4.

複選題5.(AB) This sustained, higher normal is expected to be further bolstered by continuous advancements in charging infrastructure and the gradual reduction of battery production costs. (AC) Experts caution that significant work is still required to reassure skeptical buyers that zero-emission vehicles can seamlessly integrate into their daily lifestyles. (AD) In response to this geopolitical turmoil, automotive executives have successfully lobbied governments to permanently ban the production of all internal combustion engines. (AE) These specific platforms have documented inquiry surges exceeding fifty percent within a single month, underscoring a rapid recalibration of buyer preferences. (BC) The primary motivation for this transition is no longer financial savings, but rather a sudden, collective awakening to the severe environmental impacts of carbon emissions. (BD) As the financial burden of fossil fuels intensifies, the comparative affordability of grid-supplied electricity has suddenly illuminated the economic viability of plug-in alternatives. (BE) Consequently, motorists are demonstrating a newfound willingness to absorb higher upfront investments in exchange for the promise of drastically reduced operational expenses over time.  (CD) Despite these undeniable economic pressures, the overall registration of battery-powered cars plummeted to a record low right before the outbreak of hostilities.

1. Find the angle between the body diagonals of a cube. (A) (B)(C) (D)

2. Find the divergence of the function v = s(2+sin²φ)ŝ + s sinφ cosφ φ̂ + 3zẑ (A)5 (B)10 (C)8 (D)3

3. One of these is a possible electrostatic field. Which one? (A)E = k[xyx̂ + 2yzŷ + 3xzẑ] (B)E = k[y²x̂ + (2xy + z²)ŷ + 2yzẑ] (C)E = k[yx̂ + x²ẑ] (D)E = k[yzx̂ + y²ŷ + z²ẑ]

4. The potential at the surface of a sphere (radius R) is given by V₀=k cos3θ where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density σ(θ) on the sphere. (Assume there's no charge inside or outside the sphere.) (A)cosθ[140cos²θ - 93] (B)cosθ[140cos²θ] (C)cosθ[140cosθ - 93] (D)cosθ[140cos²θ - 93]

5. A certain coaxial cable consists of a copper wire, radius a, surrounded by a concentric copper tube of inner radius c. The space between is partially filled (from b out to c) with material of dielectric constant εᵣ, as shown. Find the capacitance per unit length of this cable. (A)(B) (C)(D)

6. A steady current I flows down a long cylindrical wire of radius a. Find the magnetic field outside the wire if the current is uniformly distributed over the outside surface of the wire. (A)0 (B)(C)(D)

7. An infinitely long circular cylinder carries a uniform magnetization M parallel to its axis. Find the magnetic field (due to M) inside the cylinder. (A)2μ₀M (B)μ₀M (C)0 (D)μ₀M/4π

8. A square loop of wire, of side a, lies midway between two long wires, 3a apart, and in the same plane. (Actually, the long wires are sides of a large rectangular loop, but the short ends are so far away that they can be neglected.) A clockwise current I in the square loop is gradually increasing: dI/dt = k (a constant). Find the emf induced in the big loop. Which way will the induced current flow? (A), counterclockwise (B), clockwise (C), counterclockwise (D), clockwise

9. A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap in the wire, of width ω≪a, forms a parallel-plate capacitor, as shown in figure. Find the magnetic field in the gap, at a distance s ＜ a from the axis. (A)(B) (C)(D)

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