Question

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

Answer 1

Hint: From the concept of Stefan-Boltzmann radiation law, we can say that the radiant energy of the sun is directly proportional to the product of square of surface area and fourth power of absolute temperature of the sun. We will use this relationship to find out the ratio final radiant to its initial value.

Complete step by step answer:
Given:
Initial temperature of the sun is \[{T_1} = T\].
Final temperature of the sun is \[{T_2} = 2T\].
Initial radius of the sun is \[{R_1} = R\].
Final radius of the sun is \[{R_2} = 2R\].
We have to evaluate the ratio of final radiant energy to the initial radiant energy \[\dfrac{{{E_2}}}{{{E_1}}}\].
From the concept of Stefan-Boltzmann’s radiation law, we can write the expression for initial radiant energy of the sun.
\[{E_1} = \sigma {A_1}T_1^4\]
Here \[\sigma \] is the Stefan-Boltzmann constant of radiation.
As the value of Stefan-Boltzmann law is constant so we can replace it with the sign of proportionality in the expression of radiant energy.
\[{E_1} \propto {A_1}T_1^4\]……(1)
We can consider sun as a spherical body of radius R whose surface area is given as:
\[A = 4\pi {R^2}\]
Rearranging the above expression to establish a proportional relation between surface area and radius of the sun.
\[\begin{array}{l}
A \propto {R^2}\\
\Rightarrow A = k{R^2}
\end{array}\]
This means that surface area of the sun is directly proportional to its radius.
Substitute \[kR_1^2\] for \[{A_1}\] in equation (1).
\[\begin{array}{l}
{E_1} \propto \left( {kR_1^2} \right)T_1^4\\
\Rightarrow {E_1} \propto R_1^2T_1^4
\end{array}\]……(2)
Again using the concept of Stefan-Boltzmann’s radiation law, we can write the expression for final radiant energy of the sun.
\[{E_2} \propto {A_2}T_2^4\]
Substitute \[kR_2^2\] for \[{A_2}\] in the above expression.
\[\begin{array}{l}
{E_2} \propto \left( {kR_2^2} \right)T_2^4\\
\Rightarrow {E_2} \propto R_2^2T_2^4
\end{array}\]
It is given that the final temperature of the sun is twice of its initial value and its final radius is also twice of its initial radius. Therefore, substitute \[2T\] for \[{T_2}\] and \[2R\] for \[{R_2}\] in the above expression.
\[\begin{array}{l}
{E_2} \propto {\left( {2R} \right)^2}{\left( {2T} \right)^4}\\
\Rightarrow {E_2} \propto 64\left( {{R^2}{T^4}} \right)
\end{array}\]……(3)
Divide equation (3) by equation (2).
\[\dfrac{{{E_2}}}{{{E_1}}} = 64\]
Therefore, the ratio of final radiating energy of the sun to its previous value is \[64\].

So, the correct answer is “Option D”.

Note:
Do not just substitute the temperature change in the expression of final radiant energy of the sun because radius is also increasing and radiant energy is the function of surface area which is directly proportional to radius of the run. Hence, change in radius is also to be taken into account.