First slide

Radiation

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

Moderate

A

$T_{o} \sqrt{\frac{R}{2 d}}$
B

$T_{o}$ $\sqrt{\frac{2 R}{d}}$
C

$T_{o} \sqrt{\frac{R}{d}}$
D

$T_{o} {(\frac{R}{d})}^{\frac{1}{4}}$

Solution

r = Radius of planet
$T_{1} = T e m p e r a t u r e o f p l a n e t a t w h i c h e n e r g y i s r a d i a t e d P o w e r r a d i a t e d b y S u n, P = 4 π R^{2} σ {T_{0}}^{4}$
Energy received by planet = $\frac{P}{4 {πd}^{2}} \times {πr}^{2} = \frac{R^{2} {T_{0}}^{4} {πr}^{2} σ}{d^{2}}$
Energy radiated by planet = $4 π^{2} {σT}_{1}^{4}$
For thermal equilibrium,
$\frac{R^{2} {T_{0}}^{4} π r^{2} σ}{d^{2}} = 4 {πr}^{2} {σT}_{1}^{4}$
$R^{2} {T_{0}}^{4} = 4 d^{2} {T_{1}}^{4}$
$T_{1} = T_{o} \sqrt{\frac{R}{2 d}}$

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