Question

A spherical solid black body of radius 'r' radiates power 'H' and its rate of cooling is 'C'. If density is constant then which of the following is/are true.

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

Answer 1

The correct option is B $H\propto r^2$ and $c\propto \frac{1}{r}$

$H=\frac{e\sigma T^{4}A}{t}(A=4\pi r^2)\Rightarrow H\propto r^{2}C=\frac{e\sigma A{T}^4\times r/3}{m\times S\times r/3}\{A\cdot \frac{r}{3}=V;\frac{m}{V}=\rho \}C=\frac{e\sigma T^4}{\rho \times S\times r/3}\Rightarrow C\propto \frac{1}{r}$