Two spherical black bodies have radii ' $R_1$ ' and ' $R_2$ '. Their surface temperatures are $T_1 K$ and $T_2 K$ respectively. If they radiate the same power, the ratio $\frac{\mathrm{R}_1}{\mathrm{R}_2}$ is

The power radiated by a spherical black body follows the Stefan-Boltzmann law: $P = \sigma A T^4$, where $\sigma$ is the Stefan-Boltzmann constant, $A$ is the surface area, and $T$ is the absolute temperature.

For a sphere, $A = 4\pi R^2$, so $P = 4\pi\sigma R^2 T^4$.

Given two spheres with different radii and temperatures radiating equal power, set the expressions equal:
$4\pi\sigma R_1^2 T_1^4 = 4\pi\sigma R_2^2 T_2^4$

Canceling the common factor $4\pi\sigma$ yields $R_1^2 T_1^4 = R_2^2 T_2^4$.

Solving for the ratio $R_1/R_2$:
$\left(\frac{R_1}{R_2}\right)^2 = \left(\frac{T_2}{T_1}\right)^4$
$\frac{R_1}{R_2} = \left(\frac{T_2}{T_1}\right)^2$

The final answer is $\boxed{\text{D}}$.