Q3.2-20E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

From theoretical considerations, it is known that light from a certain star should reach Earth with intensity l₀ . However, the path taken by the light from the star to Earth passes through a dust cloud, with absorption coefficient 0.1/light-year. The light reaching Earth has intensity 1/2 l₀. How thick is the dust cloud? (The rate of change of light intensity with respect to thickness is proportional to the intensity. One light-year is the distance travelled by light during 1 yr.)

Step-by-Step Solution

Answer

The thickness of the dust cloud is 6.93 light years.

Step1: Analyzing the given statement

Given that the rate of change of light intensity with respect to thickness is proportional to the intensity.

Let the intensity be I. Therefore, $\frac{\mathbf{dI}}{\mathbf{dt}}\mathbf{\propto }\mathbf{I}$  Also given that the light from a certain star should reach Earth with intensity  $I_0$. The absorption coefficient is 0.1/light-year. The intensity of light reaching the earth is  $\frac{1}{2}{I}_0$. We have to find the thickness of dust cloud.

Step2: Determining the relation for the intensity of light with the help of given proportionality relation, to solve the question

Given,  

$$\begin{gathered} \frac{dI}{I}\propto I \\ \frac{dI}{I}=-\lambda I \end{gathered}$$

 where, $\lambda$ is the constant of proportionality.

$\frac{dI}{I}=-\lambda dt$    

Integrating both sides,

 $\ln I=-\lambda t+\ln I_0$           

where, $\ln I_0$ is an arbitrary constant.   

$$\begin{gathered} \ln I-\ln I_0=-\lambda t \\ \ln \left(\frac{I}{I_0}\right)=-\lambda t \\ \frac{I}{ I_0}=e^{-\lambda t} \\ I=I_0{e}^{-\lambda t} \end{gathered}$$   

$\mathit{I}\mathbf{=}{\mathit{I}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{\lambda }\mathbf{t}}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\left(1\right)$           

 Hence, the intensity of light, when the thickness is t, given by the relation   $\mathit{I}\mathbf{=}{\mathit{I}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{\lambda }\mathbf{t}}$.

Step3: Determining the thickness of the dust cloud

The intensity of light reaching the earth is $\frac{1}{2}{I}_0$ ,

Therefore, from equation (1),

 $$\begin{gathered} \frac{I_0}{2}=I_0 e^{-\lambda t} \\ \frac{1}{2}=e^{-\lambda t} \\ {e}^{-\lambda t}=2 \\ \lambda t=\ln 2 \end{gathered}$$

Given that the absorption coefficient is 0.1/light-year i.e.,  $\lambda =0.1$ 

Therefore,

 $$\begin{gathered} (0.1)t=\ln 2 \\ t=\frac{\ln 2}{0.1} \\ t=6.93 light years \end{gathered}$$

  Hence, the thickness of the dust cloud is 6.93 light years.