Review problems for test 3 It is important to keep in mind that these problems are just a set of typical problems representing the materials covered. HW problems are also very useful review materials. Problem 1 A uniform plane wave in free-space is given by [V/m] zjyxeujuzEπ4)63()(−+=r(1) What is the frequency of this wave? (2) Determine the magnetic field ()Hrrr (3) Determine the displacement current. (4) What is the time-domain solution(,)Ertrr? (5) What is the polarization of this wave? Problem 2 A uniform plane wave propagates in a non-magnetic dielectric material whose 4, 0.2[S/m]rεσ== at the frequency of 200 MHz. (1) Determine the loss tangent for the dielectric. (2) Calculate the attenuation constant α and the wave number β. (3) Calculate the wave impedance η. (4) What is the phase velocity, wavelength, and skin depth of the media? Problem 3 Two half spaces share a boundary at z=0, as shown. The upper half space has 101 013, 2 , 0εεμ μσ===, and the lower half space has 202 024 , 1.6 , 0εεμ μσ===. A TE wave incident from the upper half space to the interface at an incident angle of . 030(1) Determine the transmission angle tθ. (2) Determine the reflection coefficient. iθ tθ 12Problem 4 The wave impedance generally defined as εμη/= . We can measure the reflection coefficient to determine the material parameters. To this end, we send a uniform plane wave normally from air to the material (flat interface with air), and use some instruments to record the reflection coefficient. Assume 0,0μμσ== for the material, and ε may be complex. If the recorded reflection coefficient (normal incidence) is R=-0.47, determine the relative dielectric permittivity rε of the material. Air 00,, 0rεεε μ σ== 0jzeβ− 0jzReβ z z=0 Problem 5 A light source is shine isotropically is submerged at a depth below the surface of water. How far in the x-direction (both positive and negative) can the observer (on the water surface) go and still see the light? (Assume that the water surface is flat and the water is lossless with d81rε= ). x LL3m081, , 0rεμμσ== = Air Problem 6 A uniform plane wave in a lossy material is given by [V/m], zjzxeuzEπ46.23)(−−=rzjzyeuzHπ46.202.0)(−−=r [A/m] (1) What is the amplitude of the wave impedance for the material? (2) What is the time-average power density in general? Problem 7 A uniform plane wave propagates in a lossy media in the z-direction. The wave amplitude is 4 [V/m] at z=0. Calculate the total power dissipation in the box region specified by: ? Assume that the attenuation constant is 01,0 2,01xyz≤≤ ≤≤ ≤≤.50.4α= [1/m], and the wave impedance is /3180 [ ]jeπη=Ω .Problem 8 A uniform plane wave is normally incident on to an air-glass interface (from the air side). The glass permittivity (dielectric constant) is 04εε=, and the frequency of the wave is 900 MHz. A slab is inserted between the two materials to serve as the quarter-wave match slab. Find the dielectric constant and the thickness of this matching slab. Problem 9 (1) Write the Maxwell equation in frequency domain for a source free space (2) Write the boundary conditions for electric field and magnetic field. (3) If , find the corresponding magnetic field using Maxwell equation )34(4)43(),,(yxjyxeujuzyxE−−+=πr(4) Find the time average power density for the wave expression in (3) (5) What is the polarization of the wave in (3) (6) The direction of the wave in (3) is . If the wave is in free-space, what is the phase constant yxuu 6.08.0ˆ−=β?=β Since ,34 yxr −=⋅rrβ Hence ,3,4 −==yxββ What is the phase velocity in x and y directions, respectively? Problem 10 A current source radiation field in free space is given by θθπθcossin)(61reuCzErj−=r [V/m], (1) If the radiation at r=10m and is 0.1 V/m, what is the constant o45=θ1C(2) What is the frequency of this wave? (3) What is the angle of maximum radiation? (4) What is the half power beam width (HPBM) in degree? (5) What is the associated magnetic field? (6) What is the electric field at a distance of r=20m? (7) Sketch the radiation pattern for the range of o180