Q: A planet is at an average distance d from the sun and its average surface temperature is T. Assume that the planet receives energy only from the sun, and loss energy only through radiation from its surface. Neglect atmospheric effects. If T ∝ d ^{-n} , the value of n is

(a) 2

(b) 1

(c) 1/2

(d) 1/4

Ans: (c)

Sol: Let R = radius of planet

P = power radiated by the sun.

Energy received by planet per sec.

$ \displaystyle = (\frac{P}{4\pi d^2})\times 4 \pi R^2 $

Energy radiated by planet per sec.

=(4 π R^{2} )σ T^{4}

For thermal equilibrium,

$ \displaystyle 4 \pi R^2 (\sigma T^4)= (\frac{P}{4\pi d^2})\times 4 \pi R^2 $

$ \displaystyle T^4 \propto \frac{1}{d^2} $

$ \displaystyle T \propto d^{-1/2} $

Hence , n = 1/2