Two spherical bodies A (radius 6 cm) and $B$ (radius 18 cm) are at temperature $T_{1}$ 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 ?

Question

Dr. Abubacker · Accepted Answer

(a) $\lambda_{m} \propto \frac{1}{T}$
$	herefore\ \ \ \frac{\lambda_{A}}{\lambda_{B}} = \frac{T_{B}}{T_{A}} = \frac{500}{1500} = \frac{1}{3}$
$E \propto T^{4}\ (where\ A = surface\ area = 4\pi R^{2})$
$	herefore\ \ \ E \propto T^{4}R^{2}$
$\frac{E_{A}}{E_{B}} = \left( \frac{T_{A}}{T_{B}} ight)^{4}\left( \frac{R_{A}}{R_{B}} ight)^{2}$
$\ \ \ \ \ \ \ \ \ \  = (3)^{4}\left( \frac{16}{18} ight)^{2} = 9$

Explanation

Reviewed by Dr. Abubacker

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