Question

In a given direction, the intensities of the scattered light by a scattering substance for two beams of light are in the ratio of $256:81.$ The ratio of the frequency of the first beam to the frequency of the second beam is ?

• A. $64:127$
• B. $21:123$
• C. $64:27$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

Option (D):

Step 1: Given data:

The ratio of the intensities of the scattered light by a scattering substance for two beams of light is given as= $256:81.$

The ratio of the frequency of the first beam to the frequency of the second beam is=?

Step 2: Formula applied:

According to the Rayleigh scattering effect, the amount of scattering of the light is inversely proportional to the fourth power of the wavelength ,which can be given as:

$$\begin{gathered} \mathrm{I}\mathrm{\alpha }\frac{1}{{\left(\mathrm{\lambda }\right)}^4} \\ \mathrm{or},\mathrm{I}\mathrm{\alpha }{\left(\mathrm{f}\right)}^4 \end{gathered}$$

Where,

$\mathrm{I}=$Inetnsity of scattered light,

$\mathrm{\lambda }=$wavelenghth of scattered light

$\mathrm{f}=$ Frequency of scattered light

$$\begin{gathered} \therefore \frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}={\left(\frac{{\mathrm{f}}_1}{{\mathrm{f}}_2}\right)}^4 \\ \Rightarrow \frac{{\mathrm{f}}_1}{{\mathrm{f}}_2}={\left(\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}\right)}^4 \end{gathered}$$

Step 3: Calculation of the frequency ratio:

Putting the given data in the above formula, the ratio of the frequencies can be calculated as:

$$\begin{gathered} \frac{{\mathrm{f}}_1}{{\mathrm{f}}_2}={\left(\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}\right)}^{\frac{1}{4}} \\ \Rightarrow \frac{{\mathrm{f}}_1}{{\mathrm{f}}_2}={\left(\frac{{}_{256}}{81}\right)}^{\frac{1}{4}} \\ \Rightarrow \frac{{\mathrm{f}}_1}{{\mathrm{f}}_2}=\frac{4}{3} \end{gathered}$$

Hence, the ratio of the frequency of the first beam to the frequency of the second beam is found to be $4:3$, and none of the given options are correct.