Question

What is the wavenumber of a beam of light in air if its frequency is $14\times {{10}^{14}}Hz$.

& A.\,4.667\times {{10}^{8}}{{m}^{-1}} \\\ & B.\,5.667\times {{10}^{6}}{{m}^{-1}} \\\ & C.\,8.667\times {{10}^{6}}{{m}^{-1}} \\\ & D.\,4.667\times {{10}^{6}}{{m}^{-1}} \\\ \end{aligned}$$

Answer 1

Compute the value of the wavelength of the light beam present in air by dividing the speed of light in the air by the given value of the frequency of the light beam. The wave number is the inverse of the wavelength. So, by reversing the value of the wavelength, we will get the value of the wave number.
Formula used:

& c=f\lambda \\\ & k=\dfrac{1}{\lambda } \\\ \end{aligned}$$ **Complete answer:** From the given information, we have the data as follows. The frequency of a beam of light in air is,$$14\times {{10}^{14}}Hz$$. As the beam of light is in the air according to the given question statement. Thus, we will be using the general formula that relates the frequency and wavelength of the light with the speed of light in the air. Consider the formula that relates the parameters of the wave, that is, frequency and the wavelength of the light with the speed of light in air. $$c=f\lambda $$ The speed of light in air is equal to the product of the frequency and the wavelength of the light. We know the value of the speed of light in air and the value of the frequency of the beam of light in air is given, so, substitute these values to compute the value of the wavelength of the light. So, we have, $$\begin{aligned} & 3\times {{10}^{8}}=14\times {{10}^{14}}\times \lambda \\\ & \Rightarrow \lambda =\dfrac{3\times {{10}^{8}}}{14\times {{10}^{14}}} \\\ & \therefore \lambda =2.142\times {{10}^{-7}}m \\\ \end{aligned}$$ Now consider the formula that relates the parameters of the wave, that is, the wavelength of the light with the wave number of the beam of light. $$k=\dfrac{1}{\lambda }$$ The wave number of a light beam is equal to one by the wavelength of the light. Just now we have obtained the value of the wavelength of the light beam in air , so, substitute the value to compute the value of the wave number of the light. So, we have, $$\begin{aligned} & k=\dfrac{1}{2.142\times {{10}^{-7}}} \\\ & \therefore k=4.667\times {{10}^{6}}{{m}^{-1}} \\\ \end{aligned}$$ $$\therefore $$ The value of the wave number of the beam of light in air is $$4.667\times {{10}^{6}}{{m}^{-1}}$$. **Thus, option (D) is correct.** **Note:** The wavelength and the wave number are inverse to each other. So, we can say that, with the increase in wave number, the frequency of the wave increases and vice versa. Whereas, with the increase in wavelength, the frequency of the wave decreases and vice versa.