41. A cylindrical rod of length L and radius R is insulated on its surface. Its thermal conductivity k varies with temperature as k(T) = αT. If the two ends of the rod are maintained at T₁ and T₂, the heat current dQ/dt is proportional to ______.
(A) ln(T₁/T₂)
(B)
(C) T₁ - T₂
(D) T₁² - T₂²
(E) T₁³ - T₂³

42. The figure represents a closed cycle for an ideal gas (the figure is not drawn to scale). The change in the internal energy of the gas as it moves from a to c along the path abc is -500 J. As the gas moves from state c to state d, 400 J of heat is transferred to the system. An additional 200 J of heat is absorbed by the gas as it moves from d back to a. What is the work done on the gas as it moves from c to d? (A) -100 J (B) 100 J (C) -300 J (D) 300 J (E) -700 J

43. A heat engine operates between a high-temperature reservoir at 500 K and a low-temperature reservoir at 300 K. In each cycle, the engine absorbs 1000 J of heat from the high-temperature reservoir and performs 500 J of work. Which of the following statements regarding this engine is correct? (A) It violates the first law of thermodynamics but not the second. (B) It violates the second law of thermodynamics but not the first. (C) It violates both the first and second laws of thermodynamics. (D) It violates neither law and is theoretically possible. (E) There is insufficient information to determine if any laws are violated.

44. A gas is at 200 K. If we wish to double the root-mean-square speed of the molecules of the gas, to what value must we raise its temperature? (A) 283 K (B) 400 K (C) 566 K (D) 800 K (E) 1130 K

45. A long coaxial cable consists of a solid inner conductor of radius a carrying current I and a thin outer shell of radius b carrying return current I. The magnetic field at distance r (a ＜ r ＜ b) is ______. (Given: permeability constant μ₀) (A) 0 (B) μ₀I/(2πa) (C) μ₀I/(2πr) (D) μ₀Ir/(2πab) (E) μ₀I/(2πb)

46. A small AC generator is used to power motion sensors on a treadmill. The generator coil rotates in a uniform magnetic field and must produce an emf of 373 V. How many turns of wire are required? (Given: Magnetic field B = 0.60 T, Coil area A = 0.15 m², Rotation frequency f = 60.0 Hz) (A) 8 (B) 11 (C) 14 (D) 17 (E) 20

47. When operating on a 120-V circuit, an electric heater receives 1.30 × 10³ W of power, a toaster receives 1.00 × 10³ W, and an electric oven receives 1.54 × 10³ W. If all three appliances are connected in parallel on a 120-V circuit and turned on, what is the total current drawn from an external source? (A) 24.0 A (B) 32.0 A (C) 40.0 A (D) 48.0 A (E) none of the above

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

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