Two solid spheres are heated to the same temperature and allowed to cool under identical conditions. Assume that all the surfaces have the same emissivity and ratios
of their radii, specific heats and densities are respectively $1 : α, 1 : β, 1 : γ$ . Then the ratio of initial rates of loss of heat and initial rates of fall of temperature is

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$α^{2} : 1, αβγ : 1$

$αβγ : 1, 1 : α^{2}$

$1 : α^{2}, αβγ : 1$

$α^{2} : 1, 1 : αβγ$

answer is B.

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Detailed Solution

$\begin{array}{l} \frac{dQ}{dt} = σAe (T^{4} - T_{0}^{4}) \Rightarrow \frac{{(\frac{dQ}{dt})}_{1}}{{(\frac{dQ}{dt})}_{2}} = \frac{A_{1}}{A_{2}} = {(\frac{R_{1}}{R_{2}})}^{2} \\ \frac{dθ}{dt} = \frac{σAe}{ms} (T^{4} - T_{0}^{4}) \Rightarrow \frac{{(\frac{dθ}{dt})}_{1}}{{(\frac{dθ}{dt})}_{2}} = (\frac{A_{1}}{A_{2}} \times \frac{m_{2}}{m_{1}} \times \frac{s_{2}}{s_{1}}) \\ \frac{{(\frac{dθ}{dt})}_{1}}{{(\frac{dθ}{dt})}_{2}} = \frac{R_{2}}{R_{1}} \times \frac{ρ_{2}}{ρ_{1}} \times \frac{s_{2}}{s_{1}} \end{array}$