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)

$\left(\frac{T_{2}}{T_{1}}\right)^{2}$

Answer 2

The correct option is(B): $\left(\frac{T_{2}}{T_{1}}\right)^{2}$

Power radiated from black body is given by,
$E = \frac{dQ}{dt} = \sigma AT^{4}$
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 $\frac{r_{1}}{r_{2}} \left(\frac{T_{1}}{T_{2}}\right)$