Question

Assume that a medium has ε.9εο, σ-0.1 S!m, and μ. μο. A uniform plane wave at 6 GHz propagates in this medium in the direction. Find the ratio of the electric field magnitudes at x 0 and x-2 m. Also, find the phase difference between electric fields at these points.

Answer 1

For uniform plane wave the electric field in phasor form E(x) = E_0 e^-alpha x e^-j beta x The ration of magnitude at x = 0 and x = 2m is Vertical-bar E(0)/E(2) Vertical-bar = E_0 e^-alpha.0/E_0 e^-alpha.2 = 1/e^-2 alpha = e^2 alpha where alpha = omega squareroot mu epsilon/2 [squareroot 1 + (sigma/omega e)^2 - 1] alpha = 2 pi f squareroot mu_0 times g times epsilon_0/2 [squareroot 1 + (0.1/2 pi times f times g times epsilon_0)^2 - 1] put f = 6 GHz = 6 times 10^9 and mu_0 = 4 pi times 10^-7, epsilon_0 = 8.854 times 10^-12 so alpha = 6.278 Vertical-bar E(0)/E(z) Vertical-bar = e^2 times 6.278 = 2.838 times 10^5 and (b) the phase difference between electric field at x = 0 and x = 2 is angle E(0) - angle E(2) = -beta times 0 - (-beta.2) = 2 beta \r\n phase difference = 2 beta where beta = omega squareroot mu epsilon/2 [squareroot 1 + (sigma/omega epsilon)^2 + 1] = 377.3003 Phase difference = 754.6 rad