Question

An electromagnetic wave, going through a vacuum is described by $E=E0\sin (kx-\omega t)$. Which of the following is independent of wavelength?

• A. k
• B. $\omega$
• C. $\frac{k}{\omega }$
• D. $\omega k$

Answer 1

The correct option is C

$\frac{k}{\omega }$

Electromagnetic wave

1. An electromagnetic wave is transverse in nature (example: light).
2. We know. an electromagnetic wave consists of a time-varying electric field $E$and a time-varying magnetic field $B$.
3. The velocity of the electromagnetic wave is $c=3\times {10}^{8}m/s$, which is defined by the form, $c=\nu \lambda$, where, $\lambda$ is the wavelength of the wave and $\nu$ is the frequency of the wave.

Frequency and wavelength

1. Frequency is the number of complete waves in unit time. Frequency is defined as $\nu =\frac{\omega }{2\pi }$, where, $\omega$ is the angular frequency.
2. Wavelength is the distance between two successive wave peaks. wavelength is defined by the form, $\lambda =\frac{2\pi }{k}$, where, k is the wave number.

Explanation for the correct option

Option (C) : $\frac{k}{\omega }$

As we know, the frequency of a wave, $\nu =\frac{\omega }{2\pi }$.

So, $\nu =\frac{\omega }{2\pi }...........\left(1\right)$

and $k=\frac{2\pi }{\lambda }..............\left(2\right)$

So,

$$\begin{gathered} \frac{k}{\omega }=\frac{{\displaystyle \frac{2\pi }{\lambda }}}{2\pi \nu }=\frac{2\pi }{\lambda \times 2\pi \nu }=\frac{1}{\lambda \times \nu }=\frac{1}{c} \\ or\frac{k}{\omega }=\frac{1}{c} \end{gathered}$$

Explanation for incorrect options

Option (A) : k

We know, wave number, $k=\frac{2\pi }{\lambda }$, which is dependent on wavelength, $\lambda$.

Option (B) : $\omega$

Again, angular frequency,

$$\begin{gathered} \omega =2\pi \nu \\ or\omega =\frac{2\pi c}{\lambda } \end{gathered}$$

Which is dependent on wavelength, $\lambda$.

Option (D) : $\omega k$

Now,

$$\begin{gathered} \omega k=2\pi \nu \times \frac{2\pi }{\lambda } \\ or\omega k=4{\pi }^2\times \frac{\nu }{\lambda } \end{gathered}$$

It is also dependent on wavelength, $\lambda$.

Hence, option (C) is correct.