Question

blem 6. Electromagnetic plane waves in a lossy medium tric field of an electromagnetic plane wave traveling in a lossy medium can be written as The where z is the distance, t is time, and fthe frequency. Forf- 1 GHz, it is found by measurement that the amplitude of the electric field is attenuated by a factor of 3 after the wave travels 100 m (ie, to 1/3 of the amplitude at: 0 when it arrives at : -100 m). (1) Find a. (2) Suppose the conductivity of the medium is a, the dielectric constant is s, and the relative permeability is f,-1 for the medium. Notice that ε and μ μμ. Assuming σ<< 2rfe prove that simple, approximate expressions for a and Bcan be (3) Assume &-2.5. Recall that V6/e,-37742 Using the above approximation and in (1), find ơ Use 12.5-1.58. found (4) Does the ơ found in (3) Justify the use of the approximation, Explain your answer. For quick estim ation, you may find ~ (1/36π)-10-9 F/m useful. (5) (Optional, for bonus) What is the phase difference between the magnetic field and the electric field of this wave at this frequency f 1 GHz?

Answer 1

E^bar (z, t) = E_0 e^- alpha z cos [2 pi ft - beta z)x^cap rightarrow (1) f = 1 GHz and E^bar (0, t) = E_0 cos (2 pi ft - beta z)x^cap (1)& E^bar (100, t) = E_0 e^-100 alpha cos (2 pi ft - beta z)x^cap from eq'n(1) = 1/3 E_0 cos (2 pi ft - beta z)x^cap (Given in problem) implies e^-100 alpha = 1/3 implies - 100 alpha = ln(1./3) implies alpha = 1/100 ln(3) implies alpha approximatelyequalto 0.01099 N_p/m rightarrow (2) (2) sigma /2 pi + elementof < < 1 implies sigma /w elementof < < 1[because w = 2 pi f] alpha = w squareroot mu elementof /2[squareroot 1 = (sigma /w elementof)^2 - 1] (general equation for a lossy medium) (1 + x)^1/2 approximatelyequalto 1 + 1/2x for x < < 1 (because Binomial theorem) [1 + (sigma /w elementof)^2]^1/2 approximatelyequalto 1 + 1/2(sigma /w elementof)^2 rightarrow (3) (By putting x = (sigma /w elementof)^2 in above equation) implies alpha = w squareroot mu elementof /2(1 + 1/2(sigma /w elementof)^2 - 1 rightarrow (4)