Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits times the power emitted from B. The ratio of their wavelengths and at which the peaks occur in their respective radiation curves is

Question

Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits $10^{4}$ times the power emitted from B. The ratio $(\frac{λ_{A}}{λ_{B}})$ of their wavelengths $λ_{A}$ and $λ_{B}$ at which the peaks occur in their respective radiation curves is [2015]

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(2)

$From (i) Stefan-Boltzmann law,$ $(\frac{λ_{A}}{λ_{B}})$ $and (ii) Wien's displacement law,$ $λ_{m} \times T = constant$

$\frac{P_{A}}{P_{B}} = \frac{A_{A}}{A_{B}} \frac{T_{A}^{4}}{T_{B}^{4}} = \frac{A_{A}}{A_{B}} \times \frac{λ_{B}^{4}}{λ_{A}^{4}}$

$∴$ $\frac{λ_{A}}{λ_{B}} = {[\frac{A_{A}}{A_{B}} \times \frac{P_{B}}{P_{A}}]}^{\frac{1}{4}}$ $= {[\frac{R_{A}^{2}}{R_{B}^{2}} \times \frac{P_{B}}{P_{A}}]}^{\frac{1}{4}}$ $= {[\frac{400 \times 400}{10^{4}}]}^{\frac{1}{4}}$

$∴$ $\frac{λ_{A}}{λ_{B}} = 2$

Solution

Q.
Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits $10^{4}$ times the power emitted from B. The ratio $(\frac{λ_{A}}{λ_{B}})$ of their wavelengths $λ_{A}$ and $λ_{B}$ at which the peaks occur in their respective radiation curves is [2015]

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Solution

Q. Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits 104 times the power emitted from B. The ratio (λAλB) of their wavelengths λA and λB at which the peaks occur in their respective radiation curves is [2015]

Ans. (2) From (i) Stefan-Boltzmann law, P=σAT4 and (ii) Wien's displacement law, λm×T=constant PAPB=AAABTA4TB4=AAAB×λB4λA4 ∴ λAλB=[AAAB×PBPA]14=[RA2RB2×PBPA]14=[400×400104]14 ∴ λAλB=2

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Q.
Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits $10^{4}$ times the power emitted from B. The ratio $(\frac{λ_{A}}{λ_{B}})$ of their wavelengths $λ_{A}$ and $λ_{B}$ at which the peaks occur in their respective radiation curves is [2015]