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Frequency of Oscillations in LC Circuit

Three bodies ...

Question

Three bodies $A, B a n d C$ have equal surface area and thermal emissivities in the ratio $e_{A} : e_{B} : e_{C} = 1 : \frac{1}{2} : \frac{1}{4}$ . All the three bodies are radiating at same rate. Their wavelengths corresponding to maximum intensity are $λ_{A}, λ_{B} a n d λ_{C}$ respectively and their temperatures are $T_{A}, T_{B} a n d T_{C}$ on kelvin scale, then select the incorrect statement :

A

\sqrt{T_{A} T_{C}} = T_{B}

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B

\sqrt{λ_{A} λ_{C}} = λ_{B}

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C

\sqrt{e_{A} T_{A}} \sqrt{e_{C} T_{C}} = e_{B} T_{B}

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D

\sqrt{e_{A} λ_{A} T_{A} . e_{B} λ_{B} T_{B}} = e_{C} λ_{C} T_{C}

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Solution

The correct option is B $\sqrt{λ_{A} λ_{C}} = λ_{B}$
$e_{A} : e_{B} : e_{C} = 1 : \frac{1}{2} : \frac{1}{4} (1)$
For same power of radiation,
$P_{A} = P_{B} = P_{C} \Rightarrow e_{A} σ A T_{A}^{4} = e_{B} σ A T_{B}^{4} = e_{C} σ A T_{C}^{4} \Rightarrow T_{A}^{4} = \frac{1}{2} T_{B}^{4} = \frac{1}{4} T_{C}^{4} \Rightarrow T_{A} = \frac{1}{2^{\frac{1}{4}}} T_{B} = \frac{1}{4^{\frac{1}{4}}} T_{C} \Rightarrow T_{B} = \sqrt{T_{A} T_{C}} (2)$
From $(1)$ and $(2)$ ,
$\sqrt{e_{A} T_{A} e_{C} T_{C}} = e_{B} T_{B} (3) λ_{A} T_{A} = λ_{B} T_{B} = λ_{C} T_{C} (4) \sqrt{λ_{A} λ_{C}} = λ_{B}$

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