12. A TE wave propagating in a dielectric-filled rectangular metallic waveguide of unknown permittivity has dimension a = 5 cm and b = 2 cm. If the x component of its electric field is given by
E_x = -10 cos(40πx) sin(100πy) sin(2.4π × 10¹⁰t - 52.9πz) (V/m). Which following statement is correct:
(A) The mode number is TE₂₃
(B) εᵣ of the filled dielectric material is 2
(C) The cutoff frequency of the TE mode is below 11 GHz
(D) None of the above

1. Which of the following statements is true? (A) Dynamic electromagnetic forces satisfy Newton's third law (B) Static electromagnetic forces satisfy momentum conservation (C) Given the electric field E =  and charge q, the physical meaning of qA is energy. (D) All of the above are true

2. Which of the following cannot support TEM modes (ignoring the fringing effect): (A) Two-wire line (B) Rectangular waveguide (C) Coaxial cable/line (D) Microstrip line

3. Which of the following statements about the vector potential A is always true? (A) ∇ × A = B (B) ∇ · A = 0 (C) ∇²A = -μ₀J (D) None of the above

4. There is a dielectric-filled parallel-planar metallic waveguide oriented in the way that guided waves propagate along z direction, and the separation d of two parallel is along y axis. Ignoring the fringing effect, the following field component of fundamental mode supported by the waveguide is non-zero: (A) Ez (B) Ex (C) Hy (D) Hx

5. In the complex notation (also called the phasor notation), we can express a monochromatic plane wave as E = E₀ei(k·r-ωt). How do we calculate the time-averaged Poynting vector in this notation? (A) E × H (B) Re[E × H] (C) Re[E × H*] (D) None of the above

6. Consider a plane wave traveling in a lossy material. The electric field is given by E = E₀ei(kz-ωt), where k equals 1 + 0.1i (μm⁻¹). The intensity of the plane wave can be expressed as I = I₀e-αz, where α is the absorption coefficient. Please calculate α. (A) 2 (μm⁻¹) (B) 0.2 (μm⁻¹) (C) 0.1 (μm⁻¹) (D) None of the above

7. If A = r cosφ r̂ + sinφ φ̂, evaluate ∮ A · dl around the path shown in the following figure. (A) 0 (B)(C) (D) None of the above

8. The finite sheet 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 on the z = 0 plane has a surface charge density ρs = xy(x² + y² + 25)3/2 nC/m². Evaluate the electric field at (0, 0, 5).(A) -3 x̂ - 3 ŷ + 22.5 ẑ V/m (B) -3 x̂ - 3 ŷ + 11.25 ẑ V/m (C) -1.5 x̂ - 1.5 ŷ + 22.5 ẑ V/m (D) -1.5 x̂ - 1.5 ŷ + 11.25 ẑ V/m

9. In the following figure, two conducting cones of infinite extent with θ₁ and θ₂ are separated by an infinitesimal gap at R = 0. If 0 ＜ θ₁ ＜ θ₂ ＜ , V(θ₁) = 0, and V(θ₂) = 10 V, evaluate the potential distribution V(θ) between the cones. (A) 10V (B) 10 V (C) 10 [ln(tan θ cot θ₁)] / [ln(tan θ₂ cot θ₁)] V (D) 10 [ln(tan 2θ cot 2θ₁)] / [ln(tan 2θ₂ cot 2θ₁)] V

10. Consider a plane wave that is incident obliquely on an interface between two dielectric media of refractive indices of nᵢ and nₜ, as shown in the figure. We express the incident wave and transmitted wave as Eᵢ = E₀ᵢe^{i(kᵢ·r-ωt)} and Eₜ = E₀ₜe^{i(kₜ·r-ωt)}, respectively. Here, kᵢ, kᵣ, and kₜ are the wave vectors of the incident wave, reflected wave, and transmitted wave, respectively. These wave vectors are all on the same plane. Which one of the following relations is correct? (A) kᵢcosθᵢ = kₜcosθₜ (B) kᵢsinθᵢ = kₜsinθₜ (C) kᵢnᵢsinθᵢ = kₜnₜsinθₜ (D) None of the above

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