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)
$450=4\pi \sigma \left(5 0 0\right)^{4}\left(0 .12\right)^{2}$ ...(i)
Case (ii)
$\frac{E}{t}=?,T=1000K,r=0.06m$ ...(ii)
Dividing Eq. (ii) by Eq. (i), we get
$\frac{\left(E / t\right)}{450}=\frac{\left(1000\right)^{4} \left(0 .06\right)^{2}}{\left(500\right)^{4} \left(0 .12\right)^{2}}=\frac{2^{4}}{2^{2}}=4$
$\Rightarrow \frac{E}{t}=450\times 4=1800W$