A uniform plane wave in air with H =4 sin (ωt-5x) ay A/m is normally incident on a plastic region with parameters ε = 4ε0, µ=µ0,σ=0. Determine the total electric field in air, and calculate the average power density in the plastic region using Poynting theorem.

To solve this problem, we need to follow these steps:

Step 1: Determine the Electric Field in Air The relationship between the electric and magnetic fields in a plane wave is given by the equation E = ηH, where η is the intrinsic impedance of the medium. For air, η = η0 = 377 ohms. Therefore, the electric field in air is E = ηH = 377 * 4 sin (ωt-5x) = 1508 sin (ωt-5x) V/m.

Step 2: Calculate the Average Power Density in the Plastic Region The average power density (S_avg) in a medium is given by the Poynting theorem as S_avg = 1/2 * Re[E x H*], where E is the electric field, H* is the complex conjugate of the magnetic field, and Re[] denotes the real part of the complex number.

However, since the plastic region is lossy (σ ≠ 0), we need to calculate the complex Poynting vector. The complex Poynting vector is given by S = E x H*.

First, we need to find the electric field in the plastic region. The relationship between the electric and magnetic fields in a plane wave is given by the equation E = ηH, where η is the intrinsic impedance of the medium. For the plastic, η = sqrt((jωµ)/(σ+jωε)) = sqrt((jωµ0)/(0+jω4ε0)) = j50 ohms. Therefore, the electric field in the plastic is E = ηH = j50 * 4 sin (ωt-5x) = 200j sin (ωt-5x) V/m.

Then, we calculate the complex Poynting vector S = E x H* = 200j sin (ωt-5x) x 4 sin (ωt-5x) = 800j sin^2 (ωt-5x) W/m^2.

Finally, the average power density in the plastic region is the real part of the complex Poynting vector, which is S_avg = Re[S] = 0 W/m^2, because the imaginary part of a complex number is zero.