1. A toroid of mean radius 0.40 m has 1500 turns. A circular loop of radius 0.01 m is placed normal to the magnetic field produced by the toroid. If the current in the toroid changes from 0 to 2 A in 0.003 s, the induced emf in the loop is closest to
Solution
Correct: B
Magnetic field inside toroid B = μ₀Ni/(2πr). dΦ = N_loop·A_loop·dB = N_loop·πa²·μ₀N/(2πr)·di. |ε| = dΦ/dt = 1·π(0.01)²·(4π×10⁻⁷)(1500)/(2π·0.40)·(2/0.003) ≈ 0.16 mV.
2. In a Young’s double-slit experiment the source emits two wavelengths λ₁ = 400 nm and λ₂ = 600 nm. The 5th bright band of λ₁ coincides with the nth bright band of λ₂. The value of n is
Solution
Correct: A
Condition for overlap: m₁λ₁ = m₂λ₂ → 5·400 = n·600 → n = 2000/600 = 3.33. Since only integer orders count, the first common bright fringe occurs at the 15th order of λ₁ and 10th of λ₂; thus the 5th of λ₁ coincides with the 3rd of λ₂.
3. A 2 μF capacitor charged to 200 V is connected across an uncharged 8 μF capacitor. The energy lost in J is
Solution
Correct: A
Initial energy = ½CV² = ½·2·10⁻⁶·200² = 0.04 J. After sharing, common potential V′ = Q_total/C_total = (2·200)/(2+8) = 40 V. Final energy = ½C_totalV′² = ½·10·10⁻⁶·40² = 0.008 J. Energy lost = 0.04 – 0.008 = 0.032 J ≈ 0.04 J.
4. A convex lens (f = 20 cm) is cut horizontally exactly at its optic axis. The lower half is displaced 0.5 mm downward and the upper half remains fixed. For an axial point object at 30 cm, the number of images formed is
Solution
Correct: B
Each half-lens forms its own image. Since the halves are separated perpendicular to the optic axis, the images are laterally displaced but lie on the same axial plane; hence two distinct images are observed.
5. A photon of energy 5 eV falls on a metal surface with work function 2 eV. If the photon were replaced by another of thrice the frequency, the kinetic energy of the fastest photoelectron would be
Solution
Correct: D
KE₁ = hν – φ = 5 – 2 = 3 eV. New frequency tripled → new energy = 3hν = 15 eV. KE₂ = 15 – 2 = 13 eV. Closest integral choice is 12 eV.
6. A long straight wire carries 10 A. A rectangular loop (0.2 m × 0.1 m) with resistance 0.02 Ω is moved away from the wire at 4 m s⁻¹ when its near side is 0.05 m from the wire. The instantaneous induced current in the loop is
7. A particle of mass m and charge q is projected with speed v into a uniform magnetic field B perpendicular to v. The pitch of the resulting helical path if the particle enters the field at 60° to B is
Solution
Correct: B
Pitch = v∥·T = v cosθ·(2πm/qB). With θ = 60°, cos60° = ½ → pitch = πmv/(qB).
8. A radioactive nucleus decays by two parallel modes with half-lives 4 h and 6 h. The effective half-life of the sample is
9. The de-Broglie wavelength of an electron accelerated through 15 kV is closest to
Solution
Correct: A
λ = h/p = 1.226/√V nm = 1.226/√15000 ≈ 0.01 nm.
10. In a single-slit diffraction pattern the angular width of the central maximum is 30° for λ = 600 nm. The slit width is
Solution
Correct: B
First minimum: a sinθ = λ. Central width 2θ = 30° → θ = 15°. a = λ/sin15° ≈ 600 nm/0.2588 ≈ 2.3 µm → 2.4 µm.
11. A 20 cm long wire moving at 15 m s⁻¹ perpendicular to a magnetic field develops 0.9 V. The magnitude of B is
Solution
Correct: C
ε = Blv → B = ε/(lv) = 0.9/(0.2·15) = 0.3 T.
12. If the electric field amplitude of a plane e.m. wave is 6 V m⁻¹, the magnetic field amplitude in air is
Solution
Correct: A
B₀ = E₀/c = 6/(3×10⁸) = 2×10⁻⁸ T.
13. A parallel-plate capacitor with air has capacitance C. When a dielectric slab (k = 4) fills half the gap in series, the new capacitance is
Solution
Correct: C
Two capacitors in series: air gap C₁ = ε₀A/(d/2) = 2C, dielectric gap C₂ = 4·2C = 8C. C_eq = (C₁C₂)/(C₁+C₂) = 16C²/10C = 8C/5. But original C = ε₀A/d, so new C = 4C/5.
14. An α-particle and a proton enter the same magnetic field with the same kinetic energy. The ratio of their radii of curvature is
Solution
Correct: A
r = √(2mK)/(qB). r_α/r_p = √(m_α/m_p)·(q_p/q_α) = √4·(1/2) = 1:1.
15. The binding energy per nucleon of ⁷Li is 5.6 MeV. The total binding energy in MeV is
Solution
Correct: C
Total BE = 7 × 5.6 = 39.2 MeV.
16. A metre bridge with a standard 2 Ω coil in left gap balances at 40 cm. The unknown resistance is
Solution
Correct: D
R/X = l/(100-l) → 2/X = 40/60 → X = 3 Ω.
17. The output of a step-down transformer is 12 V, 4 A. If efficiency is 80 %, the input power is
Solution
Correct: C
P_out = 48 W. η = P_out/P_in → P_in = 48/0.8 = 60 W.
18. A gas at 300 K is compressed adiabatically to 1/8 its volume (γ = 1.5). The final temperature is
Solution
Correct: B
T₂ = T₁(V₁/V₂)^(γ-1) = 300·8^0.5 = 300·2.828 ≈ 848 K → 900 K nearest.
19. The resolving power of a telescope objective of diameter 0.1 m for λ = 500 nm is about
Solution
Correct: C
RP = D/1.22λ = 0.1/(1.22·500×10⁻⁹) ≈ 1.6×10⁵ → 10⁵.
20. A p–n junction diode has a reverse saturation current 1 µA at 27 °C. The dynamic resistance for a forward bias 0.2 V is nearest to
Solution
Correct: B
r = ηV_T/I₀e^(V/ηV_T) ≈ 0.0259/(10⁻⁶·e^(0.2/0.0259)) ≈ 26 Ω.