A spherical black body with a radius of $$12\,cm$$  radiates $$450\,W$$  power at 500 $$K.$$ If the radius were halved and the temperature doubled, the power radiated in watt would be

Solution :
The energy radiated per second by a black body is given by Stefan's Law
$$\frac{E}{t} = \sigma {T^4} \times A,$$   where $$A$$ is the surface area.
$$\frac{E}{t} = \sigma {T^4} \times 4\,\pi {r^2}$$    ($$\because $$ For a sphere, $$A = 4\pi {r^2}$$   )
Case (i) : $$\frac{E}{t} = 450,T = 500\,K,r = 0.12\,m$$
$$\therefore \,\,450 = 4\,\pi \sigma {\left( {500} \right)^4}{\left( {0.12} \right)^2}\,\,\,\,.....\left( {\text{i}} \right)$$
Case (ii) : $$\frac{E}{t} = \,? ,T = 1000\,K,r = 0.06\,m$$
$$\therefore \,\,\frac{E}{t} = 4\,\pi \sigma {\left( {1000} \right)^4}{\left( {0.06} \right)^2}\,\,\,.....\left( {{\text{ii}}} \right)$$
Dividing (ii) and (i), we get
$$\eqalign{ & \frac{{\frac{E}{t}}}{{450}} = \frac{{{{\left( {1000} \right)}^4}{{\left( {0.6} \right)}^2}}}{{{{\left( {500} \right)}^4}{{\left( {0.12} \right)}^2}}} \cr & = \frac{{{2^4}}}{{{2^2}}} \cr & = 4 \cr & \Rightarrow \,\,\frac{E}{t} = 450 \times 4 \cr & = 1800\,W \cr} $$