Q. 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

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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

Solution:

Power radiated from black body is given by,

$E = frac{dQ}{dt} = sigma AT^{4} $

So, $frac{E_{1}}{E_{2}} = frac{A_{1}}{A_{2}} timesfrac{T_{1^{4}}}{T_{2^{4}}} = frac{r_{1^{2}}}{r_{2^{2}}} times frac{T_{1^{4}}}{T_{2^{4}}}$

Here , $E_{1} = E_{2}$

$therefore , , , 1 = frac{r_{1^{2}}}{r_{2^{2}}} times frac{T_{1^{4}}}{T_{2^{4}}} $ or $frac{r_{1}}{r_{2}} left(frac{T_{1}}{T_{2}}right)$