A planet radiates heat at a rate proportional to the fourth power of its surface temperature T. If such a steady temperature of the planet is due to the exactly equal amount of heat received from the sun then which of the following statement is true?

Moderate

A

The planet's surface temperature varies inversely as the distance of the sun
B

The planet's surface temperature varies directly as the square of its distance of the sun
C

The planet's surface temperature varies inversely as the square root of its distance of the sun
D

The planet's surface temperature is proportional to the fourth power is distance from the sun.

Solution

Rate of loss of energy by unit area of the planet = ${σT}^{4}$ , where $σ$ is the Stefan's constant. Let Q be the total energy emitted by the sun in every second. If d is the distance of the planet from sun, then Q falls uniformly over the inner surface of the sphere of radius d. Rate of gain of heat by unit area of radius d. Rate of gain of heat by unit area of planet = $\frac{Q}{4 {πd}^{2}}$
For steady temperature of plant
${σT}^{4} = \frac{Q}{4 {πd}^{2}}$
$T^{4} = \frac{Q}{4 {πσd}^{2}} or T = {(\frac{Q}{4 {πσd}^{2}})}^{\frac{1}{4}}$
or $T \propto \frac{1}{\sqrt{d}}$

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