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 of their wavelengths λA and λB at which the peaks occur in their respective radiation curves is

Engineering

Physics

Radiation

Question

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 $(\frac{λ_{A}}{λ_{B}})$ of their wavelengths λ_A and λ_B at which the peaks occur in their respective radiation curves is

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r_A = 400 r_B

P_A = 10⁴ P_B

P_A = σ (4πr²_A) T_A⁴

P_B = σ (4πr_B²) T_B⁴

^{$\frac{P_{A}}{P_{B}} = {(400)}^{2} {(\frac{T_{A}}{T_{B}})}^{4} = 10^{4}$}

^{${(\frac{T_{A}}{T_{B}})}^{4} = \frac{10^{4}}{400 \times 400} = \frac{1}{4 \times 4} = \frac{1}{24}$}

^{$\frac{T_{A}}{T_{B}} = \frac{1}{2}$ Now $λ \propto \frac{1}{T}$}

^{$\frac{λ_{A}}{λ_{B}} = \frac{T_{B}}{T_{A}} = 2$}