An electromagnetic wave of frequency 10 MHz, propagating in a medium, has the magnitude of its propagation vector 0.5 m−1. The refractive index of the medium is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2.39
The propagation vector (wave number) is \( k = 2\pi/\lambda \), and the wave speed in the medium is \( v = \omega/k \). The refractive index is the ratio of the speed in vacuum to the speed in the medium.
\( \displaystyle n = \frac{c}{v} = \frac{ck}{\omega} = \frac{ck}{2\pi f} \)
\( f = 10\,\mathrm{MHz} = 1.0\times10^{7}\,\mathrm{Hz} \)
\( k = 0.5\,\mathrm{m^{-1}} \)
\( \displaystyle v = \frac{2\pi f}{k} = \frac{2 \times 3.14 \times 1.0\times10^{7}\,\mathrm{Hz}}{0.5\,\mathrm{m^{-1}}} \approx 1.256\times10^{8}\,\mathrm{m\,s^{-1}} \)
\( \displaystyle n = \frac{3\times10^{8}\,\mathrm{m\,s^{-1}}}{1.256\times10^{8}\,\mathrm{m\,s^{-1}}} \)
\( \displaystyle n \approx 2.39 \)