Match List-I with List-II (For an electromagnetic wave of frequency \( \nu \) propagating in a medium, where symbols have their usual meaning)
| List-I | List-II | ||
|---|---|---|---|
| (Physical quantity) | (Expression) | ||
| (A) | Refractive index of a medium | (I) | \( h\nu c^{-1} \) |
| (B) | Speed | (II) | \( 2\pi\nu c^{-1} \) |
| (C) | Propagation constant | (III) | \( ck\omega^{-1} \) |
| (D) | Momentum | (IV) | \( \omega k^{-1} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (III), (B) - (IV), (C) - (II), (D) - (I)
For a wave of frequency \( \nu \), angular frequency \( \omega = 2\pi\nu \) and propagation constant (wave number) \( k = 2\pi/\lambda \), the speed in the medium is \( v = \nu\lambda = \omega/k \). Each List-I quantity follows from these relations.
(A) Refractive index of a medium — this matches (III) \( ck\omega^{-1} \). The refractive index is the ratio of the speed in vacuum to the speed in the medium.
\( \displaystyle n = \frac{c}{v} = \frac{c}{\omega/k} = \frac{ck}{\omega} \)
(B) Speed — this matches (IV) \( \omega k^{-1} \). The wave travels one wavelength in one period.
\( \displaystyle v = \nu\lambda = \frac{\omega}{2\pi}\cdot\frac{2\pi}{k} = \frac{\omega}{k} \)
(C) Propagation constant — this matches (II) \( 2\pi\nu c^{-1} \). With \( \lambda = c/\nu \), as the expression in List-II is written with \( c \):
\( \displaystyle k = \frac{2\pi}{\lambda} = \frac{2\pi\nu}{c} \)
(D) Momentum — this matches (I) \( h\nu c^{-1} \). A photon of energy \( E = h\nu \) carries momentum \( p = E/c \).
\( \displaystyle p = \frac{h\nu}{c} = \frac{h}{\lambda} \)