This problem concerns two spherical black bodies that emit identical rates of heat per second. The Stefan-Boltzmann Law states that the power radiated \( P \) by a spherical black body is calculated as:
\[ P = \sigma A T^4 \]
In this formula, \( \sigma \) represents the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the black body's temperature.
For a spherical shape, the surface area is expressed as \( A = 4\pi R^2 \). Consequently, the power radiated by each sphere can be represented as:
\[ P_1 = \sigma (4\pi R_1^2) T_1^4 \]
\[ P_2 = \sigma (4\pi R_2^2) T_2^4 \]
Given that both bodies emit heat at the same rate per second, we can set \( P_1 \) equal to \( P_2 \):
\[ \sigma (4\pi R_1^2) T_1^4 = \sigma (4\pi R_2^2) T_2^4 \]
The constants \(\sigma\) and \(4\pi\) can be eliminated from both sides of the equation:
\[ R_1^2 T_1^4 = R_2^2 T_2^4 \]
Rearranging this equation yields:
\[ \left(\frac{T_1}{T_2}\right)^4 = \left(\frac{R_2}{R_1}\right)^2 \]
Taking the fourth root of both sides results in:
\[ \frac{T_1}{T_2} = \left(\frac{R_2}{R_1}\right)^{1/2} \]
If we then assume the relationship \( T_1 = \sqrt{2} T_2 \), this implies:
\[ \frac{T_1}{T_2} = \sqrt{2} \]
Therefore, it follows that \(\left(\frac{R_2}{R_1}\right)^{1/2} = \sqrt{2}\).
Since both sides of the equation are equivalent, the established relation \( T_1 = \sqrt{2} T_2 \) is confirmed as correct.