduces the effective number of active gaps) or forgetting that only the spaces between oppositely connected plates store energy. Final Answer: (n - 1) A

Q: Magnetization in a two-state dipole model A material contains N magnetic dipoles per m^3. If Np dipoles per m^3 are aligned parallel to the applied field and Na dipoles per m^3 are aligned anti-parallel, what is the magnetization M (A·m per m^3) of …

Introduction / Context: Magnetization M is the magnetic dipole moment per unit volume. In simple paramagnetic or diamagnetic models with two preferred orientations (parallel and anti-parallel to an applied field), M can be computed directly from the net alignment of microscopic dipoles. This question reinforces that direct counting idea. Given Data / Assumptions: Total dipole number density: N dipoles per m^3. Np aligned parallel to the field; Na aligned anti-parallel. Each dipole has the same magnetic moment magnitude m0 (A·m^2). Only the z-component along the field matters for M (sign handled by direction). Concept / Approach: By definition, magnetization M = (vector sum of dipole moments) / volume. If Np dipoles contribute +m0 each and Na contribute −m0 each along the field, the net dipole moment per unit volume is (Np − Na) * m0. Hence M equals that difference times the single-dipole moment magnitude. The remaining fraction (N − Np − Na) is either zero or oriented randomly giving negligible average along the field in this simplified two-state picture. Step-by-Step Solution: Assign signs: parallel → +m0, anti-parallel → −m0.Sum per unit volume: m_net/V = Np * m0 + (−Na * m0) = (Np − Na) * m0.Therefore magnetization: M = (Np − Na) * m0. Verification / Alternative check: When Np = Na, M = 0 (no net magnetization), which matches physical intuition. When all dipoles are parallel (Na = 0), M = N * m0, the saturation value in this idealized model. Why Other Options Are Wrong: (b) ignores opposing alignment; (c) gives saturation regardless of actual alignment; (d) is merely the negative of the correct expression and would reverse the direction convention. Common Pitfalls: Forgetting to include the sign of anti-parallel dipoles or assuming an average projection of m0 when the model explicitly counts only two orientations. Final Answer: M = (Np - Na) * m0

Q: Order of resistivity for intrinsic semiconductors at room temperature (≈300 K) The resistivity of intrinsic semiconductors at room temperature is typically of the order of:

Introduction / Context: Intrinsic semiconductors such as ultrapure silicon and germanium conduct via thermally generated electron–hole pairs. Their resistivity lies between that of good metals and insulators. Knowing the correct order of magnitude helps distinguish intrinsic from doped semiconductor behavior and from metals or dielectrics in circuit design and materials selection. Given Data / Assumptions: Temperature near 300 K (room temperature). Intrinsic (undoped) material; no intentional donors/acceptors. Macroscopic bulk sample; no nanostructure effects. Concept / Approach: Intrinsic resistivity ρ depends on intrinsic carrier concentration n_i and carrier mobilities: σ = q (n_i μ_n + n_i μ_p) = q n_i (μ_n + μ_p), with ρ = 1/σ. For Si at 300 K, n_i ≈ 10^10 cm^-3; with mobilities of order 10^3 cm^2/V·s (combined), σ is small, giving ρ around 10^3 Ω·m. Ge has much higher n_i, hence lower ρ, yet still many orders of magnitude higher than metals and far lower than typical insulators (10^8–10^12 Ω·m). Step-by-Step Solution: Recognize intrinsic carrier generation → relatively low σ compared with doped material.Estimate order: Si intrinsic ρ ~ 10^3 Ω·m; Ge is lower but not in metal-like regime.Choose the closest order-of-magnitude option: 1000 Ω-m. Verification / Alternative check: Device and materials handbooks tabulate intrinsic Si resistivity in the 10^2–10^4 Ω·m range near 300 K, consistent with the chosen order. Why Other Options Are Wrong: 1 Ω·m and 10^-3 Ω·m are characteristic of poor or good conductors respectively; 10^6 Ω·m corresponds to good insulators, not intrinsic semiconductors at room temperature. Common Pitfalls: Confusing intrinsic values with lightly doped silicon (which can be far more conductive) or with amorphous insulators. Final Answer: 1000 Ω-m

Q: Electric dipole moment of a neutral system For a system of point charges Qi located at position vectors ri, the electric dipole moment is p = Σ (Qi * ri). What must be true of the charges Qi for the system to be electrically neutral?

Introduction / Context: Dipole moment is a first-order measure of charge separation. For arbitrary charge distributions, neutrality (zero net charge) is a separate condition from the presence of a dipole moment. This question clarifies the neutrality condition for a set of discrete point charges when discussing multipole expansions in electromagnetics. Given Data / Assumptions: Point charges Qi at positions ri. Dipole moment defined as p = Σ Qi ri. Neutral system means net charge is zero. Concept / Approach: Net charge Q_total = ΣQi. Electrical neutrality requires Q_total = 0, independent of the actual value of the dipole moment p. A neutral system can still possess a nonzero dipole (e.g., +q and −q separated by vector d), and conversely a non-neutral system can have any p but is not neutral. Therefore, the necessary and sufficient condition for neutrality is ΣQi = 0. Step-by-Step Solution: Write neutrality condition: Q_total = ΣQi.Impose neutrality: Q_total = 0 → ΣQi = 0.Note independence: p = Σ Qi ri can be nonzero even when ΣQi = 0. Verification / Alternative check: Example: charges +q at r = +d/2 and −q at r = −d/2 give ΣQi = 0 and p = q d (nonzero), illustrating neutrality with finite dipole moment. Why Other Options Are Wrong: (a) or (b) alone would imply nonzero net charge unless complemented by opposite charges; (d) also yields non-neutral systems. Neutrality depends on the sum, not on individual signs being all the same. Common Pitfalls: Assuming that zero dipole moment is required for neutrality; that condition relates to charge symmetry, not to net charge. Final Answer: The algebraic sum ΣQi must be zero

Q: Bohr quantization for electron orbits If m, v, and r are the electron mass, orbital speed, and orbit radius, and h is Planck’s constant, what is the Bohr quantization condition for the allowed circular orbits in the hydrogen atom?

Introduction / Context: Bohr’s model introduces the postulate that the orbital angular momentum of an electron in a hydrogen-like atom is quantized in integer multiples of ħ = h / (2π). Although superseded by full quantum mechanics, this condition still yields correct energy levels for hydrogen and is foundational in the historical development of atomic theory. Given Data / Assumptions: Electron moves in a circular orbit under Coulomb attraction. Non-relativistic speeds; single electron atom. Quantization postulate applies to angular momentum. Concept / Approach: Bohr’s postulate: L = m v r = n ħ = n h / (2π), where n = 1, 2, 3, …. Only those orbits with angular momentum equal to an integer multiple of ħ are allowed. This leads to quantized radii r_n and energy levels E_n that match observed spectral lines via the Rydberg formula. Step-by-Step Solution: Angular momentum L = m v r.Quantization: L = n ħ = n h / (2π).Therefore allowed condition: m v r = n h / (2π). Verification / Alternative check: Using m v r = n h / (2π) and Coulomb force balance m v^2 / r = k e^2 / r^2 gives r_n ∝ n^2 and E_n ∝ −1/n^2, in agreement with hydrogen spectra. Why Other Options Are Wrong: (b) misses the 2π factor; (c) uses wrong mechanical quantity; (d) inverts the variables and is dimensionally inconsistent. Common Pitfalls: Confusing angular momentum quantization with de Broglie standing wave condition; they are equivalent but require careful use of 2π. Final Answer: m v r = n h / (2π)

Question 1

Ionic Polarization – Does It Occur in Rare (Noble) Gases? The statement claims: “In a rare (noble) gas, some atoms carry excess positive or negative charge; an electric field shifts positive ions relative to negative ions. This is ionic polarization…

Accepted Answer

Introduction / Context: Polarization mechanisms in dielectrics include electronic, ionic, orientation (dipolar), and interfacial (space-charge) polarization. Ionic polarization specifically refers to relative displacement of positive and negative ions in ionic solids (e.g., NaCl) when an electric field is applied. The statement mixes this concept with the nature of rare gases, which are monatomic and non-ionic under normal conditions. Given Data / Assumptions: Medium: a rare (noble) gas such as He, Ne, Ar at ordinary conditions. No pre-existing ions; atoms are neutral and monatomic. Linear, low-field dielectric response considered. Concept / Approach: In rare gases, the dominant polarization mechanism is electronic polarization: the electric field slightly displaces the electron cloud relative to the nucleus in each atom, inducing a dipole moment. Ionic polarization, by definition, requires ions arranged in a lattice so that entire ions shift relative to each other under the field. Since a rare gas is not an ionic crystal, the described mechanism does not apply. Only under ionization (plasma) conditions would free ions exist, which is a different physical regime, not ordinary dielectric polarization. Step-by-Step Solution: Identify the medium: neutral monatomic gas → no ionic lattice.Recall ionic polarization definition: relative motion of positive and negative ions in an ionic solid.Conclude: the correct mechanism in rare gases is electronic polarization, not ionic polarization. Verification / Alternative check: Measured relative permittivities of noble gases are very close to 1 and scale with atomic polarizability (electronic), confirming the absence of ionic contributions in the gas phase. Why Other Options Are Wrong: “True” options would require an ionic medium; that is not the case for rare gases. Common Pitfalls: Confusing “induced dipole” (electronic polarization) with “ionic polarization”; assuming any positive/negative charge separation mechanism qualifies as ionic regardless of the material class. Final Answer: False

Question 2

Bohr’s Quantization – Angular Momentum Condition (True/False) According to Bohr’s postulate for the hydrogen-like atom, only those circular orbits are stable for which the electron’s angular momentum equals an integer multiple of ħ (that is, L = n *…

Accepted Answer

Introduction / Context: Bohr’s atomic model introduced the quantization of angular momentum to explain discrete spectral lines of hydrogen. Although superseded by full quantum mechanics, the postulate captures the essence of stationary orbits and energy quantization and remains a cornerstone in the historical development of atomic theory. Given Data / Assumptions: Hydrogen-like atom, electron in a circular orbit. Bohr’s quantization condition postulated. Non-relativistic, old quantum theory context. Concept / Approach: Bohr postulated that only orbits satisfying L = nħ are permitted, where n is a positive integer, ħ = h/(2π). This leads directly to quantized radii and energies (En ∝ −1/n^2) and explains the Rydberg series. While modern quantum mechanics replaces “orbit” with stationary states, the allowed angular momenta still come in quantized units. Step-by-Step Solution: State the postulate: L = n * h / (2π) = nħ.Infer that only such orbits are stable (stationary) in the Bohr picture.Therefore, the statement is true. Verification / Alternative check: Using Coulomb force balance plus the quantization condition reproduces hydrogen spectral lines via the Rydberg formula, confirming consistency of the postulate. Why Other Options Are Wrong: False: contradicts the central Bohr assumption. Temperature or relativistic caveats are irrelevant to the original postulate’s validity within its model. Common Pitfalls: Equating Bohr orbits with exact quantum states in multi-electron atoms; overlooking that in quantum mechanics angular momentum quantization arises from boundary conditions on wavefunctions, not postulates about classical orbits. Final Answer: True

Question 3

Nichrome – Typical Application in Electrical Engineering Nichrome (a nickel–chromium alloy) is commonly used for which of the following applications?

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Introduction / Context: Material selection in electrical systems balances resistivity, temperature coefficient, oxidation resistance, and mechanical properties. Nichrome, an alloy of nickel and chromium (often with iron), is widely used as a resistance heating element due to its stable high-temperature behavior and corrosion resistance in air. Given Data / Assumptions: Nichrome composition provides high resistivity and oxidation resistance. Application environments include elevated temperatures and repeated thermal cycling. Electrical performance must be stable over long service intervals. Concept / Approach: Nichrome’s resistivity is higher than that of copper or aluminum, making it suitable where heat generation (I^2R losses) is desired in a compact form. It forms a protective oxide layer at high temperature, preventing rapid degradation. Hence, it is ideal for heater coils, toasters, furnaces, and lab heaters. In contrast, overhead lines demand low-resistivity, lightweight conductors (e.g., aluminum, ACSR), and lamp filaments typically use tungsten for its very high melting point and emissivity. Step-by-Step Solution: Match material properties (high resistivity, oxidation resistance) to the function (controlled heating).Identify typical use-case: heater coils.Exclude alternatives: overhead lines (low resistance); lamp filaments (tungsten). Verification / Alternative check: Manufacturer datasheets list nichrome wire for heating elements; power system standards specify Al/ACSR conductors; lighting standards specify tungsten filaments. Why Other Options Are Wrong: Overhead lines: efficiency loss would be excessive due to high resistivity. Lamp filaments: tungsten outperforms nichrome at incandescence temperatures. “All of the above” contradicts distinct material requirements. Common Pitfalls: Assuming any conductive wire suits all purposes; ignoring oxidation resistance at red-heat conditions. Final Answer: Heater coils (electric heating elements)

Question 4

Magnetic Response Comparison – Silicon Steel vs Cast Iron A silicon steel specimen shows flux density B = 0.5 T when subjected to magnetic field strength H = 200 A/m. For a cast iron specimen under the same H, the resulting B would be:

Accepted Answer

Introduction / Context: For magnetic materials, B = μ0 (H + M) or equivalently B = μ H in linear regions, where μ is the absolute permeability. Different ferromagnetic materials have different μ–H characteristics (B–H curves). Silicon steels are engineered to have high permeability and low hysteresis loss; cast irons generally have lower permeability and higher losses due to graphite/flakes or microstructure. Given Data / Assumptions: Same applied field H = 200 A/m to two different materials. Observed B in silicon steel is 0.5 T. Compare typical μ for cast iron vs silicon steel in a comparable magnetization region. Concept / Approach: Under the same H, the material with higher μ develops a higher B. Silicon steels are optimized for transformer/alternator cores and typically exhibit higher initial and maximum permeability than cast irons. Therefore, for identical H, cast iron will generally show a lower B than silicon steel (except in unusual saturation or microstructural anomalies not implied here). Step-by-Step Solution: Compare relative permeabilities: μr(silicon steel) > μr(cast iron).For fixed H, B ∝ μ ⇒ B_cast-iron < B_silicon-steel.Thus, B_cast-iron < 0.5 T. Verification / Alternative check: Typical B–H curves confirm silicon steels reach higher B at the same H than cast iron before saturation. Why Other Options Are Wrong: 0.5 T equal value would require equal permeability, unlikely here. More than 0.5 T contradicts lower μ of cast iron. “Cannot say” ignores standard comparative material behavior. Common Pitfalls: Confusing permeability trends at different H regions; overlooking microstructural dependence but assuming identical H still favours higher-μ materials. Final Answer: Less than 0.5 T

Question 5

Magnetic–Electric Analogy – Permeability Correspondence In electromagnetic–electrical analogies, magnetic permeability μ is most closely analogous to which electrical property?

Accepted Answer

Introduction / Context: Field analogies help engineers intuit equations by mapping magnetic quantities to electrical counterparts. In simple linear media, relationships B = μ H and J = σ E have similar forms, allowing permeability μ to be compared with electrical conductivity σ in terms of how strongly the medium responds to an applied field. Given Data / Assumptions: Linear, isotropic media for both electric and magnetic cases. Constitutive relations: B = μ H; J = σ E. Analogy focuses on proportionality constants governing response strength. Concept / Approach: The medium “gain” from field to flux/current is set by μ (magnetic) and σ (electric). Higher μ yields more magnetic flux density per unit magnetic field, just as higher σ yields more current density per unit electric field. By this mapping, μ ↔ σ and, correspondingly, magnetic reluctivity (1/μ) ↔ electrical resistivity (ρ = 1/σ). Step-by-Step Solution: Write constitutive pairs: B = μ H; J = σ E.Identify analog coefficients: μ corresponds to σ.Therefore, permeability is analogous to conductivity. Verification / Alternative check: In magnetic circuit analogies: Φ = F / ℜ with ℜ ∝ 1/μ mirrors Ohm’s law I = V / R with R ∝ 1/σ, again mapping μ to σ. Why Other Options Are Wrong: Resistivity corresponds to magnetic reluctivity (inverse of μ), not μ itself. Retentivity and coercivity are hysteresis characteristics without a direct linear analogue to σ. Common Pitfalls: Mixing up direct and inverse analogies; conflating hysteresis properties with constitutive constants. Final Answer: Conductivity

Question 6

Bohr–de Broglie Standing-Wave Condition (True/False) In the Bohr–Sommerfeld / de Broglie picture for quantized circular orbits, the circumference of an allowed electron orbit equals an integer multiple of the electron’s wavelength (2πR = nλ). True o…

Accepted Answer

Introduction / Context: De Broglie associated a wavelength λ = h/p with matter particles. The Bohr–Sommerfeld quantization condition can be viewed as a standing-wave requirement: only those orbits are stable where an integer number of wavelengths fits around the circumference, ensuring constructive interference and a stationary state. Given Data / Assumptions: Circular orbit of radius R. Wavelength λ = h/p for the electron. Old quantum theory framework (pre-Schroedinger), used as an intuitive model. Concept / Approach: The standing-wave condition is 2πR = nλ for integer n. This is equivalent to quantized angular momentum L = nħ because p = m v and L = m v R; substituting λ = h/(m v) gives m v R = nħ. Thus, both the de Broglie and Bohr views are consistent for circular orbits in hydrogen-like systems. Step-by-Step Solution: Impose standing-wave condition: 2πR = nλ.Use λ = h/p, p = m v ⇒ 2πR = n h / (m v).Rearrange to get m v R = n h / (2π) = nħ, the Bohr condition. Verification / Alternative check: The derived energy levels E_n from this quantization match the Rydberg series for hydrogen, validating the physical picture within its domain. Why Other Options Are Wrong: “False” choices contradict the established equivalence in the old quantum model. Ground-state-only or spin–orbit caveats are not part of the standing-wave criterion. Common Pitfalls: Confusing the heuristic Bohr–de Broglie model with exact quantum mechanical orbitals; the standing-wave idea is an analogy but correctly leads to L = nħ for circular motion. Final Answer: True

Question 7

Low-temperature behavior of aluminium (Al) At a temperature of about 1 K, what is the electrical state of aluminium?

Accepted Answer

Introduction / Context: Many pure metals exhibit a transition to a superconducting state at sufficiently low temperature. Aluminium (Al) is a classic example used in low-temperature physics laboratories and cryogenic electronics. This question checks your understanding of the qualitative behavior of aluminium near absolute zero and the concept of a critical temperature for superconductivity. Given Data / Assumptions: Material: high-purity aluminium bulk sample. Temperature considered: about 1 K (well below room temperature). Magnetic field is negligible, and sample is not carrying large transport currents (so we do not suppress superconductivity). Concept / Approach: Superconductivity is characterized by zero DC resistance and the expulsion of magnetic flux (Meissner effect) below a critical temperature Tc. For aluminium, Tc is approximately 1.2 K under zero magnetic field. Thus, at about 1 K (which is below Tc), aluminium is in the superconducting state. Above Tc, it behaves as an ordinary (good) conductor with finite resistivity; it is never an insulator or photoconductor in the usual sense. Step-by-Step Solution: Identify Tc for aluminium: Tc,Al ≈ 1.2 K (order-of-magnitude fact from materials data).Compare temperature: 1 K < 1.2 K → below critical temperature.Conclude: aluminium enters superconducting phase → zero DC resistivity, Meissner effect present. Verification / Alternative check: Transport measurements show a sharp drop of resistivity to instrument noise floor as T passes below Tc. Magnetization measurements show flux exclusion, confirming superconductivity. Why Other Options Are Wrong: Conductor: true only above Tc; at 1 K aluminium is more than a mere conductor. Insulator/photoconductor: aluminium is a metal, not a semiconductor; photoconduction is not the governing effect. Common Pitfalls: Confusing superconductivity with just “very low resistance” or assuming only alloys become superconducting. Purity and magnetic field do affect Tc slightly, but the qualitative answer remains unchanged. Final Answer: superconductor

Question 8

Multi-plate parallel-plate capacitor – dependence of capacitance on number of plates For a multi-plate capacitor with n identical plates (all of area A) arranged so that alternate plates are connected together to form two interleaved sets, the total…

Accepted Answer

Introduction / Context: Interleaved multi-plate capacitors are used to obtain large capacitance in compact volumes. Understanding how total capacitance scales with the number of plates helps in preliminary design and quick estimation for filters, timing circuits, and energy storage modules. Given Data / Assumptions: There are n equally spaced, identical conducting plates of area A. Alternate plates are connected together to form two terminals. Dielectric is uniform; edge effects (fringing) are neglected. Concept / Approach: With alternate plates strapped, each adjacent pair of plates constitutes one elementary capacitor. The number of such gaps equals the number of inter-plate spaces between oppositely connected plates. For n plates, the count of effective capacitors in parallel is (n − 1). Capacitors in parallel add directly, so the total capacitance C_total is proportional to (n − 1) times the capacitance of one gap, and also proportional to the plate area A (with all other parameters fixed). Step-by-Step Solution: Number of effective dielectric gaps = n − 1.Each gap has capacitance C_gap ∝ ε * A / d (ε and spacing d fixed).Parallel addition: C_total = (n − 1) * C_gap → C_total ∝ (n − 1) * A. Verification / Alternative check: Check small cases: for n = 2 (a single parallel-plate capacitor), (n − 1) = 1 → expected. For n = 3, there are two gaps in parallel → doubling the capacitance, consistent with the formula. Why Other Options Are Wrong: n A counts plates rather than gaps. (n + 1) A and A / n have no basis in the standard interleaved topology with alternate plates shorted. Common Pitfalls: Accidentally wiring adjacent plates together (which reduces the effective number of active gaps) or forgetting that only the spaces between oppositely connected plates store energy. Final Answer: (n - 1) A

Question 9

Magnetization in a two-state dipole model A material contains N magnetic dipoles per m^3. If Np dipoles per m^3 are aligned parallel to the applied field and Na dipoles per m^3 are aligned anti-parallel, what is the magnetization M (A·m per m^3) of …

Accepted Answer

Introduction / Context: Magnetization M is the magnetic dipole moment per unit volume. In simple paramagnetic or diamagnetic models with two preferred orientations (parallel and anti-parallel to an applied field), M can be computed directly from the net alignment of microscopic dipoles. This question reinforces that direct counting idea. Given Data / Assumptions: Total dipole number density: N dipoles per m^3. Np aligned parallel to the field; Na aligned anti-parallel. Each dipole has the same magnetic moment magnitude m0 (A·m^2). Only the z-component along the field matters for M (sign handled by direction). Concept / Approach: By definition, magnetization M = (vector sum of dipole moments) / volume. If Np dipoles contribute +m0 each and Na contribute −m0 each along the field, the net dipole moment per unit volume is (Np − Na) * m0. Hence M equals that difference times the single-dipole moment magnitude. The remaining fraction (N − Np − Na) is either zero or oriented randomly giving negligible average along the field in this simplified two-state picture. Step-by-Step Solution: Assign signs: parallel → +m0, anti-parallel → −m0.Sum per unit volume: m_net/V = Np * m0 + (−Na * m0) = (Np − Na) * m0.Therefore magnetization: M = (Np − Na) * m0. Verification / Alternative check: When Np = Na, M = 0 (no net magnetization), which matches physical intuition. When all dipoles are parallel (Na = 0), M = N * m0, the saturation value in this idealized model. Why Other Options Are Wrong: (b) ignores opposing alignment; (c) gives saturation regardless of actual alignment; (d) is merely the negative of the correct expression and would reverse the direction convention. Common Pitfalls: Forgetting to include the sign of anti-parallel dipoles or assuming an average projection of m0 when the model explicitly counts only two orientations. Final Answer: M = (Np - Na) * m0

Question 10

Order of resistivity for intrinsic semiconductors at room temperature (≈300 K) The resistivity of intrinsic semiconductors at room temperature is typically of the order of:

Accepted Answer

Introduction / Context: Intrinsic semiconductors such as ultrapure silicon and germanium conduct via thermally generated electron–hole pairs. Their resistivity lies between that of good metals and insulators. Knowing the correct order of magnitude helps distinguish intrinsic from doped semiconductor behavior and from metals or dielectrics in circuit design and materials selection. Given Data / Assumptions: Temperature near 300 K (room temperature). Intrinsic (undoped) material; no intentional donors/acceptors. Macroscopic bulk sample; no nanostructure effects. Concept / Approach: Intrinsic resistivity ρ depends on intrinsic carrier concentration n_i and carrier mobilities: σ = q (n_i μ_n + n_i μ_p) = q n_i (μ_n + μ_p), with ρ = 1/σ. For Si at 300 K, n_i ≈ 10^10 cm^-3; with mobilities of order 10^3 cm^2/V·s (combined), σ is small, giving ρ around 10^3 Ω·m. Ge has much higher n_i, hence lower ρ, yet still many orders of magnitude higher than metals and far lower than typical insulators (10^8–10^12 Ω·m). Step-by-Step Solution: Recognize intrinsic carrier generation → relatively low σ compared with doped material.Estimate order: Si intrinsic ρ ~ 10^3 Ω·m; Ge is lower but not in metal-like regime.Choose the closest order-of-magnitude option: 1000 Ω-m. Verification / Alternative check: Device and materials handbooks tabulate intrinsic Si resistivity in the 10^2–10^4 Ω·m range near 300 K, consistent with the chosen order. Why Other Options Are Wrong: 1 Ω·m and 10^-3 Ω·m are characteristic of poor or good conductors respectively; 10^6 Ω·m corresponds to good insulators, not intrinsic semiconductors at room temperature. Common Pitfalls: Confusing intrinsic values with lightly doped silicon (which can be far more conductive) or with amorphous insulators. Final Answer: 1000 Ω-m

Question 11

Electric dipole moment of a neutral system For a system of point charges Qi located at position vectors ri, the electric dipole moment is p = Σ (Qi * ri). What must be true of the charges Qi for the system to be electrically neutral?

Accepted Answer

Introduction / Context: Dipole moment is a first-order measure of charge separation. For arbitrary charge distributions, neutrality (zero net charge) is a separate condition from the presence of a dipole moment. This question clarifies the neutrality condition for a set of discrete point charges when discussing multipole expansions in electromagnetics. Given Data / Assumptions: Point charges Qi at positions ri. Dipole moment defined as p = Σ Qi ri. Neutral system means net charge is zero. Concept / Approach: Net charge Q_total = ΣQi. Electrical neutrality requires Q_total = 0, independent of the actual value of the dipole moment p. A neutral system can still possess a nonzero dipole (e.g., +q and −q separated by vector d), and conversely a non-neutral system can have any p but is not neutral. Therefore, the necessary and sufficient condition for neutrality is ΣQi = 0. Step-by-Step Solution: Write neutrality condition: Q_total = ΣQi.Impose neutrality: Q_total = 0 → ΣQi = 0.Note independence: p = Σ Qi ri can be nonzero even when ΣQi = 0. Verification / Alternative check: Example: charges +q at r = +d/2 and −q at r = −d/2 give ΣQi = 0 and p = q d (nonzero), illustrating neutrality with finite dipole moment. Why Other Options Are Wrong: (a) or (b) alone would imply nonzero net charge unless complemented by opposite charges; (d) also yields non-neutral systems. Neutrality depends on the sum, not on individual signs being all the same. Common Pitfalls: Assuming that zero dipole moment is required for neutrality; that condition relates to charge symmetry, not to net charge. Final Answer: The algebraic sum ΣQi must be zero

Question 12

Bohr quantization for electron orbits If m, v, and r are the electron mass, orbital speed, and orbit radius, and h is Planck’s constant, what is the Bohr quantization condition for the allowed circular orbits in the hydrogen atom?

Accepted Answer

Introduction / Context: Bohr’s model introduces the postulate that the orbital angular momentum of an electron in a hydrogen-like atom is quantized in integer multiples of ħ = h / (2π). Although superseded by full quantum mechanics, this condition still yields correct energy levels for hydrogen and is foundational in the historical development of atomic theory. Given Data / Assumptions: Electron moves in a circular orbit under Coulomb attraction. Non-relativistic speeds; single electron atom. Quantization postulate applies to angular momentum. Concept / Approach: Bohr’s postulate: L = m v r = n ħ = n h / (2π), where n = 1, 2, 3, …. Only those orbits with angular momentum equal to an integer multiple of ħ are allowed. This leads to quantized radii r_n and energy levels E_n that match observed spectral lines via the Rydberg formula. Step-by-Step Solution: Angular momentum L = m v r.Quantization: L = n ħ = n h / (2π).Therefore allowed condition: m v r = n h / (2π). Verification / Alternative check: Using m v r = n h / (2π) and Coulomb force balance m v^2 / r = k e^2 / r^2 gives r_n ∝ n^2 and E_n ∝ −1/n^2, in agreement with hydrogen spectra. Why Other Options Are Wrong: (b) misses the 2π factor; (c) uses wrong mechanical quantity; (d) inverts the variables and is dimensionally inconsistent. Common Pitfalls: Confusing angular momentum quantization with de Broglie standing wave condition; they are equivalent but require careful use of 2π. Final Answer: m v r = n h / (2π)

Question 13

Capacitance of an isolated conducting sphere in vacuum What is the capacitance C of a single isolated conducting sphere of radius R in vacuum (permittivity ε0)?

Accepted Answer

Introduction / Context: An isolated conducting sphere is a fundamental electrostatics configuration. Its capacitance defines how much charge it can store per unit potential relative to infinity. This result is widely used in high-voltage engineering, lightning protection, and as a building block for more complex capacitance calculations. Given Data / Assumptions: Perfect conductor shaped as a sphere of radius R. Surrounded by vacuum (permittivity ε0), reference potential at infinity. Electrostatic steady state; no nearby conductors or dielectrics. Concept / Approach: For a charged isolated sphere, the electric field outside is identical to that of a point charge at the center. The potential at the surface relative to infinity is V = (1 / (4π ε0)) * (Q / R). The capacitance is C = Q / V, yielding C = 4π ε0 R, linear in the radius. Step-by-Step Solution: Write surface potential: V = (1/(4π ε0)) * Q / R.Capacitance definition: C = Q / V.Compute: C = Q / [(1/(4π ε0)) * Q / R] = 4π ε0 R. Verification / Alternative check: Dimensional check: ε0 has units C^2/(N·m^2); multiplying by length R gives farads, as required. The result also matches limiting cases for large R (larger C) and small R (smaller C). Why Other Options Are Wrong: 2π ε0 R is off by a factor 2; 4π ε0 R^2 has wrong length dependence; 4π lacks units and parameters. Common Pitfalls: Confusing the sphere–plane or sphere–shell capacitance with the isolated sphere value; nearby conductors change the effective capacitance. Final Answer: 4π ε0 R

Question 14

Thermistor applications in electronics Thermistors (temperature-sensitive resistors) are commonly used for which of the following purposes?

Accepted Answer

Introduction / Context: Thermistors exhibit a strong, predictable change of resistance with temperature. Negative temperature coefficient (NTC) thermistors decrease in resistance with rising temperature, while positive temperature coefficient (PTC) types increase. Their sensitivity enables a broad range of sensing and control applications in consumer, industrial, and medical electronics. Given Data / Assumptions: Standard NTC or PTC thermistors operated within rated temperature ranges. Appropriate conditioning circuitry (bridge, linearization, or threshold detection) is available. Ambient environments do not exceed device limits. Concept / Approach: Because resistance varies strongly and monotonically with temperature, thermistors can measure temperature (convert R(T) to T), control temperature (feedback to heaters/coolers), and provide over-temperature alarms (threshold comparators). PTC thermistors also serve as resettable protectors and inrush current limiters, while NTC thermistors are common in precision sensing and compensation networks. Step-by-Step Solution: Measurement: use a Wheatstone bridge or ADC with calibration curve to convert resistance to temperature.Control: implement closed-loop control using thermistor as sensor for thermostats or PID systems.Alarm: set a threshold; if R crosses the calibrated value, drive indicators or protection relays. Verification / Alternative check: Application notes from sensor manufacturers demonstrate accuracy from ±0.1 to ±0.5 °C after calibration, confirming suitability across measurement, control, and alarm roles. Why Other Options Are Wrong: Each single option lists only one capability; thermistors perform all of them, making the comprehensive choice correct. Common Pitfalls: Self-heating due to measurement current can bias readings; always minimize current or correct for it. Nonlinearity requires linearization or table-based conversion. Final Answer: all of the above

Question 15

Hysteresis loss in ferromagnetic cores – frequency dependence Hysteresis power loss in a magnetic core operated at fixed flux density waveform amplitude is:

Accepted Answer

Introduction / Context: Core losses in transformers and inductors comprise hysteresis loss and eddy-current loss. Distinguishing their frequency dependence is essential for material selection and for estimating heat dissipation over operating ranges. Given Data / Assumptions: Magnetic core cycled sinusoidally at peak flux density B_max within the material's normal operating range. Waveform and B_max are held constant while frequency f varies. Hysteresis is characterized by a loop area proportional to B_max^n (n ≈ 1.6–2 for many steels). Concept / Approach: The energy dissipated per cycle due to hysteresis equals the loop area in the B–H plane and is approximately k_h * B_max^n per unit volume (Steinmetz form). Power loss equals energy per cycle times the number of cycles per second, so P_h ∝ f * B_max^n. At fixed B_max, this reduces to P_h ∝ f, i.e., proportional to frequency. Step-by-Step Solution: Energy per cycle per volume: W_h ≈ k_h * B_max^n.Cycles per second: f.Power loss per volume: P_h = f * W_h ≈ f * k_h * B_max^n.With B_max constant → P_h ∝ f. Verification / Alternative check: Measurements on silicon steel and ferrites show linear frequency scaling of hysteresis loss over practical ranges when B_max is held constant, while eddy-current loss scales as f^2. Why Other Options Are Wrong: (b) ignores cycling rate; (c) corresponds to eddy-current loss, not hysteresis; (d) states the general Steinmetz form but the question fixes B_max, reducing it to (a). Common Pitfalls: Mixing hysteresis and eddy losses or comparing at fixed applied voltage rather than fixed B_max (which changes the scaling). Final Answer: proportional to frequency

Question 16

Electrical resistivity of rubber (order of magnitude) What is a typical order-of-magnitude value for the resistivity of rubber at room temperature?

Accepted Answer

Introduction / Context: Rubber is widely used as an electrical insulator in cables, gloves, and protective coatings. Knowing the order of magnitude of its resistivity helps engineers judge leakage currents, insulation resistance, and safety margins in low- and high-voltage applications. Given Data / Assumptions: Room temperature and dry conditions (moisture greatly lowers surface resistance). Bulk resistivity of vulcanized or synthetic rubber compounds. DC measurement; AC losses and dielectric constants are secondary here. Concept / Approach: Insulating polymers typically have resistivities ranging from 10^8 to well above 10^12 Ω·m depending on formulation and environment. Rubber, especially in dry clean conditions, lies at the high end of this range. Therefore, values around 10^12 Ω·m are representative for order-of-magnitude estimation. Step-by-Step Solution: Identify rubber as a good insulator → very large resistivity.Compare candidate orders: 10^2 Ω·m (conductive), 10^-2 Ω·m (metal-like), 5 × 10^8 Ω·m (insulating but lower than typical high-grade rubber), 5 × 10^12 Ω·m (strong insulator).Choose the upper-range insulating value consistent with typical data sheets: 5 × 10^12 Ω·m. Verification / Alternative check: Material handbooks show rubber and PTFE in the 10^11–10^14 Ω·m bracket under dry conditions, confirming the selected order. Why Other Options Are Wrong: 10^2 Ω·m and 10^-2 Ω·m are far too low (approach semiconducting or metallic behavior). 5 × 10^8 Ω·m is possible for contaminated or damp surfaces but is not the typical high-quality bulk value. Common Pitfalls: Ignoring humidity and contamination, which reduce surface resistance dramatically; always specify environmental conditions for precise design. Final Answer: 5 × 10^12 Ω-m

Question 17

Atomic structure in classical terms In the classical (non-quantum) picture of electrons and the nucleus, how would you best describe the internal space of an atom, considering the relative sizes of the nucleus and the electron orbits/clouds?

Accepted Answer

Introduction / Context: The question probes your intuition about atomic structure from a classical perspective. Long before modern quantum mechanics refined our view, simple size arguments already suggested that atoms are mostly empty space: a tiny, massive nucleus surrounded by electrons occupying comparatively vast regions. Understanding this helps explain why matter can be compressible at nuclear scales and why scattering experiments observe what they do. Given Data / Assumptions: Classical representation is intended (no need for detailed quantum wavefunctions). Nuclear radius is orders of magnitude smaller than the atomic radius. Electrons are treated as occupying a cloud or orbits surrounding the nucleus. Concept / Approach: Compare length scales. A typical atomic radius is about 10^-10 m, while a typical nuclear radius is roughly 10^-15 m. The ratio is about 10^5 in linear dimension, which translates to about 10^15 in volume. Therefore, the dense material content (nucleus) occupies an almost negligible fraction of the atom’s volume, while the electron cloud fills the rest with extremely low mass density. Step-by-Step Solution: Define atomic radius ~ 10^-10 m; nuclear radius ~ 10^-15 m.Compute volume ratio: (10^-10 / 10^-15)^3 = (10^5)^3 = 10^15.Interpretation: the nucleus occupies ~1 part in 10^15 of the atomic volume.Hence, classically, most of an atom’s volume is empty relative to its massive nucleus. Verification / Alternative check: Rutherford’s alpha-particle scattering showed most alpha particles pass through thin metal foils with minimal deflection, implying large empty regions and a tiny, dense nucleus responsible for rare large-angle scattering. Why Other Options Are Wrong: “Essentially full” contradicts volume ratios; temperature and material choice do not change the scale separation between nucleus and atom enough to make the atom “full”. Common Pitfalls: Confusing electron cloud probability density with a classical “solid filling”; overlooking the extreme size difference between nucleus and atom. Final Answer: an atom is essentially empty

Question 18

Macroscopic (bulk) electromagnetic properties Which of the following are examples of macroscopic, bulk material quantities used in electrical and magnetic engineering?

Accepted Answer

Introduction / Context: Engineers often characterize materials using macroscopic parameters averaged over many atoms: electrical conductivity, relative permeability µr, and relative permittivity (dielectric constant) εr. These bulk properties drive circuit, antenna, insulation, and magnetic core design, without requiring atom-by-atom detail. Given Data / Assumptions: Macroscopic quantities are continuum averages applicable at scales much larger than atomic dimensions. Conductivity σ quantifies current density response to electric field. Relative permeability µr characterizes magnetic response; dielectric constant εr characterizes electric polarization response. Concept / Approach: Each listed parameter describes a constitutive relation connecting field and flux quantities in bulk media: J = σE for conduction, B = µ0µrH for magnetics, and D = ε0εrE for dielectrics. These are textbook examples of macroscopic properties used directly in design calculations. Step-by-Step Solution: Recognize conductivity as a bulk transport parameter (A/m^2 per V/m).Recognize µr as the dimensionless scaling of magnetic permeability.Recognize dielectric constant εr as the dimensionless scaling of permittivity.Since all are macroscopic, select “all of the above”. Verification / Alternative check: Handbooks list σ, µr, εr for materials like copper, ferrites, and polymers, confirming their macroscopic status and everyday engineering use. Why Other Options Are Wrong: Choosing only one or two ignores that all listed items are classic macroscopic descriptors. Common Pitfalls: Confusing microscopic carrier mobility or atomic polarizability with these bulk, averaged properties. Final Answer: all of the above

Question 19

Magnetization–field relation and the range of µr Assertion (A): For a magnetic material, M = (µr − 1) H, where M is magnetization (A/m), H is magnetic field intensity (A/m), and µr is relative permeability. Reason (R): The relative permeability µr c…

Accepted Answer

Introduction / Context: The magnetization–field relationship in linear media connects magnetization M to applied field H through magnetic susceptibility χm. Since µr = 1 + χm, we can write M in terms of µr. The assertion and reason test both the correct formula and the physically allowed range of µr. Given Data / Assumptions: Linear, isotropic magnetic material in the small-field regime. Magnetization M is dipole moment per unit volume (A/m). Definitions: B = µ0µrH and M = χm H with µr = 1 + χm. Concept / Approach: From M = χm H and µr = 1 + χm, it follows directly that M = (µr − 1) H, so the assertion is correct. The reason claims µr can range from 0 to very large numbers. While µr can indeed be very large in certain ferromagnets, it cannot be zero; diamagnets have µr slightly less than 1, paramagnets slightly greater than 1. Hence the lower bound “0” is unphysical, making the reason statement false. Step-by-Step Solution: Start with M = χm H.Use µr = 1 + χm ⇒ χm = µr − 1.Therefore M = (µr − 1) H (assertion true).Evaluate reason: practical µr values are near 1 for dia/paramagnets and up to 10^3–10^5 for ferromagnets, but never 0 ⇒ reason false. Verification / Alternative check: Materials data confirm µr < 1 for diamagnets (~0.999), slightly > 1 for paramagnets, and large for ferromagnets; no standard material has µr = 0. Why Other Options Are Wrong: Claiming the reason as correct would endorse an impossible µr = 0; claiming A false would contradict standard definitions. Common Pitfalls: Confusing absolute permeability µ with relative µr; overlooking that “any value from 0” is a sweeping, incorrect lower bound. Final Answer: A is true but R is false

Question 20

Core choice for small, high-frequency inductors For compact inductors operating at high frequency (RF range), which core material is most commonly used to obtain adequate inductance while limiting losses?

Accepted Answer

Introduction / Context: Designing small, high-frequency coils requires balancing inductance per turn against core losses. The core material strongly influences Q factor, saturation behavior, and physical size. Air cores minimize core losses but have very low inductance density. This question asks which material is most commonly used when the design requires both small size and high-frequency operation. Given Data / Assumptions: Target: high-frequency operation (from HF through VHF/UHF depending on formulation). Need compact size (limited volume), so higher permeability than air is desirable. Losses (eddy current and hysteresis) must be controlled at the operating frequency. Concept / Approach: Ferrites are ceramic, high-resistivity magnetic materials with frequency-dependent complex permeability. Their high resistivity suppresses eddy currents, and moderate relative permeability provides significant inductance in a small volume. Powdered iron cores are useful at lower RF but typically exhibit higher loss at elevated frequencies compared with suitable ferrite grades. Steel cores have prohibitive eddy-current loss at RF. Air cores offer lowest core loss but poor inductance density, limiting miniaturization when L must be large. Step-by-Step Solution: Identify need for compactness ⇒ higher µr than air is advantageous.Choose high-resistivity magnetic material to curb eddy currents ⇒ ferrite.Reject powdered iron/steel for higher RF losses; reject air when size must be minimized for a given L.Therefore, ferrite is the most common choice. Verification / Alternative check: RF inductor catalogs show abundant ferrite core offerings (toroids/rod cores) tailored for specific frequency bands; design notes highlight ferrite’s balance of µr and loss tangent. Why Other Options Are Wrong: Air cores require many turns for the same L (not compact); powdered iron has higher core loss at high RF; steel is unsuitable due to eddy currents and hysteresis. Common Pitfalls: Confusing “very-high-frequency air-core coils” (used where tiny inductance and highest Q are needed) with the general need for compact high-L inductors, where ferrite dominates. Final Answer: ferrite

Materials and Components Questions