Question 647: Thermodynamics & KTG

Question

Two black spheres $\mathrm{P} \& \mathrm{Q}$ have radii in the ratio $4: 3$. The wavelength of maximum intensity of radiation are in the ratio $4: 5$ respectively. The ratio of radiated power by P to Q is

Accepted Answer

Wien's Displacement Law:
$$ \begin{aligned} & \lambda_{\max } \propto \frac{1}{\mathrm{~T}} \ \Rightarrow & \frac{\mathrm{~T}_{\mathrm{P}}}{\mathrm{~T}_Q}=\frac{\lambda_{\max Q}}{\lambda_{\max , \mathrm{P}}}=\frac{5}{4} \end{aligned} $$
By Stefan - Boltzmann Law
Power radiated by a black body
$$ \begin{aligned} & P=\sigma A T^4=\sigma\left(4 \pi R^2ight) T^4 \ \Rightarrow \quad & P \propto R^2 T^4 \ 	herefore \quad & \frac{P_P}{P_Q}=\left(\frac{R_P}{R_Q}ight)^2\left(\frac{T_P}{T_Q}ight)^4=\frac{16}{9} 	imes \frac{625}{256}=\frac{625}{144} \end{aligned} $$
$	herefore \quad$ The ratio of radiated power by P to Q is $\frac{625}{144}$

Answer

$\frac{625}{144}$

Answer

$\frac{125}{81}$

Answer

$\frac{25}{9}$

Answer

$\frac{5}{3}$

Thermodynamics & KTG

Two black spheres \mathrm{P} \& \mathrm{Q} have radii in the ratio 4: 3. The wavelength of maximum intensity of radiation are in the ratio 4: 5 respectively. The ratio of radiated power by P to Q is

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