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Question

Assuming the sun to be a spherical body of radius R at a temperature TK, Evaluate the total radiant power incident on the Earth. (r is the distance between the sun and the earth, R0 is the radius of Earth and σ is Stefan’s Constant)


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Solution

Step 1: Given parameters

Sphericalradius=R

Temperature=T

DistancebetweentheSunandtheEarth=r

RadiusofEarth=R0

Step 2: Calculate the total Radiant Power incident on the Earth

Assuming the sun as a perfect black body, energy radiated per sec by the sun using Stefan's law is given by,
P=σAT4

Where, Ais the area of the sun, P is energy radiated per sec, σ is Stefan’s Constant, and,T is the temperature in kelvin.

P=σ×4πR2T4...1

The intensity of this power at the Earth's surface is
I=P4πr2=σ×4πR2T44πr2=σR2T4r2​,
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 πr02, hence total radiant power as received by the Earth is:

PE=πr02I=πr02σR2T4r2

Hence, the total radiant power as received by the Earth is PE=πr02σR2T4r2.


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