Two spheres ' $\mathrm{S}_1$ ' and ' $\mathrm{S}_2$ ' have same radii but temperatures are ' $\mathrm{T}_1$…
Two spheres ' $\mathrm{S}_1$ ' and ' $\mathrm{S}_2$ ' have same radii but temperatures are ' $\mathrm{T}_1$ ' and ' $\mathrm{T}_2$ ' respectively. Their emissive power is same and emissivity is in the ratio $1: 4$. Then the ratio ' $\mathrm{T}_1$ ' to ' $\mathrm{T}_2$ ' is
$1: \sqrt{2}$
$1: 2$
$2: 1$
$\sqrt{2}: 1$
Solution
The correct option is (D).
Concept:
Emissivity(e)
$=\frac{(\text { rate of radiation emitted thourgh a unit area of material) }}{\text { (rate of radiation emitted through a unit area of blackbody) }}$
Stefan-Boltzmann law states that the amount of radiation emitted by a black body per unit area $\varepsilon$ is directly proportional to the fourth power of the temperature $\mathrm{T}$.
$\varepsilon=\sigma \mathrm{T}^4$, where $\sigma$ is the Stefan coefficient.
Therefore, $\varepsilon_1=\mathrm{e}_1 \sigma \mathrm{T}_1^4$ and $\varepsilon_2=\mathrm{e}_2 \sigma \mathrm{T}_2^4$.
It is given that emmisive power of the both spheres of equal radius is the same.
So, $\mathrm{e}_1 \sigma \mathrm{T}_1^4=\mathrm{e}_2 \sigma \mathrm{T}_2^4$.
$\frac{\mathrm{T}_1}{\mathrm{~T}_2}=\left(\frac{\mathrm{e}_2}{\mathrm{e}_1}\right)^{1 / 4}=\left(\frac{4}{1}\right)^{1 / 4}$
$\frac{T_1}{T_2}=\frac{\sqrt{2}}{1}$