Heat and Thermodynamics 3 Question 12
12. Three discs, $A, B$ and $C$ having radii $2 \mathrm{~m}, 4 \mathrm{~m}$ and $6 \mathrm{~m}$ respectively are coated with carbon black on their outer surfaces. The wavelengths corresponding to maximum intensity are $300 \mathrm{~nm}, 400 \mathrm{~nm}$ and $500 \mathrm{~nm}$, respectively. The power radiated by them are $Q_A, Q_B$ and $Q_C$ respectively
(2004, 2M)
(a) $Q_A$ is maximum
(b) $Q_B$ is maximum
(c) $Q_C$ is maximum
(d) $Q_A=Q_B=Q_C$
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Answer:
Correct Answer: 12. (b)
Solution:
- $Q \propto A T^{4}$ and $\lambda _m T=$ constant
Hence,
$$ \begin{aligned} & Q \propto \frac{A}{\left(\lambda _m\right)^{4}} \\ & \text { or } \quad Q \propto \frac{r^{2}}{\left(\lambda _m\right)^{4}} \\ & Q _A: Q _B: Q _C=\frac{(2)^{2}}{(3)^{4}}: \frac{(4)^{2}}{(4)^{4}}: \frac{(6)^{2}}{(5)^{4}} \\ & =\frac{4}{81}: \frac{1}{16}: \frac{36}{625}=0.05: 0.0625: 0.0576 \end{aligned} $$
i.e. $Q _B$ is maximum.
