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Q.
Two black bodies at temperatures $327^{\circ} C$ and $427^{\circ} C$ are kept in an evacuated chamber at $27^{\circ} C$. The ratio of their rates of loss of heat are
Rate of loss of heat by a black body through radiation is
where, $ E=\sigma A\left(T^{4}-T_{S}^{4}\right)$
$\therefore \frac{E_{1}}{E_{2}} =\frac{T_{1}^{4}-T_{S}^{4}}{T_{2}^{4}-T_{S}^{4}}$
$=\frac{(327+273)^{4}-(27+273)^{4}}{(427+273)^{4}-(27+273)^{4}}$
$=\frac{(600)^{4}-(300)^{4}}{(700)^{4}-(300)^{4}}$
$=\frac{10^{8}\left(6^{4}-3^{4}\right)}{10^{8}\left(7^{4}-3^{4}\right)}$
$=\frac{\left(\left(6^{2}\right)^{2}-\left(3^{2}\right)^{2}\right)}{\left(\left(7^{2}\right)^{2}-\left(3^{2}\right)^{2}\right)}$
$= \frac{\left(6^{2}-3^{2}\right)\left(6^{2}+3^{2}\right)}{\left(7^{2}-3^{2}\right)\left(7^{2}+3^{2}\right)} $
$\left[\because\left(a^{2}-b^{2}\right)=(a+b)(a-b)\right]$
$=\frac{(36-9)(36+9)}{(49-9)(49+9)}$
$=\frac{(27)(45)}{(40)(58)}$
$=\frac{243}{464}$