Arrange the formulas of $\beta$ in the order where plane wave is travelling in lossy dielectric, free space, good conductor, loss-less dielectric respectively. A. $\beta = \omega \sqrt{\frac{\mu \epsilon}{2} [\sqrt{1 + (\frac{\sigma}{\omega \epsilon})^2} +1]}$ B. $\beta = \frac{\omega}{C}$ C. $\beta = \omega \sqrt{\mu \epsilon}$ D. $\beta = \sqrt{\frac{\omega \mu \sigma}{2}}$ Choose the correct answer from the options given below:
Identifying Phase Constant Formulas
The question asks to arrange the formulas for the phase constant ($\beta$) for plane wave propagation in four different media: lossy dielectric, free space, good conductor, and lossless dielectric.
- Formula A: $\beta = \omega \sqrt{\frac{\mu \epsilon}{2} [\sqrt{1 + (\frac{\sigma}{\omega \epsilon})^2} +1]}$ represents the phase constant in a lossy dielectric medium, characterized by the conductivity $\sigma$.
- Formula B: $\beta = \frac{\omega}{C}$ represents the phase constant in free space, where $C$ is the speed of light ($c = 1/\sqrt{\mu_0 \epsilon_0}$).
- Formula C: $\beta = \omega \sqrt{\mu \epsilon}$ is the formula for the phase constant in a lossless dielectric medium.
- Formula D: $\beta = \sqrt{\frac{\omega \mu \sigma}{2}}$ is the approximate formula for the phase constant in a good conductor at high frequencies ($\sigma \gg \omega \epsilon$).
Ordering Beta Formulas by Medium
The required order is: Lossy dielectric, Free space, Good conductor, Lossless dielectric.
Matching the formulas to this order:
- Lossy dielectric: A
- Free space: B
- Good conductor: D
- Lossless dielectric: C
Therefore, the correct sequence is A, B, D, C.
Important Questions from Electromagnetic Wave Propagation
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A plane wave is propagating in a uniform medium with propagation constant \((0.6\pi + j\,0.4\pi)\) its wavelength is
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Reason (R) : Throughout the ionosphere, there are several layers in which ionization density either reaches maximum or remains almost constant. It depend on the intensity of sun. It also depends upon the atmospheric pressure.
Ultra High Frequency (UHF) spectrum is defined as :
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Reason (R) : The amount of energy depends on the intensity of the light rays. The energy associated with each photon is proportional to the frequency.
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