Heat and Thermodynamics 3 Question 30
30. Two spherical bodies $A$ (radius $6 cm$ ) and $B$ (radius $18 cm$ ) are at temperatures $T _1$ and $T _2$, respectively. The maximum intensity in the emission spectrum of $A$ is at $500 nm$ and in that of $B$ is at $1500 nm$. Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by $A$ to that of $B$ ?
(2010)
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Solution:
- $\lambda _m \propto \frac{1}{T}$
$\therefore \frac{\lambda _A}{\lambda _B}=\frac{T _B}{T _A}=\frac{500}{1500}=\frac{1}{3}$
$E \propto T^{4} A \quad$ (where, $A=$ surface area $=4 \pi R^{2}$ )
$\therefore \quad E \propto T^{4} R^{2}$
$\frac{E _A}{E _B}=\left(\frac{T _A}{T _B}\right)^{4}\left(\frac{R _A}{R _B}\right)^{2}=(3)^{4}\left(\frac{6}{18}\right)^{2}=9$
$\therefore$ Answer is 9 .
