Kathmandu University · Course Code PHYS 102

Electricity, Magnetism & Modern Physics

Master Study Package — Built from 15 End-Semester Papers (Jan 2018 – Mar 2025)

Topic Frequency Analysis Predicted Questions High-Yield Notes Common Pitfalls 40+ MCQs with Solutions

Contents

Phase 1 — Analysis & Study Guide
1.1 Topic Frequency & Weightage Table
1.2 Most Repeated & Predicted Questions
1.3 High-Yield Notes (Formula Sheet)
1.4 Common Pitfalls Table
Phase 2 — Master MCQ Bank & Answer Key

1.1 Topic Frequency & Weightage Breakdown

How this was built: Every Section B/C question (3–5 marks each) across the 2018–2025 papers was tallied by topic. Tier 1 topics appeared in 10+ of 15 papers — treat these as non-negotiable.

Topic	Priority	Freq. (papers)	Typical Question Form
Electric dipole — potential & field (short/physical dipole, coordinate-free form)	TIER 1	~12/15	Derive V and E of a dipole; show coordinate-free form
Electric field of a charged disc / ring at axial point z	TIER 1	~10/15	Find E at distance z above disc of radius R, check R→∞ and z≫R limits
Bound charge in cavity of polarized dielectric → Clausius-Mossotti	TIER 1	~9/15	Show E(center) = P/3ε₀; derive Clausius-Mossotti equation
Mutual inductance — Neumann formula & Reciprocity theorem	TIER 1	~9/15	Derive Neumann formula; state/prove reciprocity theorem
Poynting theorem & continuity equation for EM energy	TIER 1	~9/15	State & prove Poynting theorem; show ∇·S = −∂/∂t(Uem+Umech)
Magnetic vector potential (straight wire & solenoid)	TIER 1	~8/15	Define A; derive A = (μ₀I/4π)∫dl/r; find B on solenoid axis via θ₁,θ₂
Maxwell's correction to Ampère's law + Maxwell's equations in matter/vacuum	TIER 1	~8/15	How did Maxwell fix Ampère's law? Derive Maxwell's eqns / wave equation
Raman Effect — experimental setup & quantum explanation	TIER 1	~8/15	Schematic diagram + energy-level (quantum) explanation of Stokes/anti-Stokes
Energy density of E and B fields (½ε₀E², B²/2μ₀)	TIER 1	~8/15	Derive energy stored in solenoid / coaxial cable / capacitor-like system
Radioactivity: half-life, decay constant, binding energy, Q-value & threshold energy	TIER 1	~9/15	Derive T½=(ln2)Δt/ln(R₀/R); compute B.E./nucleon; derive Eth=Q(1+Ma/Mx)
Electrostatic PE of a charge configuration (n point charges / spherical shell)	TIER 2	~7/15	Derive W for 3–4 charges; generalize to n charges; energy of charged sphere
Vector calculus toolkit (div, curl, Laplacian, identities)	TIER 2	~7/15	MCQs + "show curl of gradient/div of curl = 0"; compute div/curl of given F
Ampère's law application — B field of a wire with volume current J = kr	TIER 2	~5/15	Find B inside & outside wire for non-uniform current density
Motional EMF / flux rule (moving loop in B field)	TIER 2	~5/15	Define motional emf; prove flux rule for arbitrary loop/velocity
Hysteresis loop & hysteresis loss = area of B–H curve	TIER 2	~5/15	Sketch loop; show loss/cycle = ∮H·dB
Superconductivity — Meissner effect & critical field	TIER 2	~5/15	Explain Meissner effect; Bc(T)=Bc0[1−(T/Tc)²] numericals
Dia-, para-, ferromagnetism (incl. Langevin's theory)	TIER 2	~4/15	Explain the three terms; derive χ for diamagnetic material (Langevin)
Magnetic dipole: torque & energy in uniform B	TIER 2	~4/15	Derive N=m×B, U=−m·B; max/min energy conditions
Work-energy theorem of electrodynamics	TIER 3	~3/15	State and prove dW/dt = ∫(E·J)dτ
Self-inductance of coaxial cable / solenoid	TIER 3	~4/15	Find energy stored, then self-inductance L from U=½LI²
Nuclear reactor working principle	TIER 3	~2/15	Labeled diagram + working principle description

1.2 Most Repeated & High-Probability Predicted Questions

A. Verbatim / Near-Verbatim Repeats (appeared 3+ times)

#	Question (as it recurs)	Marks
1	Find the electric field a distance z above the center of a flat circular disc of radius R carrying uniform surface charge σ. Check R→∞ and z≫R.	5
2	A spherical cavity of radius R is made inside a uniformly polarized dielectric medium. Find E at the center due to bound charges; hence obtain the Clausius–Mossotti equation.	5
3	Define mutual inductance. Derive the Neumann formula and state the reciprocity theorem.	3–5
4	State and prove Poynting's theorem; show that S satisfies ∇·S = −∂/∂t(U_em+U_mech).	5
5	Explain the Raman Effect with a schematic diagram and give its quantum-mechanical explanation with an energy-level diagram.	5
6	How did Maxwell correct Ampère's law? Derive Maxwell's equations in a medium / derive the EM wave equation in vacuum.	3–5
7	Find the magnetic field on the axis of a tightly wound solenoid (n turns/length, current I) in terms of θ₁ and θ₂; deduce the field for an infinite solenoid.	5
8	Find the approximate potential & electric field of a physical dipole at points far away; write the coordinate-free form of E_dip.	3–5
9	Derive the half-life expression T½ = (ln2)Δt / ln(R₀/R); or solve a numerical half-life problem (e.g., radon 3.8-day half-life).	3–5
10	Derive the expression for threshold energy: E_th = Q(1 + Ma/Mx).	5
11	Using Ampère's law, find B inside and outside a long wire carrying volume current density J = kr.	3–5
12	Show that hysteresis loss per cycle equals the area enclosed by the B–H hysteresis loop.	3–5
13	Calculate the binding energy per nucleon of tritium / carbon-12 given atomic masses.	3–5
14	Show that the electric field at the center of a cavity in a uniformly polarized medium is P/3ε₀.	5
15	Show that the energy stored in a magnetic field B is U = (1/2μ₀)∫B²dτ.	3–5

B. Predicted Questions for Upcoming Exams (based on cyclical pattern)

Prediction logic: Papers alternate a fixed pool of ~28 long-answer "core" questions between OR-choices. Any topic tier-1/tier-2 above that has not appeared in the last 2 papers is due to reappear.

Derive the expression for the electric field of an infinite/finite line charge (distance z above one end) — due.
Derive work done to assemble n point charges → energy of continuous charge distribution W = (ε₀/2)∫E²dτ — due.
Derive the vector potential of an infinite solenoid — due (seen only once recently).
Explain domain theory / hysteresis with B–H curve sketch — recurring every 2–3 sessions.
Derive Q-value expression for a nuclear reaction a+X→Y+b in terms of kinetic energies/masses — high probability.
Derive self-inductance & energy stored in a coaxial cable of radii a, b — recurring every 2–3 sessions.

1.3 High-Yield Notes — Formula & Concept Sheet

Electrostatics

Coulomb / Gauss

E = (1/4πε₀)Σqᵢr̂ᵢ/rᵢ² | ∮E·da = Q_enc/ε₀ | V(r) = (1/4πε₀)∫ρ/r dτ | E = −∇V

Dipole (physical, at large r)

V_dip(r,θ) = p cosθ /4πε₀r² | E_dip = p/4πε₀r³ (2cosθ r̂ + sinθ θ̂) = (1/4πε₀r³)[3(p·r̂)r̂ − p]

Disc / Ring on axis

E_disc(z) = (σ/2ε₀)[1 − z/√(z²+R²)] k̂ → as R→∞: E = σ/2ε₀ (infinite sheet) → as z≫R: E ≈ Q/4πε₀z² (point charge)

Polarization & Cavity fields

P = ε₀χ_e E | σ_b = P·n̂, ρ_b = −∇·P | E_cavity(center) = P/3ε₀ | Clausius–Mossotti: χ_e/(χ_e+3) = Nα/3ε₀

Energy

W = (ε₀/2)∫E²dτ (all space) | W_sphere(uniform Q,R) = 3Q²/20πε₀R

Magnetostatics

Biot–Savart / Ampère

B = (μ₀I/4π)∫(dl×r̂)/r² | ∮B·dl = μ₀I_enc | A(r) = (μ₀I/4π)∫dl/r (Coulomb gauge, ∇·A=0)

Wire, J = kr (volume current)

Inside (s<R): B = μ₀ k s²/3 Outside (s>R): B = μ₀kR³/3s (apply Ampère's law with I_enc=∫J·da=∫₀ˢ kr·2πr dr)

Solenoid on axis

B = (μ₀nI/2)(cosθ₂ − cosθ₁) | Infinite solenoid: B = μ₀nI (uniform, inside only)

Self / Mutual Inductance

Neumann formula: M = (μ₀/4π)∮∮ (dl₁·dl₂)/r₁₂ | Reciprocity: M₁₂ = M₂₁ | L (solenoid, per unit length) = μ₀n²πR² | U = ½LI² = (1/2μ₀)∫B²dτ

Magnetic dipole in field B

Torque N = m×B | Energy U = −m·B (max when antiparallel, min/most negative when parallel)

Magnetic materials

Electrodynamics / Maxwell

Faraday's Law / Motional EMF

∇×E = −∂B/∂t | EMF = −dΦ/dt | Motional emf: ε=∮(v×B)·dl

Maxwell's Equations (vacuum)

∇·E=ρ/ε₀ ∇·B=0 ∇×E=−∂B/∂t ∇×B=μ₀J+μ₀ε₀∂E/∂t → Wave eqn: ∇²E = μ₀ε₀∂²E/∂t², similarly for B, with c=1/√(μ₀ε₀)

In matter

∇·D=ρ_f ∇·B=0 ∇×E=−∂B/∂t ∇×H = J_f + ∂D/∂t (Maxwell's fix: adds displacement current ∂D/∂t to Ampère's law so charge continuity holds)

Poynting Theorem

S = (1/μ₀)(E×B) (energy flux, W/m²) | dW/dt = ∫(E·J)dτ | ∇·S = −∂/∂t(U_em + U_mech), U_em = ½(ε₀E²+B²/μ₀)

Modern Physics

Radioactive decay

N=N₀e^(−λt) | T½ = ln2/λ = (ln2)Δt/ln(R₀/R) | Activity R=λN

Nuclear energetics

Binding Energy = [Zm_p+(A−Z)m_n−M_nucleus]c² (use 1u=931.5 MeV) | Q = (Σm_reactants−Σm_products)c² | Threshold energy: E_th = Q(1+M_a/M_X) (Q<0, target X at rest)

Raman Effect

Stokes lines: scattered photon loses energy to molecule (ν₀−ν_vib), longer λ, more intense | Anti-Stokes: gains energy (ν₀+ν_vib), shorter λ, less intense (fewer molecules in excited vibrational state at room T)

Superconductivity

Meissner effect: B expelled from interior (perfect diamagnet, χ=−1) below Tc | B_c(T) = B_c0[1−(T/Tc)²]

1.4 Common Pitfalls — Where Marks Are Lost

Mistake	Why marks are lost	Exact Fix
Confusing E-field direction/sign for disc/dipole problems	Sign errors in θ dependence or forgetting the ẑ vs r̂ direction cost 1–2 marks even with correct magnitude	Always state the coordinate system first (cylindrical for disc, spherical for dipole) and box the final vector, not just magnitude
Skipping boundary/limit checks (R→∞, z≫R)	Examiners explicitly award 1 mark for verifying the two limiting cases — students who compute E but skip limits lose these	Always finish disc/rod problems by explicitly plugging in the two limits and stating the physical interpretation ("reduces to infinite sheet formula")
Writing Neumann formula without stating Coulomb gauge / reciprocity proof steps	Half-derivations (just quoting the formula) get partial credit only	Always start from Φ₂=∫B₁·da₂, substitute A₁, then swap integration order to prove M₁₂=M₂₁
Mixing up ∇×H=J_f+∂D/∂t vs ∇×B=μ₀J+μ₀ε₀∂E/∂t	Using vacuum form when question says "in a medium" (or vice versa) — instant deduction	Underline the keyword "matter/medium" vs "vacuum" in the question before writing equations
Forgetting the negative sign / continuity form in Poynting theorem	Poynting theorem proofs are marked step-wise; missing the −∂U/∂t sign breaks the "law of conservation" interpretation	Explicitly write "energy leaving = decrease of stored energy", so the minus sign is justified in words, not just algebra
Not converting u (atomic mass unit) to MeV correctly in binding-energy numericals	Using 1u = 1.66×10⁻²⁷ kg then forgetting to convert to MeV via c², or misusing 931.5 MeV/u factor	Memorize 1u = 931.5 MeV/c² and always compute Δm in u first, then multiply directly by 931.5 to get MeV
Sign error in Q-value / threshold energy formula	Students forget Q is negative for endothermic reactions, making E_th come out negative	State explicitly "Q is negative for endothermic reaction" before substituting into E_th = Q(1+M_a/M_X)
Applying Gauss/Ampère's law without justifying symmetry	0.5–1 mark reserved just for "by symmetry, B is tangential and constant along the loop"	Always write one line justifying the choice of Gaussian surface / Amperian loop symmetry before integrating
Confusing Stokes vs anti-Stokes intensity/wavelength in Raman effect	A very common flip — students say Stokes has shorter λ or lower intensity	Remember: Stokes = Smaller energy = longer λ = Stronger intensity (mnemonic "S-S-S")
Treating retentivity and coercivity as the same	Loses conceptual marks in hysteresis-loop questions	Retentivity = magnetization left at H=0; Coercivity = reverse field needed to bring B to 0. State both clearly on the sketch
Vector identity sign errors (BAC-CAB rule, product rules)	MCQs use one flipped sign to test rote memorization	Learn the identities from first principles once: ∇×(fA)=f∇×A−A×∇f and ∇×(∇×A)=∇(∇·A)−∇²A — derive them once, don't just memorize
Not showing units / dimensional check in fill-in-the-blank numericals	Even correct final numbers lose marks without units (A/m, C/m², J, etc.)	Always append SI unit to every fill-in-the-blank final answer

Each card shows the topic tag, four options, and a fully worked answer. Cover the answer box with your hand and self-test first (see Active Recall plan in the strategic roadmap).

Q1Vector Calculus

The divergence of the vector function F⃗ = xe⁻ˣî + yĵ − xzk̂ is:

A. (x−1)(1+e⁻ˣ)

B. (1−x)(1+e⁻ˣ)

C. (1−x)(1−e⁻ˣ)

D. (1−x)(e⁻ˣ−1)

Answer: B. ∇·F = ∂(xe⁻ˣ)/∂x + ∂y/∂y + ∂(−xz)/∂z = (e⁻ˣ−xe⁻ˣ) + 1 + (−x) = (1−x)e⁻ˣ + (1−x) = (1−x)(1+e⁻ˣ).

Q2Gauss's Law

Point charge Q₁ lies inside sphere S₁ (radius R₁); point charge Q₂ lies outside S₁ but inside concentric sphere S₂ (radius R₂). If φ₁, φ₂ are fluxes through S₁, S₂ respectively:

A. φ₁/φ₂ = Q₁/Q₂

B. φ₁/φ₂ = Q₁/(Q₁+Q₂)

C. φ₁/φ₂ = (Q₁+Q₂)/Q₂

D. φ₁/φ₂ = Q₁R₁²/Q₂R₂²

Answer: B. By Gauss's law, φ₁ = Q₁/ε₀ (only Q₁ enclosed), φ₂ = (Q₁+Q₂)/ε₀ (both enclosed). Ratio = Q₁/(Q₁+Q₂).

Q3Electrostatics — Shells

Let V and E be the potential and field inside a spherical shell carrying uniformly distributed total charge q. Which is correct everywhere inside the shell?

A. V=0, E≠0

B. V≠0, E≠0

C. V=0, E=0

D. V≠0, E=0

Answer: D. Inside a uniform spherical shell E=0 (Gauss's law, no enclosed charge), but V is constant and equal to the (non-zero) surface potential kq/R, not zero.

Q4Dielectrics

The electric potential V (in volts) as a function of distance x (in metres) is V = 10x² + 10x − 100. The electric field at x = 2 m is:

A. 20 V/m

B. −40 V/m

C. −60 V/m

D. −50 V/m

Answer: D. E = −dV/dx = −(20x+10). At x=2: E = −(40+10) = −50 V/m.

Q5Magnetism — Materials

Which one of the following materials is an example of a paramagnetic substance?

A. Copper

B. Gold

C. Cobalt

D. Platinum

Answer: D. Platinum is paramagnetic. Copper & gold are diamagnetic; cobalt is ferromagnetic.

Q6Nuclear Physics

Which process releases energy in the Sun?

A. Fission

B. Haber's process

C. Fusion

D. Radioactivity

Answer: C. Fusion — light nuclei (H isotopes) combine to form He, releasing energy (mass defect × c²).

Q7Nuclear Decay

In the alpha decay ²³⁸₉₂U → ᴬ_Z X + α, the values of Z and A for the daughter nucleus are:

A. Z=92, A=238

B. Z=90, A=234

C. Z=93, A=238

D. Z=88, A=236

Answer: B. α = ⁴₂He removes 2 protons & 2 neutrons: Z = 92−2 = 90, A = 238−4 = 234.

Q8Inductance

The self-inductance per unit length of a long solenoid of radius R and n turns per unit length carrying current I is:

A. μ₀πnR²

B. μ₀πnIR²

C. (3/4)μ₀πn²R³

D. μ₀πn²R²

Answer: D. Flux linkage per unit length = n·(nI·μ₀·πR²)= μ₀n²IπR², so L per unit length = μ₀πn²R² (independent of I).

Q9Maxwell's Equations

Which equation is a direct consequence of ∇×H = J_f + ∂D/∂t?

A. ∇(J_f·∂D/∂t) = 0

B. ∇×(J_f+∂D/∂t) = 0

C. J_f + ∂D/∂t = 0

D. ∇·(J_f+∂D/∂t) = 0

Answer: D. Taking the divergence of both sides: ∇·(∇×H)=0 identically, so ∇·(J_f+∂D/∂t)=0 — this is exactly the continuity/consistency condition Maxwell needed to fix Ampère's law.

Q10Vector Calculus

The divergence of the position vector r⃗ = xî+yĵ+zk̂ at the point (3, 2, −4) is:

A. 0

B. 3

C. 9

D. −4

Answer: B. ∇·r⃗ = ∂x/∂x+∂y/∂y+∂z/∂z = 1+1+1 = 3, a constant independent of position.

Q11Electrostatics — Sphere

The potential difference between the center and the surface of a solid sphere (radius R, uniform volume charge density ρ) is:

A. ρR²/2ε₀

B. ρR²/3ε₀

C. ρR²/6ε₀

D. ρR²/ε₀

Answer: C. V(0) = ρR²/2ε₀, V(R) = ρR²/3ε₀. Difference = ρR²/2ε₀ − ρR²/3ε₀ = ρR²/6ε₀.

Q12EM Waves

The energy per unit time per unit area transported by electromagnetic fields is called the:

A. Polarization vector

B. Poynting vector

C. Electric displacement vector

D. Magnetization vector

Answer: B. Poynting vector, S = (1/μ₀)(E×B), with SI units W/m².

Q13Electrostatics

The work done to move a charge from one point to another on an equipotential surface is:

A. Constant

B. Infinity

C. Zero

D. Negative

Answer: C. Zero — W = qΔV, and ΔV = 0 on an equipotential surface.

Q14Ferromagnetism

The area of the B–H hysteresis curve is an indication of the:

A. Energy dissipated per cycle

B. Retentivity of the material

C. Permeability of the material

D. Susceptibility of the material

Answer: A. The loop area equals ∮H·dB, the energy lost as heat per magnetization cycle.

Q15Charged Particle Motion

Cyclotron frequency of a charged particle circling in a magnetic field is independent of:

A. Charge of the particle

B. Mass of the particle

C. Speed of the particle

D. Magnetic field strength

Answer: C. Speed. f = qB/2πm — depends only on q, B, m, not on v (radius adjusts to keep f fixed).

Q16Ferromagnetism

The magnetization left on a ferromagnetic material after removal of the magnetizing field (post-saturation) is called:

A. Coercivity

B. Retentivity

C. Permeability

D. Susceptibility

Answer: B. Retentivity (remanence) — the coercivity is instead the reverse field needed to demagnetize.

Q17Superconductivity

A superconductor (below Tc) is characterized by:

A. R=0, B=0, χ_magnetic=0

B. R=0, B=∞, χ_magnetic=0

C. R=0, B=0, χ_magnetic<0

D. R=0, B=0, χ_magnetic>0

Answer: C. Meissner effect: B expelled (B=0 inside), and the material behaves as a perfect diamagnet, χ = −1 < 0.

Q18Superconductivity — Numerical

Tin has critical temperature Tc = 3.7 K with critical field Bc0 = 0.0306 T at 0 K. The critical field at T = 2 K is:

A. 0.0102 T

B. 0.0166 T

C. 0.0217 T

D. 0.0306 T

Answer: C ≈ 0.0217 T. Bc(T)=Bc0[1−(T/Tc)²] = 0.0306×[1−(2/3.7)²] = 0.0306×(1−0.2921) = 0.0306×0.7079 ≈ 0.02166 T.

Q19Magnetic Field — Wires

Two long straight parallel wires, separated by distance d, carry equal current I in opposite directions. The magnitude of B at the midpoint between them is:

A. μ₀I/(2πd)

B. μ₀I/(πd)

C. 2μ₀I/(πd)

D. Zero

Answer: C. Each wire gives B=μ₀I/(2π(d/2))=μ₀I/(πd) at the midpoint; since currents are opposite, fields add: total = 2μ₀I/(πd).

Q20Dielectrics — Numerical

The dielectric constant of salt is 5.9. The ratio of bound charge to free charge density magnitude in a unit volume is:

A. 0.831

B. 0.169

C. 5.9

D. 4.9

Answer: A ≈ 0.831. χ_e = ε_r−1 = 4.9. |ρ_b/ρ_f| = χ_e/(1+χ_e) = 4.9/5.9 ≈ 0.831.

Q21Ferromagnetism

Domain formation is a necessary feature of:

A. Diamagnetism

B. Paramagnetism

C. Ferromagnetism

D. Superconductivity

Answer: C. Ferromagnetic materials organize into domains — regions of aligned atomic magnetic moments.

Q22Polarization Units

The SI unit of polarization (P) is:

A. C/m²

B. C·m

C. C/m³

D. A/m

Answer: A. C/m² — same units as surface charge density since σ_b = P·n̂.

Q23Molecular Spectroscopy

The vibrational spectra of a molecule fall in the far infra-red region, while the rotational spectra fall in the:

A. Ultraviolet region

B. Visible region

C. Microwave region

D. X-ray region

Answer: C. Microwave region — rotational energy spacings are smaller than vibrational, so transitions occur at longer wavelengths (microwave).

Q24Gauss's Law — Geometry

A cube is inscribed in a sphere of radius r, with a positive point charge Q at the common center. The ratio of electric flux through the sphere (Φ_sphere) to that through the cube surface (Φ_cube) is:

A. 1

B. 4/3

C. 2√3/3

D. √3/2

Answer: A. 1. By Gauss's law, total flux through any closed surface enclosing charge Q is Q/ε₀ regardless of shape — sphere and cube enclose the same Q, so their total fluxes are equal.

Q25Magnetic Materials

The superconducting state is perfectly ______ in nature.

A. Diamagnetic

B. Paramagnetic

C. Ferromagnetic

D. Ferrimagnetic

Answer: A. Diamagnetic (χ = −1, complete flux expulsion — Meissner effect).

Q26Dimensional Analysis

The dimension of 1/(μ₀ε₀) is:

A. [LT⁻¹]

B. [L²T⁻²]

C. [L⁻¹T]

D. [L⁻²T²]

Answer: B. c = 1/√(μ₀ε₀), so 1/(μ₀ε₀) = c² which has dimension [L²T⁻²].

Q27Nuclear Fission — Numerical

The energy released by fission of one ²³⁵₉₂U atom is 200 MeV. The energy released when 1 kg of ²³⁵₉₂U undergoes fission is approximately:

A. 5.13×10²⁶ MeV

B. 5.13×10²⁶ J

C. 8.20×10¹³ MeV

D. 5.13×10²³ MeV

Answer: A. Number of atoms in 1 kg = (1000 g / 235 g·mol⁻¹) × 6.022×10²³ ≈ 2.563×10²⁴ atoms. Energy = 2.563×10²⁴ × 200 MeV ≈ 5.13×10²⁶ MeV.

Q28Polarizability

The electric susceptibility of a material with dielectric constant 2.26 is:

A. 2.26

B. 1.26

C. 3.26

D. 0.26

Answer: B. 1.26. χ_e = ε_r − 1 = 2.26 − 1 = 1.26 (dimensionless).

Q29Inductance — Energy

A coil has inductance 10 H and resistance 20 Ω. If a 200 V emf is applied, the energy stored in the magnetic field once current builds to its maximum value (I_max = V/R) is:

A. 200 J

B. 250 J

C. 500 J

D. 1000 J

Answer: C. 500 J. I_max = V/R = 200/20 = 10 A. U = ½LI² = ½×10×(10)² = 500 J.

Q30Radioactive Decay — Numerical

The half-life of a radioactive substance is 693 years. Its disintegration constant λ is approximately:

A. 1×10⁻³ yr⁻¹

B. 6.93×10⁻³ yr⁻¹

C. 1×10⁻² yr⁻¹

D. 6.93×10⁻⁴ yr⁻¹

Answer: A ≈ 1×10⁻³ yr⁻¹. λ = ln2/T½ = 0.693/693 = 1.0×10⁻³ per year.

Q31Nuclear Decay

If a radioactive nuclide ᴬ_Z X decays by emitting a gamma ray only, what happens to A and Z?

A. Z changes, A unchanged

B. A and Z both unchanged

C. A changes, Z unchanged

D. Both A and Z decrease by 1

Answer: B. Gamma emission is a de-excitation of the nucleus (photon only) — no change in proton or nucleon number.

Q32Vector Identities

Which of the following vector identities is NOT correct? (A,B are vector fields, f is scalar)

A. ∇×(fA) = f∇×A − A×∇f

B. ∇×(∇×A) = ∇(∇·A) − ∇²A

C. ∇×(A×B) = B×(∇×A) − A×(∇×B)

D. ∇·(A×B) = B·(∇×A) − A·(∇×B)

Answer: C is NOT correct. The true identity is ∇×(A×B) = A(∇·B) − B(∇·A) + (B·∇)A − (A·∇)B — it involves divergence and directional-derivative terms, not curl cross products as stated in C. A, B, D are standard correct vector identities.

Q33Charged Particle Motion

A charged particle is released from rest in a region with constant, parallel E and B fields. The path of the particle is a:

A. Circle

B. Helix

C. Cycloid

D. Straight line

Answer: D. Straight line. At t=0, v=0 so the magnetic force qv×B=0; the particle accelerates only along E (which is parallel to B), so it continues moving along that single line — v stays parallel to B for all time, keeping qv×B=0 throughout.

Q34Ampère's Law — Geometry

An Amperian loop is valid (line integral calculable) only if, along the loop:

A. B is tangential and varies uniformly along the loop

B. B is tangential and constant at every point on the loop

C. B is perpendicular and varies uniformly along the loop

D. B is perpendicular and constant at every point on the loop

Answer: B. Symmetry must make B tangential (so B·dl=B dl) and of constant magnitude along the chosen loop so ∮B·dl = B×(loop length) can be solved for B.

Q35Diamagnetism — Concept

A material has atomic/molecular dipoles that are randomly oriented with zero net moment in the absence of a field, arising purely from electron spin, and the atoms/molecules contain an odd number of electrons. This material is called:

A. Diamagnetic

B. Ferromagnetic

C. Paramagnetic

D. Anti-ferromagnetic

Answer: C. Paramagnetic. Odd number of electrons → net unpaired spin moment per atom (though randomly oriented without field) — hallmark of paramagnetism, unlike diamagnetism which needs no permanent atomic moment at all.

Q36Raman Spectroscopy

In Raman spectra:

A. Stokes lines have shorter wavelength and are less intense than anti-Stokes

B. Stokes lines have longer wavelength and are more intense than anti-Stokes

C. Stokes lines have longer wavelength and are less intense than anti-Stokes

D. Stokes lines have shorter wavelength and are more intense than anti-Stokes

Answer: B. Stokes photons lose energy to the molecule (ν₀−ν_vib) → longer λ; since most molecules start in the ground vibrational state, Stokes transitions are more probable/intense than anti-Stokes.

Q37Curl of Electric Field

In the region of free space with no charge or current, the curl of curl of Faraday's induced electric field E is:

A. ε₀μ₀ ∂²E/∂t²

B. μ₀ ∂²E/∂t²

C. ε₀ ∂²E/∂t²

D. (1/ε₀μ₀) ∂²E/∂t²

Answer: A. ∇×(∇×E) = ∇×(−∂B/∂t) = −∂/∂t(∇×B) = −∂/∂t(μ₀ε₀∂E/∂t) → using ∇×(∇×E)=∇(∇·E)−∇²E=−∇²E (since ∇·E=0), we get −∇²E = −μ₀ε₀∂²E/∂t² i.e. ∇×(∇×E) = μ₀ε₀ ∂²E/∂t².

Q38Solenoid Field — Geometry

For a solenoid on-axis field, the finite-solenoid result B=(μ₀nI/2)(cosθ₂−cosθ₁) reduces to B=μ₀nI for an infinite solenoid because:

A. θ₁→0°, θ₂→180°

B. θ₁→180°, θ₂→0°

C. θ₁=θ₂=90°

D. θ₁→90°, θ₂→90°

Answer: A. As the solenoid becomes infinite, the near end subtends θ₁→0° (cosθ₁→1) and far end θ₂→180° (cosθ₂→−1): B=(μ₀nI/2)(−1−1)=−μ₀nI (sign depends on convention) → magnitude μ₀nI.

Q39Magnetic Dipole Energy

A magnetic dipole possesses maximum potential energy inside a uniform magnetic field when:

A. Magnetic moment & field are antiparallel

B. Magnetic moment & field are parallel

C. The magnetic moment is zero

D. The magnetic field is zero

Answer: A. U = −m·B = −mB cosθ. Maximum U occurs at θ=180° (antiparallel), giving U=+mB, the highest (least negative/most positive) value.

Q40Beta Decay — Concept

The electron emitted in β⁻ radiation originates from:

A. Inner orbits of atoms

B. Free electrons existing in the nucleus

C. The decay of a neutron in the nucleus

D. A photon escaping from the nucleus

Answer: C. β⁻ decay: n → p + e⁻ + ν̄ₑ (a neutron converts to a proton, emitting an electron and antineutrino) — electrons are not pre-existing in the nucleus.