A plane wave travels through air. The electric field has the form Ē(t) = 0.5 sin(@t-kx)2 [V/m] Additional measurements indicate that the wavelength is λ = 30cm. What is the direction of propagation? b. Find the propagation constant k and the frequency @. Write the electric and magnetic fields in phasor form. d. Find the magnetic field, H(t).
A plane wave travels through air. The electric field has the form Ē(t) = 0.5 sin(@t-kx)2 [V/m] Additional measurements indicate that the wavelength is λ = 30cm. What is the direction of propagation? b. Find the propagation constant k and the frequency @. Write the electric and magnetic fields in phasor form. d. Find the magnetic field, H(t).
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![**Plane Wave Propagation in Air: Electric and Magnetic Fields Analysis**
A plane wave travels through air, and its electric field is given by:
\[
\vec{E}(t) = 0.5 \sin(\omega t - kx) \hat{z} \quad [\text{V/m}]
\]
Additional measurements indicate that the wavelength is \(\lambda = 30\, \text{cm}\).
**Questions:**
a. **What is the direction of propagation?**
b. **Find the propagation constant \(k\) and the frequency \(\omega\).**
c. **Write the electric and magnetic fields in phasor form.**
d. **Find the magnetic field, \(\vec{H}(t)\).**
**Notes:**
- **Direction of Propagation:** In the expression \(\vec{E}(t) = 0.5 \sin(\omega t - kx) \hat{z}\), the wave is propagating in the direction of the \(x\)-axis.
- **Propagation Constant and Frequency:** Use the relationship \(k = \frac{2\pi}{\lambda}\) to find \(k\), and the angular frequency \(\omega = 2\pi f\) where \(f\) is the frequency.
- **Phasor Form:** Represent the electric and magnetic fields using complex notation to simplify analysis in the frequency domain.
- **Magnetic Field Calculation:** Use Maxwell’s equations to determine \(\vec{H}(t)\) given \(\vec{E}(t)\). The relationship between electric and magnetic fields in a plane wave is given by \(\vec{H} = \frac{1}{\eta} \vec{E} \times \hat{a}_{\text{propagation}}\), where \(\eta\) is the intrinsic impedance of the medium.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F584ffa08-94ad-4eb4-a87d-fd258db9f802%2Ff4539438-2d0e-4fe7-a9ea-10e528749f4d%2Fgrnyvhk_processed.png&w=3840&q=75)
Transcribed Image Text:**Plane Wave Propagation in Air: Electric and Magnetic Fields Analysis**
A plane wave travels through air, and its electric field is given by:
\[
\vec{E}(t) = 0.5 \sin(\omega t - kx) \hat{z} \quad [\text{V/m}]
\]
Additional measurements indicate that the wavelength is \(\lambda = 30\, \text{cm}\).
**Questions:**
a. **What is the direction of propagation?**
b. **Find the propagation constant \(k\) and the frequency \(\omega\).**
c. **Write the electric and magnetic fields in phasor form.**
d. **Find the magnetic field, \(\vec{H}(t)\).**
**Notes:**
- **Direction of Propagation:** In the expression \(\vec{E}(t) = 0.5 \sin(\omega t - kx) \hat{z}\), the wave is propagating in the direction of the \(x\)-axis.
- **Propagation Constant and Frequency:** Use the relationship \(k = \frac{2\pi}{\lambda}\) to find \(k\), and the angular frequency \(\omega = 2\pi f\) where \(f\) is the frequency.
- **Phasor Form:** Represent the electric and magnetic fields using complex notation to simplify analysis in the frequency domain.
- **Magnetic Field Calculation:** Use Maxwell’s equations to determine \(\vec{H}(t)\) given \(\vec{E}(t)\). The relationship between electric and magnetic fields in a plane wave is given by \(\vec{H} = \frac{1}{\eta} \vec{E} \times \hat{a}_{\text{propagation}}\), where \(\eta\) is the intrinsic impedance of the medium.
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