48. As shown in the figure, three charged particles (1, 2, and 3) are positioned along an x-axis. Particles 1 and 2 are fixed in place. Particle 3 is free to move, but the net electrostatic force exerted on it by Particles 1 and 2 happens to be zero. Given that the distance between Particles 1 and 2 is L₁₂, the distance between Particles 2 and 3 is L₂₃, and they satisfy the relationship L₁₂ = 2 L₂₃. What is the ratio of the charges q₁/q₂?

(A) 4
(B) -4
(C) 9
(D) -9
(E) -8

49. A parallel-plate capacitor has an initial capacitance of C₀ = 6 μF. The capacitor is first connected to a 100 V battery until fully charged and then disconnected from the power source. A dielectric slab with a dielectric constant κ = 3 is then inserted symmetrically between the plates. The thickness of the slab is exactly half of the plate separation distance. What is the external work involved in the process of inserting the dielectric slab? (A) 10 mJ (B) -10 mJ (C) 15 mJ (D) -15 mJ (E) 0

50. In two LRC series AC circuits, suppose circuit 1 has R₁ = 60 Ω, C₁ = 300 μF, L₁ = 0.80 H and circuit 2 has R₂ = 120 Ω, C₂ = 600 μF, L₂ = 1.60 H. Find the ratio of , where f₁ and f₂ are the resonance frequencies of circuits 1 and 2, respectively. (A) (B) (C) 1 (D)(E) 2

51. A long solenoid has 5000 turns per meter and a cross-sectional area of 2.0×10⁻⁴ m². A larger secondary coil with 500 turns, which has a cross-sectional area of 5.0×10⁻⁴ m², and a total resistance of 2π Ω is placed outside and coaxial with the central part of the solenoid. The current in the solenoid drops uniformly from 4.0 A to 0 A in a time interval of Δt = 0.1 s. What is the magnitude of the induced current in the external coil during this time interval? (Given: permeability constant μ₀ = 4π × 10⁻⁷ T·m/A) (A) 4.0 mA (B) 5.0 mA (C) 6.0 mA (D) 7.0 mA (E) 8.0 mA

52. A wire is bent into a three-quarter circular loop of radius R and carries a steady current I. It is placed in a uniform magnetic field perpendicular to the plane of the three-quarter circular loop. The magnitude of net magnetic force on the wire is ______. (A) 0 (B) πRIB (C) 2RIB (D) (3/4)πRIB (E) √2 RIB

53. A circular coil 0.05 m in radius, with 50 turns of wire, lies on a horizontal plane. It carries a counterclockwise current of 8.0 A. The coil is in a uniform 1.50 T magnetic field directed toward the +Y direction. Find the torque on the coil. (Given: Unit vectors î, ĵ and k̂ point in the directions of the positive X-, Y- and Z- axes and have magnitude of 1) (A) -0.03 π î (B) -1.50 π î (C) 0.03 π k̂ (D) 1.5 π k̂ (E) 0.03 π ĵ

54. A beam of monochromatic light with a wavelength of λ = 600 nm is incident perpendicularly onto a diffraction grating with 500 lines/mm. What is the maximum diffraction order (m) that can be observed on the screen? (Note: Consider only the orders on one side of the central maximum, excluding m = 0) (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

55. Alice directs a beam of 0.12 nm x rays at certain planes of a silicon crystal. As she increases the angle of incidence of the beam from zero, the first strong interference maximum occurs when the beam makes an angle of θ = tan⁻¹() with the planes. How far apart are the planes? (A) 0.045 nm (B) 0.075 nm (C) 0.1 nm (D) 0.15 nm (E) 0.2 nm

56. A rectangular conducting loop with resistance R lies in the same plane as a long straight wire carrying a steady current i. The nearest edge of the loop is initially at a distance a from the wire, and the loop has the width w and length L as shown in the figure. The loop is then pulled directly away from the wire with constant speed v. As the motion just begins, the induced current in the loop is ______. (Given: permeability constant μ₀) (A)(B) (C) (D) (E)

57. A thin layer of oil with an index of refraction n = 1.5 coats a glass plate with an index of refraction of 1.8. White light from air (n = 1.0) is incident normal to the film. In the reflected light: (1) Fully constructive interference occurs at a wavelength of λ₁ = 600 nm. (2) Fully destructive interference occurs at a wavelength of λ₂ = 400 nm. Assume that both wavelengths correspond to the same interference order, what is the thickness of the oil film? (A) 100 nm (B) 133 nm (C) 167 nm (D) 200 nm (E) 266 nm

58. What is the de Broglie wavelength of an electron accelerated from rest through a potential difference of 50.0 V? (A) 0.100 nm (B) 0.139 nm (C) 0.174 nm (D) 0.834 nm (E) none of the above

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