[25 points] Given an electric field E⃗(x, y, z, t) = x̂ 4 e^-α zsin(20π z - 6π× 10^9 t - π/2) mV

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[25 points] Given an electric field \(\vec{E}(x, y, z, t) = \hat{x} 4 e^{-\alpha z} \sin(20\pi z - 6\pi \times 10^9 t - \pi/2)\) mVolts/m where x, y, and z are in meters and t is in seconds, answer the following questions: (a) Is this function of the form of a propagating wave and why? State your answer in terms of the argument \(\phi\) of the function. (b) What is the phase velocity \(v_p\) and direction of this propagating wave? Is this wave propagating in air? Why or why not? (c) Is the wave time harmonic? Explain why or why not. If time harmonic and single frequency, what is the wavelength \(\lambda\), frequency f, and period T of this electric field intensity? (d) The attenuation \(\alpha\) of this wave is 2 Nepers/meter Sketch the amplitude of this wave at time t = 0 along the propagating axis (indicate this axis in the plot), from the origin at 0 meters to \(2\lambda\) meters away from the origin. Correctly label the maximum/minimum amplitudes along with 0 crossings and their locations along the propagation axis at time t = 0. (e) Calculate the vector electric field wave in phasor form \(\vec{E}\). Indicate the proper form and units.

          [25 points] Given an electric field \(\vec{E}(x, y, z, t) = \hat{x} 4 e^{-\alpha z} \sin(20\pi z - 6\pi \times 10^9 t - \pi/2)\) mVolts/m where x, y, and z are in meters and t is in seconds, answer the following questions:
(a) Is this function of the form of a propagating wave and why? State your answer in terms of the argument \(\phi\) of the function.
(b) What is the phase velocity \(v_p\) and direction of this propagating wave? Is this wave propagating in air? Why or why not?
(c) Is the wave time harmonic? Explain why or why not. If time harmonic and single frequency, what is the wavelength \(\lambda\), frequency f, and period T of this electric field intensity?
(d) The attenuation \(\alpha\) of this wave is 2 Nepers/meter Sketch the amplitude of this wave at time t = 0 along the propagating axis (indicate this axis in the plot), from the origin at 0 meters to \(2\lambda\) meters away from the origin. Correctly label the maximum/minimum amplitudes along with 0 crossings and their locations along the propagation axis at time t = 0.
(e) Calculate the vector electric field wave in phasor form \(\vec{E}\). Indicate the proper form and units.

[25 points] Given an electric field E⃗(x, y, z, t) = x̂ 4 e^-α zsin(20π z - 6π× 10^9 t - π/2) mVolts/m where x, y, and z are in meters and t is in seconds, answer the following questions:
(a) Is this function of the form of a propagating wave and why? State your answer in terms of the argument ϕ of the function.
(b) What is the phase velocity vp and direction of this propagating wave? Is this wave propagating in air? Why or why not?
(c) Is the wave time harmonic? Explain why or why not. If time harmonic and single frequency, what is the wavelength λ, frequency f, and period T of this electric field intensity?
(d) The attenuation α of this wave is 2 Nepers/meter Sketch the amplitude of this wave at time t = 0 along the propagating axis (indicate this axis in the plot), from the origin at 0 meters to 2λ meters away from the origin. Correctly label the maximum/minimum amplitudes along with 0 crossings and their locations along the propagation axis at time t = 0.
(e) Calculate the vector electric field wave in phasor form E⃗. Indicate the proper form and units.

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