Question

The earth receives solar radiation, at a rate of $8.2{\text{J cm}}^{-2}{\text{min}}^{-1}$. Assuming that the sun radiates like a blackbody, calculate the surface temperature of the sun. The angle subtended by the sun on the earth is ${0.53}^{\circ }$ and the Stefan's constant $\sigma =5.67\times {10}^{-8}$${\text{Wm}}^{-2}{\text{K}}^{-4}$ .

• A. $4700\text{K}$
• B. $5800\text{K}$
• C. $6200\text{K}$
• D. $6600\text{K}$

Answer 1

The correct option is B $5800\text{K}$


Let the radius of the Sun be $R_S$ and the distance between Sun and Earth be $R$.

Given
$\frac{2R_S}{R}\simeq {0.53}^{\circ }=0.53\times \frac{\pi }{180}\text{rad}$

$\Rightarrow \frac{2R_S}{R}=9.25\times {10}^{-3}\text{rad}$

Radiation emitted by Sun per unit time is given by Stefan's law,

$Q=\sigma A{T}^4=\sigma 4\pi R_S^2{T}^4$

At a distance $R$ from the Sun this radiation falls on an area $4\pi R^2$ around the Sun.

The radiation received per unit time per unit area on earth surface is,

$\frac{\sigma 4\pi R_S^2{T}^4}{4\pi R^2}=\sigma T^4{\left(\frac{R_S}{R}\right)}^2$

From the given data,

$\sigma T^4{\left(\frac{R_S}{R}\right)}^2=8.2{\text{J cm}}^{-2}{\text{min}}^{-1}$

$\Rightarrow 5.67\times {10}^{-8}\times T^4\times {\left(\frac{9.25\times {10}^{-3}}{2}\right)}^2=\frac{8.2}{{10}^{-4}\times 60}$

$\Rightarrow T=5794\text{K}\simeq 5800\text{K}$

Hence, option $\left(B\right)$ is correct.

Why this question? Caution: While using the relationship between, sun radius, distance between the sun and the earth and angle subtended by sun on earth, use angle in radian not in degree.