Question

Two spherical black bodies of radii $r_1$ and $r_2$ at temperatures $T_1$ and $T_2$ respectively, radiate same power. Then $\frac{r_1}{r_2}$ must be equal to

• A. ${\left(\frac{T_1}{T_2}\right)}^2$
• B. ${\left(\frac{T_2}{T_1}\right)}^2$
• C. ${\left(\frac{T_1}{T_2}\right)}^4$
• D. ${\left(\frac{T_2}{T_1}\right)}^4$

Answer 1

Answer: (B)

Power radiated from black body is given by,
$E=\frac{dQ}{dt}=\sigma A{T}^4$
So, $\frac{E_1}{E_2}=\frac{A_1}{A_2}\times \frac{T_{1^4}}{T_{2^4}}=\frac{r_{1^2}}{r_{2^2}}\times \frac{T_{1^4}}{T_{2^4}}$
Here , $E_1=E_2$
$\therefore 1=\frac{r_{1^2}}{r_{2^2}}\times \frac{T_{1^4}}{T_{2^4}}$ or $\frac{r_1}{r_2}\left(\frac{T_1}{T_2}\right)$