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

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, \(\mathrm{e}_1 \sigma \mathrm{T}_1^4=\mathrm{e}_2 \sigma \mathrm{T}_2^4\).
\(\frac{\mathrm{T}_1}{\mathrm{~T}_2}=\left(\frac{\mathrm{e}_2}{\mathrm{e}_1}\right)^{1 / 4}=\left(\frac{4}{1}\right)^{1 / 4}\)
\(\frac{T_1}{T_2}=\frac{\sqrt{2}}{1}\)