Q100 / 180 · JIPMER 2015 · Physics

The earth receives solar radiation at a rate of $8.2 \mathrm{~J} \mathrm{~cm}^{-2} \mathrm{~min}^{-1}$. If the sun radiates as the black bodies, the temperature at the surface of the sun will be (the angle subtended by sun on the earth in suppose $0.53^{\circ}$ and Stefan constant is $\sigma=5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^4$ )

  • Correct answer
    A
    5800 K
  • B
    6700 K
  • C
    8000 K
  • D
    7800 K

Solution

Let diameter of the sun is $d$ while radius of the earth is $r$ then $\begin{aligned} \frac{D}{2} & \approx 0.53 \times \frac{\pi}{180} \\ & =9.25 \times 10^{-3}\end{aligned}$ The radiation emitted per unit area of the sun is $4 \pi\left(\frac{d}{2}\right)^2 \sigma T^A=\pi D^2 \sigma T^A$ The radiation received at the earths surface per unit time per unit area is $\frac{\pi \sigma d T^4}{4 \pi r^2}=\frac{\sigma T^4}{4}\left(\frac{d}{R}\right)^2$ $\begin{array}{rlrl}\Rightarrow & & \frac{\sigma T^4}{4}\left(\frac{D}{R}\right)^2 & =82 \mathrm{Jcm}^{-2} \mathrm{~min}^{-1} \\ & \text { or } & \frac{1}{4} \times 5.67 \times 10^{-8} \times T^4 \times\left(9.25 \times 10^{-3}\right)^2 \\ & = & \frac{8.2}{10^{-4} \times 60} \mathrm{Wm}^{-2} \\ \Rightarrow & & T & =5794 \mathrm{~K} \cong 5800 \mathrm{~K}\end{array}$

Hence, the correct answer is option: 'A'.

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