4. Given two problems (Problem A, and Problem B) in the free space shown below. Assume the solution of the scalar electrical potential distribution of Problem A is Vₐ, and the solution of the scalar electrical potential distribution of Problem B is V_b. Please evaluate:
[Vₐ(x,0,z) - V_b(x,0,z)] ∇[Vₐ(x,0,z) - V_b(x,0,z)] · ŷ dxdz
 
(A)
(B)
(C) 0
(D) None of the above

1. Given the vector function A = 3x²eˣx̂ + x³y³ŷ. Please evaluate ∮ around the rectangular area from P₁(0,0,0), to P₂(3,0,0), to P₃(3,4,0), to P₄(0,4,0), and back to P₁(0,0,0). (A) 0 (B) 864 (C) 1728 (D) None of the above

2. A point charge Q is placed at the original point inside a hollow sphere, as shown in the figure below. The material of the hollow sphere is perfect electrical conductor (PEC). The total outward flux of the electric field intensity over the closed surface S (∮) is: (A) 0 (B) Q/ε₀ (C) -Q/ε₀ (D) None of the above

3. A lossy dielectrics (conductivity = σ, absolute permittivity = ε) is placed between two parallel perfect metal plates, as shown in the figure below. The areas of the metal plates are A (A = W×L). The distance between the metal plates is d and is much smaller than the length or width of the metal plates (d ＜＜ W and d ＜＜ L). Applying a voltage V₀ between the perfect metal plates, please evaluate the ratio between the storage energy and dissipated power of this structure (Estorage/Pdiss). (A) (B) (C) (D) None of the above

5. A positive point charge Q is at the center of a spherical dielectric sphere with radius of R. The material of this dielectric sphere is simple medium with relative permittivity of εᵣ. Please evaluate the equivalent polarization surface charge density (or bound surface charge density) of the surface of the sphere. (A) 0 (B) (C) (D) None of the above

6. A charge is moving in a magnetic field. Which of the following statements is true? (A) The magnetic field does no work on the charge only if the magnetic field is static. (B) The magnetic field does no work on the charge only if the magnetic field is time-dependent. (C) The magnetic field never does work on the charge. (D) None of the above.

7. The mutual inductance between two coils is M. Which of the following statements is true? (A) When the currents of two coils are doubled, their mutual inductance is halved. (B) The mutual inductance is changed when the materials of the coils are changed. (C) Mutual inductance depends on the geometry of the coils, specifically their shape, size, and the relative positions of their windings. (D) None of the above.

8. A solenoid has n turns per unit length. The self-inductance of a solenoid is proportional to (A) n (B) n² (C) n³ (D) None of the above

9. A sinusoidal electromagnetic wave in vacuum has an electric field E = E₀ sin(kz - ωt) ŷ. What is its magnetic field B? c the light speed. (A)sin(kz - ωt) x̂ (B) sin(kz - ωt) x̂ (C) cos(kz - ωt) x̂ (D) None of the above

10. Which of the following statements about the skin depth of a metal is true? (A) The skin depth increases if the frequency of the electromagnetic wave decreases. (B) The skin depth increases if the conductivity of the metal increases. (C) The skin depth does not depend on the conductivity of the metal. (D) None of the above.

11. A uniform plane wave travels in a dielectric medium. The electric field component is given by(y,z,t) = 2400π × cos(2π × 10⁸t - 8πy + 6πz + π) x̂ (V/m). Assume this dielectric medium is lossless and non-magnetic. Which of the following statements is correct? (A) The wavelength in the dielectric medium is 5 m. (B) The dielectric constant εᵣ is 15. (C) The amplitude of the magnetic field in y-direction (Hy) is 16 A/m. (D) None of the above

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