Question

An electromagnetic wave going through vacuum is described by
E = E₀ sin (kx − ωt).
Which of the following is/are independent of the wavelength?
(a) k
(b) ω
(c) k/ω
(d) kω

Answer 1

(c) k/ω

The given quantities can be expressed as:
k is given by
$k=\frac{2\pi }{\lambda }$
ω is given by
$$\begin{gathered} \omega =2\pi \nu \\ c=\nu \lambda \\ \Rightarrow \omega =2\pi \frac{c}{\lambda } \end{gathered}$$
k/ω is given by
$\frac{k}{\omega }=\frac{2\pi /\lambda }{2\pi c/\lambda }=\frac{1}{c}$

kω is given by

$k\times \omega =\frac{2\pi }{\lambda }\times \frac{2\pi c}{\lambda }=\frac{4{\pi }^{2}c}{{\lambda }^2}$
Thus, ​k/ω is independent of the wavelength.