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Chapter 11: Problem 23

If the temperature of the sun is increased from \(T\) to \(2 T\) and its radius from \(R\) to \(2 R\), then the ratio of the radiant energy received on earth to what it was previously will be (A) 4 (B) 16 (C) 32 (D) 64

Short Answer

Expert verified
The ratio of the radiant energy received on earth to what it was previously will be 16, which corresponds to option (B).

Step by step solution

01

Calculate Initial Radiant Energy

The initial radiant energy can be calculated using the Stefan-Boltzmann law, which is given by \(P1 = 4\pi R^2 \sigma T^4\). Where \(R\) is the radius, \(T\) is the temperature and \(\sigma\) is the Stefan-Boltzmann constant.
02

Calculate Final Radiant Energy

The final radiant energy is calculated, again using the Stefan-Boltzmann law, taking into account the changes mentioned in the exercise. The temperature is now \(2T\) and the radius is now \(2R\). As such, the Stefan-Boltzmann law expression becomes \(P2 = 4\pi(2R)^2\sigma(2T)^4\). After simplification, \(P2 = 64\pi R^2 \sigma T^4\).
03

Form the ratio of Final to Initial Radiant Energy

We find the ratio of the Final Radiant Energy to the Initial Radiant Energy by dividing \(P2\) by \(P1\), that is, \(\frac{P2}{P1}\). Here \(P2\) is the final radiant energy and \(P1\) is the initial radiant energy. The ratio becomes \(\frac{64\pi R^2 \sigma T^4}{4\pi R^2\sigma T^4}\).
04

Simplify the Ratio

Upon simplifying, we find that the ratio simplifies to 16 since when \(\frac{64\pi R^2 \sigma T^4}{4\pi R^2\sigma T^4}\) is simplified, the term \(\pi R^2\sigma T^4\) cancels out in the numerator and the denominator, leaving behind \(\frac{64}{4}\), which simplifies further to 16.

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