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4. A plane wave in free space has the phasor electric field Ë(y) =(jê+2)eky. (a) In what direction is the wave propagating? (b) Specify whether the polarization is linear, RHCP or LHCP and give brief justification for your answer. (c) Write the phasor expression for the magnetic field.

4. A plane wave in free space has the phasor electric field Ë(y) =(jê+2)eky. (a) In what direction is the wave propagating? (b) Specify whether the polarization is linear, RHCP or LHCP and give brief justification for your answer. (c) Write the phasor expression for the magnetic field.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### Problem 4 A plane wave in free space has the phasor electric field \(\vec{E}(y) = (j\hat{x} + \hat{z}) e^{jky}\). #### (a) In what direction is the wave propagating? To determine the direction of wave propagation, examine the argument of the exponential term in the electric field expression. The wave's phase \(e^{jky}\) indicates propagation in the negative \(y\)-direction because of the positive sign in the exponent. #### (b) Specify whether the polarization is linear, RHCP or LHCP and give brief justification for your answer. Assessing the components of the phasor electric field \(\vec{E}(y)\): - \(\hat{x}\)-component: \(j e^{jky}\) - \(\hat{z}\)-component: \(e^{jky}\) Expressing the field components in terms of cosine and sine functions: - \(\hat{x}\)-component: \(j e^{jky} = j (\cos(ky) + j \sin(ky)) = -\sin(ky) + j\cos(ky)\) - \(\hat{z}\)-component: \(e^{jky} = \cos(ky) + j \sin(ky)\) There is a 90-degree phase difference between the \(\hat{x}\)-component and the \(\hat{z}\)-component. Since the \(\hat{x}\)-component leads (has a \(j\) factor which corresponds to a \(\pi/2\) lead in phase), the wave is **Right-Hand Circularly Polarized (RHCP)**. #### (c) Write the phasor expression for the magnetic field. Using Maxwell's equations, specifically the relation between the electric field \(\vec{E}\) and the magnetic field \(\vec{H}\) in free space: \[ \vec{H} = \frac{1}{\eta} \hat{k} \times \vec{E} \] where \(\eta\) is the intrinsic impedance of free space and \(\hat{k}\) is the unit vector in the direction of wave propagation (negative \(y\)-direction). Given \(\vec{E} = (j\hat{x} + \hat{z}) e^{jky}\) and \(\hat{k} =
Transcribed Image Text:### Problem 4 A plane wave in free space has the phasor electric field \(\vec{E}(y) = (j\hat{x} + \hat{z}) e^{jky}\). #### (a) In what direction is the wave propagating? To determine the direction of wave propagation, examine the argument of the exponential term in the electric field expression. The wave's phase \(e^{jky}\) indicates propagation in the negative \(y\)-direction because of the positive sign in the exponent. #### (b) Specify whether the polarization is linear, RHCP or LHCP and give brief justification for your answer. Assessing the components of the phasor electric field \(\vec{E}(y)\): - \(\hat{x}\)-component: \(j e^{jky}\) - \(\hat{z}\)-component: \(e^{jky}\) Expressing the field components in terms of cosine and sine functions: - \(\hat{x}\)-component: \(j e^{jky} = j (\cos(ky) + j \sin(ky)) = -\sin(ky) + j\cos(ky)\) - \(\hat{z}\)-component: \(e^{jky} = \cos(ky) + j \sin(ky)\) There is a 90-degree phase difference between the \(\hat{x}\)-component and the \(\hat{z}\)-component. Since the \(\hat{x}\)-component leads (has a \(j\) factor which corresponds to a \(\pi/2\) lead in phase), the wave is **Right-Hand Circularly Polarized (RHCP)**. #### (c) Write the phasor expression for the magnetic field. Using Maxwell's equations, specifically the relation between the electric field \(\vec{E}\) and the magnetic field \(\vec{H}\) in free space: \[ \vec{H} = \frac{1}{\eta} \hat{k} \times \vec{E} \] where \(\eta\) is the intrinsic impedance of free space and \(\hat{k}\) is the unit vector in the direction of wave propagation (negative \(y\)-direction). Given \(\vec{E} = (j\hat{x} + \hat{z}) e^{jky}\) and \(\hat{k} =
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