Question

Two bodies of the same shape, size, and radiating power have emissivity 0.2 and 0.8. the ratio of their temperature is:
(A) \[\sqrt{3}:1\]
(B) $\sqrt{2}:1$
(C) $1:\sqrt{5}$
(D) $1:\sqrt{3}$

Answer 1

Hint Use the Stefan-Boltzmann equation for both the bodies and form respective equations for each body. Either equate the values of radiative power from both the equations or simply divide one equation by the other. Cancel all common terms and substitute the given values into the final equation and simplify to get the answer.
Equation used
Stefan-Boltzmann law of radiation:
$P=\dfrac{Q}{t}=\sigma eA{{T}^{4}}$
Where $P$ is the radiative power,
$Q$ is the heat transferred,
t is the time taken,
$\sigma $ is the Stefan-Boltzmann constant with a value of $5.67\times {{10}^{-8}}J/s\cdot {{m}^{2}}\cdot {{K}^{4}}$,
$e$ is the emissivity of the object,
$A$ is the surface area of the object,
$T$ is the temperature.

Complete Step by step solution
Let the emissivity of the first object be ${{e}_{1}}$ and that of the second object be ${{e}_{2}}$, and let the temperature of the first and second object be ${{T}_{1}}$ and ${{T}_{2}}$ respectively.
By using the Stefan-Boltzmann equation that gives the radiative power of a body for first and second bodies respectively, we get
$P=\sigma {{e}_{1}}A{{T}_{1}}^{4}$
$P=\sigma {{e}_{2}}A{{T}_{2}}^{4}$
As it is given in the question that the radiative powers of both the bodies are equal and that they have the same shape and size, we can take the variable for radiative power and surface area of both the bodies to be $P$ and $A$ respectively.
Dividing the radiative powers of both the bodies, we get
$\dfrac{P}{P}=\dfrac{\sigma {{e}_{1}}A{{T}_{1}}^{4}}{\sigma {{e}_{2}}A{{T}_{2}}^{4}}$
Canceling the common terms and rearranging the above equation, we get
$\dfrac{{{T}_{1}}^{4}}{{{T}_{2}}^{4}}=\dfrac{{{e}_{2}}}{{{e}_{1}}}$
Substituting the given values of emissivity, we get
$\dfrac{{{T}_{1}}^{4}}{{{T}_{2}}^{4}}=\dfrac{0.8}{0.2}$
Taking ${{{}^{1}/{}_{4}}^{th}}$ power on both sides, we get
$\dfrac{{{T}_{1}}}{{{T}_{2}}}={{\left( \dfrac{0.8}{0.2} \right)}^{{}^{1}/{}_{4}}}$
$\Rightarrow \dfrac{{{T}_{1}}}{{{T}_{2}}}=\dfrac{\sqrt{2}}{1}$
Or ${{T}_{1}}:{{T}_{2}}=\sqrt{2}:1$
$\therefore $ the ratio of the temperature of the first body to the second body is $\sqrt{2}:1$ .

That is, option (B) is the correct option.

Note
The Stefan-Boltzmann equation is an equation that tells us about the power radiated out of an object. We can find the luminosity of a star by using the Stefan-Boltzmann equation by taking the value of emissivity to be $1$ as stars are examples of a blackbody and black bodies have emissivity equal to $1$.