Explanation Calculation: Write the expression for magnetic field component of given EM wave as follows: H = 50 e − 100 x cos ( 2 π × 10 9 t − 200 x ) a y mA / m (1) Write the general form of the magnetic field component of EM wave as follows: H = H o e − α x cos ( ω t − β x ) a y A / m (2) Compare Equation (1) with Equation (2) and write the values of H o , α , β , and ω . H o = 50 mA / m α = 100 Np / m β = 200 rad / m ω = 2 π × 10 9 rad / s Find the direction of wave propagation of electric field for the given EM wave. a E = − ( a k × a H ) = − ( a x × a y ) { ∵ a H = a y ∵ a k = a x } = − a z { ∵ a x × a y = a z } = − a z Write the expression to find the electric field E as follows: E = H o | η | e − α x cos ( ω t − β x + θ η ) ( − a z ) (3) Here, H o is the magnitude of the magnetic field component at x = 0 , | η | is the magnitude of the intrinsic impedance, and θ η is the phase angle of the intrinsic impedance. Consider the expression for phase constant for a nonmagnetic medium. β = ω μ ε Rewrite the expression for nonmagnetic medium as follows: β = ω μ o ε o ε r { ∵ μ = μ o } Rearrange the expression as follows: ε r = ( 1 μ o ε o ) ( β ω ) 2 Substitute 4 π × 10 − 7 H / m for μ o , 10 − 9 36 π F / m for ε o , 200 rad / m for β , and 2 π × 10 9 rad / s for ω . ε r = [ 1 ( 4 π × 10 − 7 H / m ) ( 10 − 9 36 π F / m ) ] ( 200 rad / m 2 π × 10 9 rad / s ) 2 ≅ 91.1890 Consider the expression for attenuation constant. α = ω μ ε 2 [ 1 + ( σ ω ε ) 2 − 1 ] (4) Here, σ is the conductivity. Consider the expression for the expression for phase constant. β = ω μ ε 2 [ 1 + ( σ ω ε ) 2 + 1 ] (5) Consider 1 + ( σ ω ε ) 2 as x . x = 1 + ( σ ω ε ) 2 (6) Rewrite the Equation (4) as follows: α = ω μ o ε o ε r 2 ( x − 1 ) Substitute 100 Np / m for α s, 2 π × 10 9 rad / s for ω , 4 π × 10 − 7 H / m for μ o , 10 − 9 36 π F / m for ε o , and 91

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ISBN: 9780190698669

Author: Sadiku

Publisher: Oxford University Press

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Chapter 10, Problem 14P

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Chapter 10 Solutions

Elements of Electromagnetics

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