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. 450
• B. 1000
• C. 1800
• D. 225

Answer 1

The correct option is C 1800

Given,
$r_1=12cm,T_1=500K,Q_1=450W$
$r_2=r_1/2$ and $T_2=2T_1$

As we know, $Q=\in \sigma A{T}^4$
$\begin{array}{}\therefore Q_1=\in \sigma A_{1}T{{}_1}^4& \text{(1)}\end{array}$
$\begin{array}{}\therefore Q_2=\in \sigma A_{2}T{{}_2}^4& \text{(2)}\end{array}$
Dividing eq.(1) by eq. (2)
$\frac{Q_1}{Q_2}=\frac{\in \sigma A_1{T}_1^4}{\in \sigma A_2{T}_1^4}$
As, $(A\propto r^2)$
$\frac{Q_1}{Q_2}=\frac{A_1{T}_1^4}{A_2{T}_2^4}\Rightarrow \frac{r_1^2{T}_1^4}{r_2^2{T}_2^4}$
$\frac{Q_1}{Q_2}=\frac{4r_1^2{T}_1^4}{16r_1^2{T}_1^4}$
$\frac{Q_1}{Q_2}=\frac{1}{4}$
$Q_2=4Q_1\Rightarrow 4\times 450=1800W$