A uniform plane wave propagating in a medium has

\(\vec E = 2e^{-α z} \sin (10^8 t - β z) \hat a_y \ V/m\)

if the medium is characterized by ϵ_r = 1, μ_r = 20 and σ = 3 S/m. The value of α and β are respectively.

Concept:

The waves are used for the transportation of energy or information.

Characteristics of a plane wave in good conductors.

In a good conductor σ >> ωϵ 

σ ≈ ∞, ϵ = ϵ₀, μ = μ₀μ_r  

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

\(v=\frac{\omega}{\beta}=\sqrt {\frac{2\omega}{\mu \sigma}}\)

\(\lambda = \frac{2\pi}{\beta}\)

\(\eta = \sqrt {\frac{\omega \mu}{\sigma}}\angle 45^\circ\)

Calculation:

Given ϵr = 1, μr = 20, σ = 3 S/m and 

\(\vec E = 2e^{-α z}sin(10^8 t -β z)\widehat {a_y}\)

\(\frac{σ}{ω ϵ}=\frac{3}{10^8 \times \frac{10^{-9}}{36\pi}}\)

\(\frac{σ}{ω ϵ}=360\pi \times 3 \)

\(\frac{σ}{ω ϵ} >> 1\)

Hence the given medium is a good conductor.

In a good conductor the values of α and β are:

\(α = β = \sqrt {\frac{ω μ σ}{2}}=\sqrt {\frac{10^8 \times 20 \times 4\pi \times 10^{-7}\times 3}{2}}\)

α = β = 61.39