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  4. Two metal spheres have radii r and 2 r and they emit thermal radiation with maximum intensities at wavelengths λ and 2 λ respectively. The respective ratio of the radiant energy emitted by them per second will be

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Q. Two metal spheres have radii $r$ and $2 r$ and they emit thermal radiation with maximum intensities at wavelengths $\lambda$ and $2 \lambda$ respectively. The respective ratio of the radiant energy emitted by them per second will be

Thermal Properties of Matter

Solution:

$T \propto \frac{1}{\lambda}$
[wien's displacement law]
So, $\frac{T_{1}}{T_{2}}=\frac{\lambda_{2}}{\lambda_{1}}=\frac{2 \lambda}{\lambda}=2$
and $H=e A \sigma T^{4}$
$ \Rightarrow H \propto A T^{4}$
$\frac{H_{1}}{H_{2}}=\frac{4 \pi r^{2}}{4 \pi(2 r)^{2}} \times \frac{T_{1}^{4}}{T_{2}^{4}}=\frac{1}{4} \times(2)^{4}=4$
$\therefore H_{1}: H_{2}:: 4: 1$