Question

An electromagnetic wave going through vacuum is described by $E=E_{0} \sin (k x-\cot )$. Which of the following is independent of the wavelength? (A) \(\omega\) (B) \((\mathrm{k} / \mathrm{c})\) (C) \(\mathrm{k}_{\mathfrak{e}}\) (D) \(\mathrm{k}\)

Short answer

None of the given options seem to be independent of wavelength. However, if we exclude option (C) as it is poorly defined in this context, we can conclude that all other options depend on the wavelength.

Step-by-step solution

Write given equation in a more readable form

First, let's rewrite the given equation in a more readable form using latex notation: \(E = E_0 \sin(kx - \omega t)\)

Identify the relationships between variables

Wave number (k) and wavelength (λ) are related by: \(k = \frac{2\pi}{\lambda}\) Angular frequency (ω) and wave speed (c) are related as: \(\omega = ck\)

Analyze the given options with respect to wavelength

Now let's analyze each option to see if it depends on the wavelength: (A) \(\omega\) By using the relationship between \(\omega\), c, and k, we can see that: \(\omega = c \cdot \frac{2\pi}{\lambda}\) Here, \(\omega\) depends on \(\lambda\). (B) \(\frac{k}{c}\) By using the relationship between wave number and wavelength, we can express this option in terms of wavelength: \(\frac{k}{c} = \frac{\frac{2\pi}{\lambda}}{c}\) Here, this expression depends on \(\lambda\). (C) \(\mathrm{k}_{\mathfrak{e}}\) Option (C) seems to introduce an unknown variable \(\mathrm{k}_{\mathfrak{e}}\), which is not useful in this context. We cannot judge if this is independent of wavelength or not. (D) \(k\) From the relationship between wave number (k) and wavelength (λ): \(k = \frac{2\pi}{\lambda}\) Here, k is dependent on \(\lambda\).

Choose the correct option

Based on our analysis, none of the given options seem to be independent of wavelength, so we cannot give a definitive answer. However, if we exclude option (C) as it is poorly defined in this context, we can conclude that all other options depend on the wavelength.