If εr = 9, µ=µ0 for the medium in which a wave with frequency f= 0.3GHz is propagating, calculate propagation constant, intrinsic impedance and phase velocity of the medium when i) σ =0 and ii) σ= 10 mho/m

Sure, let's solve this step by step.

First, we need to know the values of the constants ε0 and µ0. The permittivity of free space ε0 is approximately 8.85 x 10^-12 F/m and the permeability of free space µ0 is 4π x 10^-7 H/m.

i) When σ = 0:

1. Propagation constant (γ): The propagation constant is given by the formula γ = √(jωµ(σ+jωε)), where ω = 2πf. Since σ = 0, the formula simplifies to γ = √(jωµjωε) = jω√(µε). Substituting the given values, we get γ = j2π(0.3 x 10^9)√((4π x 10^-7)(9 x 8.85 x 10^-12)) = j18.85 rad/m.

2. Intrinsic impedance (η): The intrinsic impedance is given by the formula η = √(jωµ/σ+jωε). Since σ = 0, the formula simplifies to η = √(jωµ/jωε) = √(µ/ε). Substituting the given values, we get η = √((4π x 10^-7)/(9 x 8.85 x 10^-12)) = 376.73 Ω.

3. Phase velocity (v): The phase velocity is given by the formula v = ω/β, where β is the phase constant. Since γ = α + jβ and α = 0, we have β = γ. So, v = ω/β = 2π(0.3 x 10^9)/18.85 = 1 x 10^8 m/s.

ii) When σ = 10 mho/m:

1. Propagation constant (γ): Now, we use the original formula γ = √(jωµ(σ+jωε)). Substituting the given values, we get γ = √(j2π(0.3 x 10^9)(4π x 10^-7)(10+j2π(0.3 x 10^9)(9 x 8.85 x 10^-12))) = 1.77 + j18.85 rad/m.

2. Intrinsic impedance (η): We use the original formula η = √(jωµ/σ+jωε). Substituting the given values, we get η = √((j2π(0.3 x 10^9)(4π x 10^-7))/(10+j2π(0.3 x 10^9)(9 x 8.85 x 10^-12))) = 18.97 - j1.89 Ω.

3. Phase velocity (v): Again, v = ω/β. But now, β = Im(γ) = 18.85. So, v = 2π(0.3 x 10^9)/18.85 = 1 x 10^8 m/s. The phase velocity remains the same because it only depends on the medium and not on the conductivity.