Two metallic spheres \(P\) and \(Q\) are made of same material have same smoothness but the weight of \(P\)…
Two metallic spheres \(P\) and \(Q\) are made of same material have same smoothness but the weight of \(P\) is 8 times that of \(Q\). If the two are heated to same temperature and left to cool, then the ratio of rate of cooling of \(Q\) to that of \(P\) is
4
8
2
1
Solution
Given, weight of material \(P\) is 8 times the weight of material \(Q\).
i.e., \(\quad m_p=8 m_Q\)
From stefan's law,
the rate of heat radiation from an object is given by
\(\frac{\Delta Q}{\Delta t}=e \sigma A T^4\)...(i)
where, \(e=\) emissivity, \(\sigma=\) Stefan's constant
\(A=\text { area and } T=\text { Temperature }\)
But, \(\Delta Q=m c \Delta T\)...(ii)
\(\therefore\) From Eqs. (i) and (ii), we get
\(\begin{aligned}
\frac{m c \Delta T}{\Delta t} & =e \sigma A T^4 \\
\frac{\Delta T}{\Delta t} & =\frac{e \sigma A T^4}{m c} \quad \ldots (iii)
\end{aligned}\)
But, \(\quad m=\frac{4}{3} \pi r^3 \cdot \sigma\)
\(\therefore \quad r=\left(\frac{3 m}{4 \pi \sigma}\right)^{\frac{1}{3}}\)
\(\therefore \quad A=4 \pi r^2=4 \pi\left(\frac{3 m}{4 \pi \sigma}\right)^{\frac{2}{3}}\)...(iv)
\(\therefore\) From Eqs. (iii) and (iv), we get
\(\begin{aligned}
& \frac{\Delta T}{\Delta t}=\frac{e \sigma T^4}{m c}\left[4 \pi\left(\frac{3 m}{4 \pi \sigma}\right)^{\frac{2}{3}}\right] \\
& \therefore \frac{\Delta T}{\Delta t}=k\left(\frac{1}{m}\right)^{\frac{1}{3}} \quad \text { where, } k=\left[\frac{e \sigma T^4}{c}\left(4 \pi\left(\frac{3}{4 \pi}\right)^{2 / 3}\right)\right] \\
& \therefore \frac{\left(\frac{\Delta T}{\Delta t}\right)_Q}{\left(\frac{\Delta T}{\Delta t}\right)_P}=\left(\frac{m_P}{m_Q}\right)^{\frac{1}{3}}=\left(\frac{8 m_Q}{m_Q}\right)^{\frac{1}{3}}=(8)^{\frac{1}{3}}=2
\end{aligned}\)