Question

If a black body at 400 K surrounded by atmosphere at 300 K has rate of cooling \( \propto T^4 \), the same body at 900 K, surrounded by same atmosphere, will have rate of cooling nearly

• A. \( 4T_0 \)
• B. \( 16T_0 \)
• C. \( 36T_0 \)
• D. \( 64T_0 \)

Hint:

The rate of heat radiated by a black body is proportional to the fourth power of its temperature. This means that small increases in temperature result in much larger increases in radiated energy.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The rate of cooling of a black body depends on its temperature and the surrounding temperature according to Stefan-Boltzmann law.
Step 2: Key Formula or Approach:
Net rate of radiation (cooling): \( R \propto (T^4 - T_s^4) \).
Step 3: Detailed Explanation:
Initially: \( T_1 = 400\text{ K} \), \( T_s = 300\text{ K} \).
\[ R_0 = k (400^4 - 300^4) = k \cdot 10^8 (4^4 - 3^4) = k \cdot 10^8 (256 - 81) = 175 k \cdot 10^8 \]
Finally: \( T_2 = 900\text{ K} \), \( T_s = 300\text{ K} \).
\[ R' = k (900^4 - 300^4) = k \cdot 10^8 (9^4 - 3^4) = k \cdot 10^8 (6561 - 81) = 6480 k \cdot 10^8 \]
Ratio:
\[ \frac{R'}{R_0} = \frac{6480}{175} \approx 37.02 \]
The nearest value in the options is \( 36 R_0 \).
Step 4: Final Answer:
The rate of cooling will be nearly \( 36 R_0 \).