Question

A plane wave of frequency 2 GHz travelling in air parallel to a flat ground plane with H parallel to the ground plane losses 3 μW / m² into the ground plane having conductivity σ = 10⁷ Ʊ / m, μ_r = ϵ_r = 1. The following data is given:

A) E = 4.106 × 10^-4 V/m

B) E = 8.98 V / m

C) H = 1.089 × 10^-6 A/m

D) H = 3.97 mA / m

Choose the most appropriate answer from the options given below:

1. (A) and (C) only
2. (A) and (D) only
3. (B) and (C) only
4. (B) and (D) only

Answer 1

Correct Answer - Option 1 : (A) and (C) only

Concept:

Total loss per unit area of the conductor surface:

\(W = \mathop \smallint \nolimits_0^\infty σ E_0^2{e^{ - 2\alpha z}}dz\)

Since σE² = energy density (W/m³)

E = E₀e^-αz

\(W = \frac{{σ \;{{\left| {{E_0}} \right|}^2}}}{{2\alpha }}\)

Calculation:

Given:

f = 2 GHz

σ =10⁷ ℧/m

\(\alpha = \sqrt {\frac{{\omega \mu \sigma }}{2}} = \sqrt {\frac{{2\pi f \times \mu \sigma }}{2}} \)

\(\alpha = \sqrt {\frac{{2\pi \times 2 \times {{10}^9} \times 4\pi \times {{10}^{ - 7}} \times {{10}^7}}}{2}} \)

α = 280992.5892

\(W = \frac{{\sigma \;{{\left| {{E_0}} \right|}^2}}}{{2\alpha }}\)

\(3 \times {10^{ - 6}} = \frac{{{{10}^7}{{\left| {{E_0}} \right|}^2}}}{{2\alpha }}\)

|E₀| = 4.106 × 10^-4 V/m

|H₀| = 1.089 × 10^-6 A/m