Assuming The Sun To Be A Spherical Body Of Radius R At A Temperature T K, 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)

Assuming the sun as a perfect black body, energy radiated per sec by the sun using Stefan’s law is given by the formulae:

P = \(\sigma AT^{4}\)

Where:

A is the area of the sun

P is energy radiated per second

= \(P = \sigma * 4\pi R^{2}T^{4}—-[1]\)

The intensity of this power at the earth’s surface is (assuming \( r > > r_{o}\) \(I = \frac{P}{4\pi r^{2}}\)

=\(I = \frac{\sigma * 4\pi R^{2}T^{4}}{4\pi R^{2}}\)

Now by substituting the values from the equation [1], we get

= \(I = \frac{\sigma * 4\pi R^{2}T^{4}}{4\pi R^{2}}\)

=\(I = \frac{\sigma R^{2}T^{4}}{R^{2}}\)

Since the earth is very far from the sun, out of the total energy radiated, a small fraction is received by the planet. Therefore, Earth can be considered as a small disc, whose radius is the radius of the planet.

The surface area of the disc is \( \pi r_{0}^{2}\), hence total radiant power as received by the earth is:

= \(P_{E} = \pi r_{0}^{2} * I\)

= \(P_{E} = \frac{\pi r_{0}^{2} * \sigma R^{2} T^{4}}{r^{2}}\)

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