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A 10-GHz plane wave traveling in free space has the E-field: E = 1 V/m: (a) Find the phase velocity, the wavelength, the propagation constant. (b) Determine the characteristic impedance of the medium. (c) Find the amplitude and direction of the magnetic field intensity if the wave is traveling in the -z direction.

A 10-GHz plane wave traveling in free space has the E-field: E = 1 V/m: (a) Find the phase velocity, the wavelength, the propagation constant. (b) Determine the characteristic impedance of the medium. (c) Find the amplitude and direction of the magnetic field intensity if the wave is traveling in the -z direction.

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**Topic: Electromagnetic Waves in Free Space** A 10-GHz plane wave traveling in free space has the electric field (E-field) represented as: \(\vec{E} = 1 \hat{x} \, \text{V/m}\). **Questions:** (a) Find the phase velocity, the wavelength, and the propagation constant. (b) Determine the characteristic impedance of the medium. (c) Find the amplitude and direction of the magnetic field intensity if the wave is traveling in the \(-z\) direction. **Analysis:** To address these topics, we need to explore fundamental concepts of wave propagation in free space. 1. **Phase Velocity (\(v_p\))**: This is the speed at which the phase of the wave propagates in space. 2. **Wavelength (\(\lambda\))**: The distance between consecutive points of the same phase in the wave, such as peaks or troughs. 3. **Propagation Constant (\(\beta\))**: Represents the phase change per unit length, related to the wavelength. 4. **Characteristic Impedance (\(Z_0\))**: The intrinsic impedance of free space, a measure of the medium’s response to the electromagnetic wave. 5. **Magnetic Field Intensity (\(\vec{H}\))**: The magnitude and directional relationship with the electric field when traveling in a specified direction, applying the right-hand rule for cross-products. Through the derivation of these parameters, students will gain a real-world understanding of electromagnetic wave behavior in various media. Detailed mathematical solutions will follow these explanations to reinforce comprehension.
Transcribed Image Text:**Topic: Electromagnetic Waves in Free Space** A 10-GHz plane wave traveling in free space has the electric field (E-field) represented as: \(\vec{E} = 1 \hat{x} \, \text{V/m}\). **Questions:** (a) Find the phase velocity, the wavelength, and the propagation constant. (b) Determine the characteristic impedance of the medium. (c) Find the amplitude and direction of the magnetic field intensity if the wave is traveling in the \(-z\) direction. **Analysis:** To address these topics, we need to explore fundamental concepts of wave propagation in free space. 1. **Phase Velocity (\(v_p\))**: This is the speed at which the phase of the wave propagates in space. 2. **Wavelength (\(\lambda\))**: The distance between consecutive points of the same phase in the wave, such as peaks or troughs. 3. **Propagation Constant (\(\beta\))**: Represents the phase change per unit length, related to the wavelength. 4. **Characteristic Impedance (\(Z_0\))**: The intrinsic impedance of free space, a measure of the medium’s response to the electromagnetic wave. 5. **Magnetic Field Intensity (\(\vec{H}\))**: The magnitude and directional relationship with the electric field when traveling in a specified direction, applying the right-hand rule for cross-products. Through the derivation of these parameters, students will gain a real-world understanding of electromagnetic wave behavior in various media. Detailed mathematical solutions will follow these explanations to reinforce comprehension.
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