Explanation Given data: The given electric field vector of the wave is E ˜ = ( x ^ + j y ^ ) 30 e − j π z / 6 V / m . Calculation: Write the standard expression for the instantaneous value of the electric field. E ( z , t ) = ℜ e [ E ˜ ( z ) e j ω t ] Here, E ( y , t ) is the instantaneous value of the electric field. ω is the angular frequency of the wave. Substitute ( x ^ + j y ^ ) 30 e − j π z / 6 for E ˜ in the above equation. E ( z , t ) = ℜ e [ ( x ^ + j y ^ ) 30 e − j π z / 6 e j ω t ] (1) Write the standard conversion from complex variable to the exponential form as, e j ω = cos ω + j sin ω Here, ω is the angular frequency. Substitute π 2 for ω in the above expression. e j π 2 = cos ( π 2 ) + j sin ( π 2 ) = 0 + j ⋅ 1 = j Substitute e j π 2 for j in equation (1). E ( z , t ) = ℜ e [ ( x ^ + y ^ e j π 2 ) 30 e − j π z / 6 e j ω t ] = ℜ e [ x ^ 30 e − j π z / 6 e j ω t + y ^ 30 e j π 2 e − j π z / 6 e j ω t ] Simplify the above equation as, E ( z , t ) = ℜ e [ x ^ 30 e j ω t − j π z / 6 + y ^ 30 e j π 2 − j π z / 6 + j ω t ] = x ^ 30 cos ( ω t − π z 6 ) + y ^ 30 cos ( ω t − π z 6 + π 2 ) Solve the above equation. E ( z , t ) = x ^ 30 cos ( ω t − π z 6 ) − y ^ 30 sin ( ω t − π z 6 ) (3) The modulus of the electric field is calculated as, | E | = ( E x 2 + E y 2 ) Here, | E | is the modulus of the electric field. E x is the component of electric field along the x - axis. E y is the component of electric field along the y - axis. Substitute 30 cos ( ω t − π z 6 ) for E x and − 30 sin ( ω t − π z 6 ) for E y in the above equation. | E | = [ ( 30 cos ( ω t − π z 6 ) ) 2 + ( − 30 sin ( ω t − π z 6 ) ) 2 ] = [ 30 2 cos 2 ( ω t − π z 6 ) + ( − 30 ) 2 sin 2 ( ω t − π z 6 ) ] = [ 30 2 cos 2 ( ω t − π z 6 ) + ( 30 ) 2 sin 2 ( ω t − π z 6 ) ] = [ 30 2 ( cos 2 ( ω t − π z 6 ) + sin 2 ( ω t − π z 6 ) ) ] Solve the above expression. | E | = 30 2 = 30 V / m Write the standard expression for the auxiliary angle of the wave. ψ 0 = arctan ( a y a x ) (4) Here, ψ 0 is the auxiliary angle of the wave. a y is the magnitude in the y - component of the electric field. a x is the magnitude of the x - component of the electric field. Write the standard relation for frequency, speed of light and wavelength. f = c λ (5) Here, f is the frequency of the wave. λ is the wavelength of the wave. c is the speed of the light, which has the standard value as 3 × 10 8 m / s . Write the standard relation for wavelength and wave number of plane wave. λ = 2 π k (6) Here, k is the wave number of the wave. Compare the equation (3) with the standard equation E ˜ = ( x ^ a x cos ( ω t − k z − δ x ) + y ^ a y cos ( ω t − k z − δ y ) ) to calculate the wave number that is, k = π 6 rad / m Substitute π 6 rad / m for k in equation (6)

7th Edition

ISBN: 8220100663659

Author: ULABY

Publisher: PEARSON

See similar textbooks

Question

Chapter 7, Problem 11P

To determine

The modulus and direction of electric field intensity at the z=0 plane at t=0, t=5 ns and t=10 ns.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7, Problem 10P

Chapter 7, Problem 12P

Students have asked these similar questions

2. A system with unity feedback is shown below. The feed-forward transfer function is G(s), where 5 . G(S) = (+1) Sketch the root locus for the variations in the values of pi. (s+P1)s R(s) C(s) G(s)

3. The following closed-loop systems in Fig. 1 and Fig. 2 operate with a damping ratio of 0.707 (=0.707). The system in Fig. 1 does not have a PI controller, while the one in Fig. 2 does. R(s): S Gain Plant R(s) + E(s) 1 C(s) K (s+1)(s+2)(s+10) Fig. 1: Closed-loop system without PI controller Compensator Plant R(s) + E(s) K(s+0.1) S 1 (s+1)(s+2)(s+10) C(s) Fig. 2: Closed-loop system with a practical PI controller a. Please use Matlab to find the intersection point between line and the root locus of the system in Fig. 1. Then find the K value and one complex closed-loop pole corresponding to the intersection point. Calculate the steady-state error. Show the Matlab code in your answer sheet. b. Please use Matlab to find the intersection point between § line and the root locus of the system in Fig. 2. Then find the K value and one complex closed-loop pole associated with the intersection point. Compare the complex closed-loop pole with the one you just found in task a. Are they very…

1. Please draw the root locus by hand for the following closed-loop system, where G(s) = s+6 = S-2 s+8 s-2' and H(s) = Find the range of K for stability using Method II in Examples 2 and 3 in Lecture 15. Input R(s) Output C(s) KG(s) H(s)

Chapter 7 Solutions

EBK FUNDAMENTALS OF APPLIED ELECTROMAGN

Ch. 7.2 - What is a uniform plane wave? Describe its...Ch. 7.2 - Since E and H are governed by wave equations of...Ch. 7.2 - If a TEM wave is traveling in the y direction, can...Ch. 7.2 - Prob. 1E Ch. 7.2 - Prob. 2E Ch. 7.2 - If the magnetic field phasor of a plane wave...Ch. 7.2 - Repeat Exercise 7-3 for a magnetic field given by...Ch. 7.3 - An elliptically polarized wave is characterized by...Ch. 7.3 - Prob. 5CQ Ch. 7.3 - The electric field of a plane wave is given by...

Ch. 7.4 - If the electric field phasor of a TEM wave is...Ch. 7.4 - The constitutive parameters of copper are = 0 = 4...Ch. 7.4 - Prob. 8E Ch. 7.4 - For a wave traveling in a medium with a skin depth...Ch. 7.5 - Prob. 6CQ Ch. 7.5 - In a good conductor, does the phase of H lead or...Ch. 7.5 - Prob. 8CQ Ch. 7.5 - Is a conducting medium dispersive or...Ch. 7.5 - Compare the flow of current through a wire in the...Ch. 7.6 - Convert the following values of the power ratio G...Ch. 7.6 - Find the voltage ratio g corresponding to the...Ch. 7 - The magnetic field of a wave propagating through a...Ch. 7 - Prob. 2P Ch. 7 - The electric field phasor of a uniform plane wave...Ch. 7 - The electric field of a plane wave propagating in...Ch. 7 - A wave radiated by a source in air is incident...Ch. 7 - The electric field of a planewave propagating in a...Ch. 7 - The magnetic field of a plane wave propagating in...Ch. 7 - A 60 MHz plane wave traveling in the x direction...Ch. 7 - Prob. 9P Ch. 7 - For a wave characterized by the electric field...Ch. 7 - Prob. 11P Ch. 7 - The magnetic field of a uniform plane wave...Ch. 7 - A linearly polarized plane wave of the form...Ch. 7 - The electric field of an elliptically polarized...Ch. 7 - Compare the polarization states of each of the...Ch. 7 - Plot the locus of E(0, t) for a plane wave with...Ch. 7 - Prob. 17P Ch. 7 - Prob. 18P Ch. 7 - In a medium characterized by r = 9, r = 1, and =...Ch. 7 - Prob. 20P Ch. 7 - Prob. 21P Ch. 7 - Prob. 22P Ch. 7 - The skin depth of a certain nonmagnetic conducting...Ch. 7 - Prob. 24P Ch. 7 - The electric field of a plane wave propagating in...Ch. 7 - The magnetic field of a plane wave propagating in...Ch. 7 - At 2 GHz, the conductivity of meat is on the order...Ch. 7 - In a nonmagnetic, lossy, dielectric medium, a 300...Ch. 7 - A rectangular copper block is 30 cm in height...Ch. 7 - Prob. 30P Ch. 7 - The inner and outer conductors of a coaxial cable...Ch. 7 - Prob. 32P Ch. 7 - The magnetic field of a plane wave traveling in...Ch. 7 - A wave traveling in a nonmagnetic medium with r =...Ch. 7 - The electric-field phasor of a uniform plane wave...Ch. 7 - Prob. 36P Ch. 7 - A wave traveling in a lossless, nonmagnetic medium...Ch. 7 - At microwave frequencies, the power density...Ch. 7 - Consider the imaginary rectangular box shown in...Ch. 7 - Repeat Problem 7.39 for a wave traveling in a...Ch. 7 - Given a wave with E=x E0 cos(t kz): (a) Calculate...Ch. 7 - Prob. 42P

Knowledge Booster

The modulus and direction of electric field intensity at the z = 0 plane at t = 0 , t = 5 ns and t = 10 ns .

Solution Summary: The author calculates the instantaneous value of the electric field at the z=0 plane.

The modulus and direction of electric field intensity at the z=0 plane at t=0, t=5 ns and t=10 ns.

Want to see the full answer?

Chapter 7 Solutions