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 $\left(\frac{\lambda_{A}}{\lambda_{B}}\right)$ of their wavelengths $\lambda_{A}$ and $\lambda_{B}$ at which the peaks occur in their respective radiation curves is

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Heat & Thermodynamics

Two spherical stars

A

and

B

emit blackbody radiation. The radius of

A

is 400 times that of

B

and

A

emits

1 0^{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

Answer: 2

By Wien's law,

λ_{m} T =

constant, so

λ_{m_{A}} T_{A} = λ_{m_{B}} T_{B}

. By Stefan-Boltzmann law, power

P = σ A T^{4}

. Given

\frac{P _{A}}{P _{B}} = 1 0^{4}

and

\frac{r _{A}}{r _{B}} = 400

\frac{P _{A}}{P _{B}} = \frac{( 4 π r _{A}^{2} ) T _{A}^{4}}{( 4 π r _{B}^{2} ) T _{B}^{4}} = (400)^{2} (\frac{T _{A}}{T _{B}})^{4} = 1 0^{4}

. This gives

(\frac{T _{A}}{T _{B}})^{4} = \frac{1}{16}

, so

\frac{T _{A}}{T _{B}} = \frac{1}{2}

. Therefore

\frac{λ _{A}}{λ _{B}} = \frac{T _{B}}{T _{A}} = 2

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Integer answer