Question

A spherical black body with a radius of $12cm$ radiates $450W$ power at $500K$. If the radius were made half and if the temperature is doubled, the power radiated in watts would be given as,
$\begin{align}
  & A.1800 \\
 & B.225 \\
 & C.450 \\
 & D.900 \\
\end{align}$

Answer 1

Hint: The power radiated by the black body is found by taking the product of the area of the black body, Stefan’s constant and the fourth power of temperature. The area is given as the area of a sphere as the black body is given as a spherical one. Now the conditions of halving the radius and making the temperature doubled, should be applied in the same equation and compare it with the previous condition. These all will help you to solve this question.

Complete step by step answer:
First of all we have to find the power radiated by the spherical body at specific conditions. It is given by the equation,
${{P}_{old}}=A\sigma {{T}^{4}}$
As the area of the sphere is given by the equation,
$A=4\pi {{r}^{2}}$
Where $r$ be the radius of the spherical body.
Substituting this in the equation of power radiated,
${{P}_{old}}=4\pi {{r}^{2}}\sigma {{T}^{4}}$
The power radiated at $500K$, given in the question is,
\[{{P}_{old}}=450W\]
Now let us look at the second case. As it is mentioned that the the radius is made half, we can write that,
\[r\to \dfrac{r}{2}\]
And also the temperature is made twice, therefore we can write that,
\[T\to 2T\]
Let us substitute this in the equation of power,
${{P}_{old}}=4\pi {{r}^{2}}\sigma {{T}^{4}}=450W$
${{P}_{new}}=4\pi {{\left( \dfrac{r}{2} \right)}^{2}}\sigma {{\left( 2T \right)}^{4}}$
This can be simplified by doing the calculations,
${{P}_{new}}=\dfrac{16\times 4\pi {{r}^{2}}\sigma {{T}^{4}}}{4}=4\times 4\pi {{r}^{2}}\sigma {{T}^{4}}$
Therefore we can write that,
${{P}_{new}}=4\times {{P}_{old}}$
As the value of power in the old condition is given as,
${{P}_{old}}=450W$
Substituting this in the equation will give the answer,
${{P}_{new}}=4\times 450=1800W$
Therefore the correct answer is given as option A.

Note:
Stefan-Boltzmann law is the law explaining the power radiated by a black body. It says that the total power emitted or radiated from a surface is proportional to the fourth power of its absolute temperature. This law is applicable only to blackbodies.