Distance between sun and Earth is $2 \times 10^{8}\, km ,$ temperature of sun $6000\, K ,$ radius of sun $7 \times 10^{5}\, km .$ If emmisivity of the earth is $0.6,$ find out the temperature of the earth in thermal equilibrium (in $k$)
AIIMS 2019, Medium
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For the condition of thermal equilibrium the energy received by earth should be equal to the energy transmitted by earth.
$\frac{T_{ s }^{4} 4 \pi R_{ s }^{2}}{4 \pi d^{2}} \times \pi R_{e}^{2}=\sigma \rho T_{e}^{4} 4 \pi R_{e}^{2}$
$T_{c}^{4}=\frac{T_{ s }^{4} R_{ s }^{2}}{4 e d^{2}}$ $....(I)$
Substitute $6000$ for $T_{s}, 7 \times 10^{8}$ for $R_{5}, 2 \times 10^{11}$ for $d$ and $0.6$ for $e$ in equation $(I).$
$T_{ e }^{4}=\frac{(6000)^{4}\left(7 \times 10^{8}\right)^{2}}{4\left(2 \times 10^{11}\right)^{2} \times 0.6}$
$=\frac{36 \times 36 \times 7 \times 7}{4 \times 4 \times 0.6} \times 10^{6}$
$=66.15 \times 108$
$T_{ e } \approx 300 K$
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