Problem 49.13 (RHK)
A convex lens 3.8 cm in diameter and focal length 26 cm produces an image of the Sun on a thin black screen the same size as the image. We have to find the highest temperature to which the screen can be raised. The effective temperature of the Sun is .
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For answering this problem we will make use of the following basic data of the Sun:
Mean radius of the Sun, ;
Mean orbital radius of the Earth about the Sun, .
The size of the image of the Sun formed by lens of focal length 26 cm will be given by the relation
The amount of energy falling per second on the thin black screen, which is of the same size as the screen, will be equal to the solar energy falling per second on the lens.
We calculate the energy radiated per second by the Sun. It is
The energy from the Sun received at Earth per square meter and per second is
The amount of solar energy that falls on a lens 3.8 cm in diameter per second will be
The highest temperature to which the black screen can be raised by focussing solar energy through the lens will be determined by . The screen acts as a black body at temperature , and at equilibrium the energy emitted per second by the screen will be equal to the energy received by it per second. Therefore,