Question

A spherical black body with a radius of $12cm$ radiates $450W$ power at $50K$. If the radius were helved and the temperature doubled, the power radiated in watts would be

• A. $225$
• B. $450$
• C. $900$
• D. $1800$

Answer 1

The correct option is D $1800$

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 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)
$\frac{E}{t}=450,T=500K,r=0.12m$,
$\therefore 450=4\pi \sigma \left(500{)}^4\right(0.12{)}^2....\left(i\right)$
Case (ii)
$\frac{E}{t}=?,T=1000K,r=0.06m...\left(ii\right)$
Dividing Eq. (ii) by Eq. (i), we get
$\frac{E/t}{450}=\frac{\left(1000{)}^4\right(0.06{)}^2}{\left(500{)}^4\right(0.12{)}^2}=\frac{2^4}{2^2}=4$
$\Rightarrow \frac{E}{t}=450\times 4=1800W$.