GATE 2015 EC – Question 65
Consider a uniform plane wave with amplitude ($E_0$) of 10 V/m and 1.1 GHz frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity ($\epsilon_r$) and permeability ($\mu_r$) as shown in the figure.
[Figure: A plane wave in air ($\eta = 120\pi\ \Omega$) with $|E_0| = 10$ V/m at 1.1 GHz meets a dielectric with $\mu_r = 1 - j2$ and $\epsilon_r = 1 - j2$. The field is measured after 10 cm inside the dielectric.]
The magnitude of the transmitted electric field component (in V/m) after it has travelled a distance of 10 cm inside the dielectric region is ________.
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Show answer and explanation
Correct answer: 0.08 to 0.12
Explanation
Because $\mu_r = \epsilon_r$, the intrinsic impedance of the dielectric equals that of air, so there is no reflection and the transmitted amplitude at the surface is the full 10 V/m. The refractive index is $\sqrt{\mu_r\epsilon_r} = 1 - j2$, so the attenuation constant is $\alpha = k_0 \times 2$ with $k_0 = \frac{2\pi \times 1.1 \times 10^9}{3 \times 10^8} = 23.04$ rad/m, so $\alpha = 46.07$ Np/m. After 0.1 m the field is $10 \times e^{-4.607} = 0.1$ V/m.