Two solid spheres of radii $R_1$ and $R_2$ made of same material and have similar surfaces. The spheres are…

Two solid spheres of radii $R_1$ and $R_2$ made of same material and have similar surfaces. The spheres are raised to the same temperature and then allowed to cool under identical conditions. Assuming spheres to be perfect conductors of heat, then the ratio of initial rates of cooling are
  1. $\begin{array}{r}R_1^2 \\ \hline R_2^2\end{array}$
  2. $\frac{R_1^4}{R_2^4}$
  3. $\frac{R_2^3}{R_1^3}$
  4. $\frac{R_2}{R_1}$

Solution

According to Stefan's law, we know that: $Q=\sigma A T^4 \times e$ where, area of sphere, $A=4 \pi R^2$. Here, temperature, thermal conductivity and emissivity of material are same. So, we get the relation: $Q \propto R^2$ Initial rate of cooling is given as: $\frac{Q_1}{Q_2}=\frac{R_1^2}{R_2^2}$

Asked in: MHT CET 2022 (07 Aug Shift 2)

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