Step 1: Understanding the Concept:
The amount of heat radiated per unit area per unit time by a body is its emissive power.
According to the Stefan-Boltzmann law, the emissive power depends on the emissivity of the surface and the fourth power of its absolute temperature.
Step 2: Key Formula or Approach:
The Stefan-Boltzmann law states:
\[ E = \epsilon \sigma T^4 \]
where $E$ is the emissive power (heat radiated per unit area per unit time), $\epsilon$ is the emissivity, $\sigma$ is the Stefan-Boltzmann constant, and $T$ is the absolute temperature.
We are given that $E_A = E_B$.
Step 3: Detailed Explanation:
For body A:
\[ E_A = \epsilon_A \sigma T_1^4 \]
For body B:
\[ E_B = \epsilon_B \sigma T_2^4 \]
Since they radiate the same amount of heat per unit area per unit time:
\[ E_A = E_B \]
\[ \epsilon_A \sigma T_1^4 = \epsilon_B \sigma T_2^4 \]
We can cancel the Stefan-Boltzmann constant $\sigma$:
\[ \epsilon_A T_1^4 = \epsilon_B T_2^4 \]
Rearrange to group the temperatures and emissivities:
\[ \frac{T_1^4}{T_2^4} = \frac{\epsilon_B}{\epsilon_A} \]
\[ \left( \frac{T_1}{T_2} \right)^4 = \frac{\epsilon_B}{\epsilon_A} \]
We are given the ratio of emissivities $\epsilon_A : \epsilon_B = 16 : 1$. Therefore, $\frac{\epsilon_B}{\epsilon_A} = \frac{1}{16}$.
Substitute this into the equation:
\[ \left( \frac{T_1}{T_2} \right)^4 = \frac{1}{16} \]
Taking the fourth root of both sides:
\[ \frac{T_1}{T_2} = \left( \frac{1}{16} \right)^{1/4} = \frac{1}{2} \]
We are given the relation $\text{T}_1 = x\text{T}_2$, which means $x = \frac{T_1}{T_2}$.
Therefore, $x = \frac{1}{2} = 0.5$.
Step 4: Final Answer:
The value of $x$ is 0.5.