Question

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

• A. $(\frac{6}{7})$
• B. ${\left(\frac{6}{7}\right)}^2$
• C. ${\left(\frac{6}{7}\right)}^3$
• D. $\frac{243}{464}$

Answer 1

Answer: (D)

Rate of loss of heat by a black body through radiation is
where, $E=\sigma A\left(T^4-T_S^4\right)$
$\therefore \frac{E_1}{E_2}=\frac{T_1^4-T_S^4}{T_2^4-T_S^4}$
$=\frac{(327+273{)}^4-(27+273{)}^4}{(427+273{)}^4-(27+273{)}^4}$
$=\frac{(600{)}^4-(300{)}^4}{(700{)}^4-(300{)}^4}$
$=\frac{10^8\left(6^4-3^4\right)}{10^8\left(7^4-3^4\right)}$
$=\frac{\left({\left(6^2\right)}^2-{\left(3^2\right)}^2\right)}{\left({\left(7^2\right)}^2-{\left(3^2\right)}^2\right)}$
$=\frac{\left(6^2-3^2\right)\left(6^2+3^2\right)}{\left(7^2-3^2\right)\left(7^2+3^2\right)}$
$\left[\because \left(a^2-b^2\right)=(a+b)(a-b)\right]$
$=\frac{(36-9)(36+9)}{(49-9)(49+9)}$
$=\frac{(27)(45)}{(40)(58)}$
$=\frac{243}{464}$