Question

The surface temperature of the sun is $T_{0}$ and it is at average distance $d$ from a planet. The radius of the sun is $R$. The temperature at which planet radiates the energy is

• A. $T_{0} \sqrt{\frac{R}{2 d}}$
• B. $T_{0} \sqrt{\frac{2 R}{d}}$
• C. $T_{0} \sqrt{\frac{R}{d}}$
• D. $T_{0}\left(\frac{R}{d}\right)^{1 / 4}$

Answer 1

Answer: (A)

$r=$ Radius of planet
$T_{1}=$ Temperature of planet at which energy is radiated
Power radiated by Sun, $P=4 \pi R^{2} \sigma T_{0}^{4}$
Energy received by planet $=\frac{P}{4 \pi d^{2}} \times \pi r^{2}=\frac{R^{2} T_{0}^{4} \pi r^{2} \sigma}{d^{2}}$
Energy radiated by planet $=4 \pi r^{2} \sigma T_{1}^{4}$ For thermal equilibrium,
$\frac{R^{2} T_{0}^{4} \pi r^{2} \sigma}{d^{2}}=4 \pi r^{2} \sigma T_{1}^{4}$
$R^{2} T_{0}^{4}=4 d^{2} T_{1}^{4}$
$T_{1}=T_{0} \sqrt{\frac{R}{2 d}}$