At our distance from the Sun, the intensity of solar radiation is 1370 W/m^2. Th

Question

At our distance from the Sun, the intensity of solar radiation is 1370 W/m^2. The temperature of the Earth is affected by the greenhouse effect of the atmosphere. This phenomenon describes the effect of absorption of infrared light emitted by the surface so as to make the surface temperature of the Earth higher than if it were airless. For comparison, consider a spherical object of radius r with no atmosphere at the same distance from the Sun as the Earth. Assume its emissivity is the same for all kinds of electromagnetic radiation and its temperature is uniform over its surface. (a) Explain why the projected area over which it absorbs sunlight is ?r^2 and the surface area over which it radiates is 4?r^2.

Explanation / Answer

At the distance the Earth is from the sun, we can treat the incoming light as planar--that is, as coming from parallel rays of light. Because of this, the cross-sectional area of the object can be used for absorption of light, which in the case of the sphere is r2. This can be visuallized by thinking about what area of light rays are being stopped by the object (ie. the shadow it casts); a light bulb on the ceiling will approximate planar rays on the floor with a small ball--look at the area of the shadow of a ping-pong ball on the floor when the light is directly overhead.

Once energy is absorbed, however, it is radiated evenly in all directions, since we are assuming that temperature is uniform across the object. Thus, we must use the surface area of a sphere, 4r2, for the radiative area.

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