Two spherical black bodies of radii ' \(R_1\) ' and ' \(\mathrm{R}_2\) ' and with surface temperature ' \(\mathrm{T}_1\) ' and ' \(\mathrm{T}_2\) ' respectively radiate the same power. The ratio of ' \(R_1\) ' to ' \(R_2\) ' will be

For any two bodies of radii \(R_1\) and \(R_2\), kept at temperatures \(T_1\) and \(T_2\), the power radiated or rate of loss of heat by them can be given as,
\(\frac{\mathrm{Q}_1}{\mathrm{Q}_2}=\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2 \times\left(\frac{\mathrm{T}_1}{\mathrm{~T}_2}\right)^4\)
In case of non-spherical bodies, \(\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2\) can be replaced by \(\left(\frac{A_1}{A_2}\right)\) where \(A_1, A_2\) are areas of the given bodies.
\(\therefore \quad\) For same power,
\(\begin{aligned}
&\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2=\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\right)^4 \\
& \therefore \quad\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)=\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\right)^2
\end{aligned}\)