Question

If the temperature of the Sun were to increase from T to 2T and its radius from R to 2R. The ratio of power radiated by it would become:
A. 64 times
B. 16 times
C. 32 times
D. 4 times

Answer 1

Hint: The problem is based on the blackbody radiation concept. Stefan Boltzmann’s law which will be necessary to solve the current problem is $E=\sigma A{{T}^{4}}.$ E refers to energy, A refers to area of the radiating body, T refers to temperature of the body and $\sigma $ refers to Stefan's constant.

Step by step solution:
Let’s start by discussing what is Stefan Boltzmann law. This law states that, the total amount of radiation (energy) emitted from a black body per unit time is proportional to the fourth power of the absolute temperature of the black body.
When the proportionality is removed to equate, then the formula changes to, the radiation energy emitted by a black body is equal to the Stefan’s constant times the area of the black body times the fourth power of the absolute temperature of the black body.
That is, $E=\sigma A{{T}^{4}}.$ Here, E is the radiation energy, $\sigma $ is the Stefan’s constant, A is the area of the black body being considered and T is the absolute temperature of the black body.
The area of sun is given by, $A=4\pi {{R}^{2}}.$
For the first case, where the Sun’s absolute temperature is T and radius of Sun is R, the radiation energy of Sun in this case is: ${{E}_{1}}=\sigma {{A}_{1}}T_{1}^{4},{{A}_{1}}=4\pi {{R}^{2}},{{T}_{1}}=T.$
Therefore, ${{E}_{1}}=\sigma (4\pi {{R}^{2}}){{T}^{4}}\Rightarrow {{E}_{1}}=\sigma 4\pi {{R}^{2}}{{T}^{4}}.$
For the second case, where the Sun’s absolute temperature is 2T and radius of Sun is 2R, the radiation energy of Sun in this case is: \[{{E}_{2}}=\sigma {{A}_{2}}T_{2}^{4},{{A}_{2}}=4\pi {{(2R)}^{2}}\Rightarrow {{A}_{2}}=4(4\pi {{R}^{2}}),{{T}_{2}}=2T.\]
Therefore, ${{E}_{2}}=\sigma (4)(4\pi {{R}^{2}}){{(2T)}^{4}}\Rightarrow {{E}_{1}}=(64)\sigma 4\pi {{R}^{2}}{{T}^{4}}.$
That is, ${{E}_{2}}=64{{E}_{1}}\Rightarrow \dfrac{{{E}_{2}}}{{{E}_{1}}}=64.$ Hence, the power radiated by Sun in the second case would increase by 64 times as compared to the first case.

Note: The Stefan Boltzmann law can also be used for emission of radiation from a hot body, which isn’t a black body. It’s important to know that a hot body is being considered, since radiation can only be released by a body, when the body is excited, hence, a hot body in this case. For a sufficiently high temperature, the radiations from a body can be approximated to black body radiation, however since, it is not a black body completely, hence a part of the radiations will be absorbed back by the hot body as well.
However, the equation of Stefan Boltzmann law changes for emission by a hot body. An additional constant known as emissivity coefficient is added to the equation. This emissivity coefficient (e) is equal to the absorptive power of the hot body, which is specific for every kind of body and ranges from 0 to 1.
Hence, the general Stefan Boltzmann law is $E=\sigma eA{{T}^{4}},$ and for a black body, e=1.