Question

Rate of emission thermal radiation of a spherical black body of radius $r$ is $H$. It rate of cooling is $C$. Then

• A. $H\propto r$
• B. $H\propto r^3$
• C. $C\propto \frac{1}{r}$
• D. $C\propto \frac{1}{r^2}$

Answer 1

Answer: (C)

$C\propto \frac{1}{r}$

According to Stefan's law,

$H=\sigma eA{T}^4=\sigma e\left(4\pi r^2\right)T^4$. . . . . . .(1)

$H\propto r^2$

$\sigma eA{T}^4=ms\frac{dT}{dt}$

$\frac{dT}{dt}=C=\frac{\sigma eA{T}^4}{ms}$

$C=\frac{\sigma e{T}^4\left(4\pi r^2\right)}{\left(\rho \frac{4}{3}\pi r^3\right)s}$ ($m=\rho \frac{4}{3}\pi r^3$)

$C\propto \frac{1}{r}$

The correct option is C.