Question 1: Thermodynamics & KTG

Question

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 $\left( {{{{\lambda _A}} \over {{\lambda _B}}}} \right)$ of their wavelengths ${{\lambda _A}}$ and ${{\lambda _B}}$ at which the peaks
occur in their respective radiation curves is

Accepted Answer

Power, $P = (\sigma {T^4}A) = \sigma {T^4}(4\pi {R^2})$
or, $P \propto {T^4}{R^2}$ ..... (i)
According to Wien's law,
$\lambda  \propto {1 \over T}$
($\lambda$ is the wavelength at which peak occurs)
$	herefore$ Eq. (i) will become,
$P \propto {{{R^2}} \over {{\lambda ^4}}}$
or, $\lambda  \propto {\left[ {{{{R^2}} \over P}} ight]^{1/4}}$
$ \Rightarrow {{{\lambda _A}} \over {{\lambda _B}}} = {\left[ {{{{R_A}} \over {{R_B}}}} ight]^{1/2}}{\left[ {{{{P_B}} \over {{P_A}}}} ight]^{1/4}}$
$ = {[400]^{1/2}}{\left[ {{1 \over {{{10}^4}}}} ight]^{1/4}} = 2$

Thermodynamics & KTG

Integer / Numeric Answer Type