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

Answer 1

The correct option is D $T_{\circ }\sqrt{\frac{R}{2d}}$

Let, $r=$ Radius of planet

$T_1$= Temperature of planet at which energy is radiated

Power radiated by Sun, $p=\pi R^2\sigma T_0^4$

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,

$\frac{R^2{T}_0^4\pi r^2\sigma }{d^2}=4\pi r^2\sigma T_1^4$

$R^2{T}_0^4=4d^2{T}_1^4$

$T_1=T_0\sqrt{\frac{R}{2d}}$