Question

Two bodies $A$ and $B$ have emissivities $0.5$ and $0.8$ respectively. At some temperatures the two bodies have maximum spectral emissive power at wavelength $8000\stackrel{o}{\text{A}}$ and $4000\stackrel{o}{\text{A}}$ respectively. The ratio of their emissive powers at these temperature is

• A. $\frac{5}{128}$
• B. $10$
• C. $\frac{5}{16}$
• D. None of these

Answer 1

The correct option is A $\frac{5}{128}$

Let, the bodies have a temperature $T_1$ and $T_2$ respectively at wavelength ${\lambda }_1=8000\stackrel{o}{\text{A}}$ and ${\lambda }_2=4000\stackrel{o}{\text{A}}$
$\therefore$ From Wien's displacement law
$\lambda T=$ Constant
$\Rightarrow {\lambda }_1{T}_1={\lambda }_2{T}_2$ or $8000\times T_1=4000T_2$
Or $\frac{T_1}{T_2}=\frac{1}{2}$
Emissive power $=e\sigma A{T}^4$
$\therefore$ Ratio of emissive powers at these temperature is $\frac{E_1}{E_2}=\frac{e_1{T}_1^4}{e_2{T}_2^4}=\frac{0.5}{0.8}\times (\frac{1}{2}{)}^4=\frac{5}{128}$
Hence, option $\left(A\right)$ matches correctly.