Question 959: Thermodynamics & KTG

Question

The maximum wavelength of radiation emitted by a star is 289.8 nm . Then intensity of radiation for the star is (Given : Stefan's constant $=5.67 	imes 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}$, Wien's constant, $b=2898 \mu \mathrm{mK}$ )

Accepted Answer

$$\begin{aligned}
&	ext { Given, maximum wavelength, }\
&\begin{aligned}
\lambda_m & =289.8 \mathrm{~nm}=289.8 	imes 10^{-9} \mathrm{~m} \
& =2898 	imes 10^{-10} \mathrm{~m}
\end{aligned}
\end{aligned}$$
Stefan's constant, $\sigma=5.67 	imes 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}$
Wien's constant, $b=2898 \mu \mathrm{mK}=2898 	imes 10^{-6} \mathrm{mK}$
According to Wien's displacement law, the maximum wavelength is given by
$$\lambda_m=\frac{b}{T} \Rightarrow T=\frac{b}{\lambda_m}\quad	ext{.... (i)}$$
Substituting given values in Eq. (i), we get
$$T=\frac{2898 	imes 10^{-6}}{2898 	imes 10^{-10}}=10^4 \mathrm{~K}\quad	ext{.... (ii)}$$
According to Stefan's law, the energy radiated from a source is given by
$$E=\sigma A e T^4\quad	ext{.... (iii)}$$
where, $A=$ area of source
$e=$ emissivity (value between 0 to 1)
The intensity of radiations emitted is equal to energy radiated from a given surface area, i.e.,
$$I=\frac{E}{A}=\sigma e T^4 \quad	ext{[from Eq. (iii)]}$$
As e is very small, so
$$I=\sigma T^4\quad	ext{.... (iv)}$$
Substituting the value of $T$ from Eq. (ii) in Eq. (iv), we get
$$\begin{aligned}
I= & \sigma\left(10^4ight)^4=5.67 	imes 10^{-8} 	imes 10^{16} \
& {\left[\because \sigma=5.67 	imes 10^{-8} 	ext { (given) }ight] } \
& =5.67 	imes 10^8 \mathrm{Wm}^{-2}
\end{aligned}$$

Answer

$5.67 	imes 10^{-12} \mathrm{Wm}^{-2}$

Answer

$10.67 	imes 10^{14} \mathrm{Wm}^{-2}$

Answer

$5.67 	imes 10^8 \mathrm{Wm}^{-2}$

Answer

$10.67 	imes 10^7 \mathrm{Wm}^{-2}$

Thermodynamics & KTG

The maximum wavelength of radiation emitted by a star is 289.8 nm . Then intensity of radiation for the star is (Given : Stefan's constant =5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}, Wien's constant, b=2898 \mu \mathrm{mK} )

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