Question

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

• A. $225$
• B. $450$
• C. $1000$
• D. $1800$

Answer 1

The correct option is D $1800$

Given,
$R_1=12\text{cm}$
$P_1=450\text{W}$
$T_1=500\text{K}$
$R_2=\frac{R_1}{2}=6\text{cm}$
$T_2=2T_1=1000\text{K}$
$P_2=?$
As we know by Stefan-Boltzmann law,
$P=e\sigma A{T}^4$
Given material is same.
$\therefore P\propto A{T}^4$
$P\propto R^2{T}^4$ $[A=\pi R^2]$
$\frac{P_1}{P_2}=\frac{R_1^2{T}_1^4}{R_2^2{T}_2^4}$
$\frac{450}{P_2}=\frac{(12{)}^2\times (500{)}^4}{(6{)}^2\times (1000{)}^4}$
$P_2=1800\text{W}$