Question

Two metallic spheres $S_1$ and $S_2$ of equal densities $\left(\rho \right)$ are made of the same material and have got identical surface finish. The mass of $S_1$ is thrice that of $S_2$. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of $S_1$ to that $S_2$ is

• A. $\frac{1}{3}$
• B. $\frac{1}{\sqrt{3}}$
• C. $\frac{\sqrt{3}}{1}$
• D. ${\left(\frac{1}{3}\right)}^{\frac{1}{3}}$

Answer 1

The correct option is D ${\left(\frac{1}{3}\right)}^{\frac{1}{3}}$

From the Stefan's law, the rate at which energy leaves the object is

$\frac{\Delta Q}{\Delta t}=e\sigma A{T}^4$

Since, $\Delta Q=mC\Delta T$, we get

$\frac{\Delta T}{\Delta t}=\frac{e\sigma A{T}^4}{mC}$

Since $m=\frac{4}{3}\pi r^3\rho$ for a sphere, we get
$A=4\pi r^2=4\pi {\left(\frac{3m}{4\pi \rho }\right)}^{\frac{2}{3}}$

Hence, $\frac{\Delta T}{\Delta t}=\frac{e\sigma T^4}{mC}\left[4\pi {\left(\frac{3m}{4\pi \rho }\right)}^{\frac{2}{3}}\right]=K{\left(\frac{1}{m}\right)}^{\frac{1}{3}}$

For the given two bodies

$\frac{{\left(\frac{\Delta T}{\Delta t}\right)}_1}{{\left(\frac{\Delta T}{\Delta t}\right)}_2}={\left(\frac{m_2}{m_1}\right)}^{\frac{1}{3}}={\left(\frac{1}{3}\right)}^{\frac{1}{3}}$

$\begin{array}{c}\text{Why this question}?\\ \text{This question checks the basic understanding}\\ \text{of Stefan's law and its application.}\end{array}$