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- Homework Statement
- The intensity of light coming from a distant star is measured using two identical instruments A and B, where A is placed in a satellite outside the Earth's atmosphere and B is placed on the

Earth's surface. The results are as follows:

For green (500nm wavelength), intensity of light at A and B (in nW) is 100 and 50 respectively.

For red (700nm wavelength), intensity of light at A and B (in nW) is 200 and x respectively.

Assuming that there is scattering, but no absorption of light in the Earth's atmosphere at these wavelengths, the value of x can be estimated as:

Options:

(a) 177

(b) 167

(c) 157

(d) 147

(e) 137

- Relevant Equations
- (1) ##I = e\sigma T^4##

(2) ## \lambda T = constant = k##

I actually am not sure what equations are relevant here but I thought these are the relevant ones.

By Stefan-Boltzman Law, the intensity absorbed by the Earth is given as ## I = e \sigma T^4## where e is the emissivity of Earth, ##\sigma## is Stefan-Boltzman constant and T is the temperature of the Earth. The values of ##I## for green and red then are ## 50-100 = -50## and ##x-200## respectively.

Now this is where I am stuck. I am not sure if what I do next is valid: By using Wien's displacement law, ##I## may be given as ##I = \frac{e\sigma k }{\lambda^4} ##. Since the wavelengths are given, I simply divide the respective intensity and obtain

\begin{equation}

\frac{x-200}{-50} =\frac{500^4}{700^4}

\end{equation}

Solving for ##x##, I get ##x \approx 187\ nW##. But this is not the right answer. The correct answer is 167 ##nW##.

Can someone please explain where I am going wrong?

**My Approach:**By Stefan-Boltzman Law, the intensity absorbed by the Earth is given as ## I = e \sigma T^4## where e is the emissivity of Earth, ##\sigma## is Stefan-Boltzman constant and T is the temperature of the Earth. The values of ##I## for green and red then are ## 50-100 = -50## and ##x-200## respectively.

Now this is where I am stuck. I am not sure if what I do next is valid: By using Wien's displacement law, ##I## may be given as ##I = \frac{e\sigma k }{\lambda^4} ##. Since the wavelengths are given, I simply divide the respective intensity and obtain

\begin{equation}

\frac{x-200}{-50} =\frac{500^4}{700^4}

\end{equation}

Solving for ##x##, I get ##x \approx 187\ nW##. But this is not the right answer. The correct answer is 167 ##nW##.

Can someone please explain where I am going wrong?