The energy radiated per second by a black body is given by Stefan's Law as  $\frac{E}{t}=\sigma T^2\times A$ where A is the surface area of the black body

$\frac{E}{t}\sigma T^4\times 4\pi r^2$

Since black body is a sphere, $(A=4\pi r^2)$.

Case(i)

$\frac{E}{t}=450,T=500K,r=0.12m,$

$\therefore 450=4\pi \sigma \left(500{)}^4\right(0.12{)}^2$                              .....(i)

Case(ii)

$\frac{E}{t}=?,T=1000K,r=0.06m,$

dividing (ii) and (I) we get,

$\frac{\frac{E}{t}}{450}=\frac{\left(1000{)}^4\right(0.06{)}^2}{\left(500{)}^4\right(0.12{)}^2}=\frac{2^4}{2^2}=4$

$\Rightarrow \frac{E}{t}=450\times 4=1800W$

(d) is the correct option.