A small object is placed at the centre of a large evacuated hollow spherical container. Assume that the container is maintained at $$0$$ K. At time t $$=$$ $$0$$, the temperature of the object is $$200$$ K. The temperature of the object becomes $$100$$ K at t $$=$$ $$t1$$ and $$50$$ K at t $$=$$ $$t2$$. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio (t2$$ $$/$$ $$t1) is $$_____$$ .

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σAe⁡$$T4$$−$$O4=msdTdt$$→∫$$dTT4$$∝∫$$dtT1=100k$$ $$1T03$$−$$1T13=kt1T2=50k$$ $$1T03$$−$$1T23=kt2t2t1=17$$

Heat transfer-Radiation

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$\begin{array}{l} σAe (T^{4} - O^{4}) = ms \frac{dT}{dt} \\ \to \int \frac{dT}{T^{4}} \propto \int dt \\ T_{1} = 100 k \frac{1}{T_{0}^{3}} - \frac{1}{T_{1}^{3}} = {kt}_{1} \\ T_{2} = 50 k \frac{1}{T_{0}^{3}} - \frac{1}{T_{2}^{3}} = {kt}_{2} \\ \frac{t_{2}}{t_{1}} = 17 \end{array}$

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Heat transfer-Radiation

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σAe⁡T4−O4=msdTdt→∫dTT4∝∫dtT1=100k 1T03−1T13=kt1T2=50k 1T03−1T23=kt2t2t1=17

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$\begin{array}{l} σAe (T^{4} - O^{4}) = ms \frac{dT}{dt} \\ \to \int \frac{dT}{T^{4}} \propto \int dt \\ T_{1} = 100 k \frac{1}{T_{0}^{3}} - \frac{1}{T_{1}^{3}} = {kt}_{1} \\ T_{2} = 50 k \frac{1}{T_{0}^{3}} - \frac{1}{T_{2}^{3}} = {kt}_{2} \\ \frac{t_{2}}{t_{1}} = 17 \end{array}$