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Question

Assuming the sun to be a spherical body of radius $$R$$ at a temperature $$T$$ K, evaluate the total radiant power incident on earth. ($$r$$ is the distance between the sun and the earth, $$r_{0}$$ is the radius of earth and $$\sigma$$ is stefans constant) :


A
4πr20R2σT4r2
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B
πr20R2σT4r2
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C
πr20R2σT44πr2
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D
R2σT4r2
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Solution

The correct option is B $$\displaystyle \frac{\pi r^{2}_{0}R^{2}\sigma T^{4}}{r^{2}}$$
Assuming sun as a perfect blackbody, energy radiated per sec by sun using Stefan's law is:

$$P=\sigma AT^4$$ (Where A is the area of the sun, P is energy radiated per second)

$$\Rightarrow P=\sigma \times4\pi R^2T^4$$.................(1)
The intensity of this power at earth's surface is (assuming $$r>>r_o$$)
$$I=\dfrac{P}{4\pi r^2}$$

$$\Rightarrow I=\dfrac{\sigma\times 4\pi R^2T^4}{4\pi r^2}$$, (Putting the value from the equation (1))

$$\Rightarrow I=\dfrac{\sigma R^2T^4}{r^2}$$

Since the earth is very far from the sun, out of the total energy radiated, a small fraction of it is received by the earth. Earth can be considered as a small disc whose radius is the radius of the earth.
The surface area of the disc is $$\pi{r_0}^2$$, hence total radiant power as received by the earth is:
$$P_E=\pi r_0^2\times I$$

$$P_E=\dfrac{\pi r_o^2\times \sigma R^2T^4}{r^2}$$

Physics

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