Question

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

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

Answer 1

Answer: (D)

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