Step 1: Understanding the Concept:
The power radiated by a black body depends on its surface area and its absolute temperature.
This relationship is given by the Stefan-Boltzmann law.
Step 2: Key Formula or Approach:
Stefan-Boltzmann Law: Total radiated power \( P = \sigma A T^4 \).
For a sphere, the surface area is \( A = 4\pi R^2 \).
Therefore, \( P = \sigma (4\pi R^2) T^4 \), which means \( P \propto R^2 T^4 \).
Step 3: Detailed Explanation:
Let the initial conditions be:
Radius \( = R \)
Temperature \( = T \)
Initial radiated power \( P_1 = P \propto R^2 T^4 \).
According to the problem, the new conditions are:
New temperature \( T' = 2T \)
New radius \( R' = 2R \)
Let the new radiated power be \( P_2 \).
Using the proportionality relation:
\[ P_2 \propto (R')^2 (T')^4 \]
Substitute the new values:
\[ P_2 \propto (2R)^2 (2T)^4 \]
Expand the terms:
\[ P_2 \propto (4R^2) \cdot (16T^4) \]
\[ P_2 \propto 64 \cdot R^2 T^4 \]
Now, compare \( P_2 \) with the initial power \( P_1 \):
\[ \frac{P_2}{P_1} = \frac{64 R^2 T^4}{R^2 T^4} = 64 \]
Therefore, \( P_2 = 64 P_1 \).
Since \( P_1 = P \), the new power is \( 64\text{ P} \).
Step 4: Final Answer:
The power radiated would be \( 64\text{ P} \).