The electric field of a plane electromagnetic wave, travelling in an unknown nonmagnetic medium is g

Question

The electric field of a plane electromagnetic wave, travelling in an unknown nonmagnetic medium is given by, $ E_{\mathrm{y}}=20 \sin \left(3 	imes 10^6 x-4.5 	imes 10^{14} \mathrm{t}ight) \mathrm{V} / \mathrm{m} $ (where $x, \mathrm{t}$ and other values have S.I. units). The dielectric constant of the medium is $\_\_\_\_$ (speed of light in free space is $3 	imes 10^8 \mathrm{~m} / \mathrm{s}$ )

Professor PC Thomas Classes · Answer

Comparing the given electric field equation with the standard wave equation $E = E_0 \sin(kx - \omega t)$, we identify the angular wave number $k = 3 	imes 10^6 	ext{ m}^{-1}$ and the angular frequency $\omega = 4.5 	imes 10^{14} 	ext{ rad/s}$. The speed of the wave in the medium is $v = \frac{\omega}{k} = \frac{4.5 	imes 10^{14}}{3 	imes 10^6} = 1.5 	imes 10^8 	ext{ m/s}$. The refractive index is $\mu = \frac{c}{v} = \frac{3 	imes 10^8}{1.5 	imes 10^8} = 2$. For a non-magnetic medium ($\mu_r \approx 1$), $\mu = \sqrt{\epsilon_r \mu_r} = \sqrt{\epsilon_r}$. Thus, the dielectric constant is $\epsilon_r = \mu^2 = 2^2 = 4$.